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Renormalization of Vector Fields Hans Koch 2 Abstract These notes cover some of the recent developments in the renormalization of quasiperi odic ows This includes skew ows over tori Hamiltonian ows and other ows on Td X RI After stating some of the problems and describing alternative approaches we focus on the de nition and basic properties of a single renormalization step A second part deals with the construction of conjugacies and invariant tori including shearless tori and non differentiable tori for critical Hamiltonians Then we discuss properties related to the spectrum of the linearized renormaliza tion transformation such as the accumulation rates for sequences of closed orbits The last part A 44 describes from f im r to I 1 rotation vectors This involves sequences of renormalization transformations that are related to continued fractions expansions in one and more dimensions Whenever appropriate the discussion of details is restricted to special cases where inessential technical complications can be avoided Expanded notes from a minicourse given at the Fields Institute in Toronto Canada November 2005 2 Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin TX 787120257 HANS KOCH Content 1 Background 11 Invariant tori 12 Two direct approaches 13 Hamiltonians 14 KAM theory 15Scales 16 Breakup of invariant tori 2 Renormalization of ows 21 Hamiltonian systems 22 Resonant and nonresonant Hamiltonians 23 The change of variables UH 24 Other vector elds 25Skewsystems 3 A single renormalization group step 31 Skew systems de nitions 32 Skew systems estimates 33 More general vector elds 34 A general elimination procedure 4 A nontrivial RG xed point 41 Observations and result 42 Strategy of proof 43 Non twist ows 5 Invariant tori 51 Some ideas and results 52 Renormalization of invariant tori 53 Existence 54 Critical invariant tori 55 Shearless tori 6 Scaling 61 Spectrum of the linearized RC transformation 62 Accumulation of periodic orbits 63 Choice of the manifold 20 64 The manifolds Zn and orbits 771 7 Sequences of RG transformations 71 Diophantine and Brjuno numbers 72 Multidimensional continued fractions 73 Composing different RC transformations 74 An invariant manifold theorem 8 Reduction of skew ows 81 A general result 82 The stable manifold 83 Conjugacy to a linear ow 84 The special case SL2R 85 Excluding hyperbolicity References H N OTHgtDDQODD H Renormalization of Vector Fields 3 1 Background 10 Disclaimer This review is primarily about methods and ideas It does not intend to give a com prehensive list of theorems on invariant tori conjugacies bifurcations and other topics covered The main focus is on renormalization group methods for Hamiltonian and other vector elds And more speci cally on methods that that implement renormalization as a dynamical system on a space of Hamiltonians or vector elds Much of the discussion is restricted to problems that l have worked on myself which is not meant to imply that other work is less important 11 Invariant tori The general goal is to describe certain asymptotic behavior like quasiperiodicity for continuous time dynamical systems a Xu 11 Here X is a vector eld on some manifold M The ow CIDX associated with X is de ned by Oio ut where u is the solution of 11 with initial condition 710 no In all cases discussed here M will be a product of the d torus Td with some other manifold B Most of the problems considered involve either invariant tori or conjugacies By an invariant torus with rotation vector w 6 Rd we mean a map F Td a M that is locally one to one and satis es Po lltlt1gtoP lltqqtw 12 In other words P de nes a semi conjugacy between a restriction to the range of P of the original ow CIDX and the linear ow 1 on the torus Some true conjugacies CIDX Uo l on 1 will be discussed as well where it is possible to conjugate CIDX to the ow for a trivial vector eld Z on all of M In order to see some of the dif culties involved in solving equation 12 it is useful to look at the differentiated version wVPXoP 13 One of the problems is that the differential operator w V is not easily invertible Its spectrum if nontrivial accumulates at zero in a way that depends on the arithmetic properties or the rotation vector w 12 Two direct approaches One way of trying to solve equation 13 is by perturbing about an approximate solution To Substituting P TO y and expanding X 0 TO y in powers of y we obtain an equation of the form vvoWv WW ZD anW 14 mgt1 4 HANS KOCH where D w V 7 DX 0 P0 lterating the map y gt gt W WW yields a formal power series for the torus also referred to as Lindstedt series This series is in general highly divergent but there are nontrivial situations where resummation techniques can be used to obtain y from its Lindstedt series 39 Interestingly the problems that one encounters are similarly to those found for Feyn man graph expansions in quantum field theory It is these expansions that lead to the development of renormalization methods 118 The divergencies can be associated with different scales in the problem and by re normalizing the expansion parameters appro priately the divergencies at any given scale cancel Applications of renormalization ideas from quantum field theory to the resummation problem for Lindstedt series can be found eg in 51 575859 In this context renormalization can be viewed as a method for deal ing with combinatorial problems and cancellations in certain highly nontrivial perturbation expansions Equation 13 also has a vague resemblance to field equations in quantum field theory In some special cases it is possible to make this connection more precise and write 13 as the Euler Lagrange equation for some functional 7 gt gt lm The modern way of ana lyzing such fields is via functional integrals Expanding these integrals in powers of small coupling constants yields the above mentioned Feynman graphs lntegrals of the same type also appear in statistical mechanics where there are usually no small parameters The approach taken in non perturbative renormalization is to perform the integration one scale at a time transforming a Lagrangian k at scale k to a new Lagrangian k1 at scale k 1 making the map k gt gt k1 a dynamical system if possible Inspired by this approach Bricmont et al have devised a renormalization scheme that applies to the problem of constructing invariant tori The formalism itself is non perturbative but in practice the analysis can be carried out only for X close to constant where y is small Similar ideas have also been applied successfully to the study of PDEs 1n the context described here renormalization can be viewed as a procedure for solving certain dif cult scale free77 problems iteratively one scale at a time 13 Hamiltonians The renormalization group approach that we will focus on later is much closer to KAM theory than to the approaches sketched above In order to simplify the discussion we will restrict our attention to Hamiltonian ows Consider M Td gtlt B where B is some open neighborhood of the origin in Rd A Hamiltonian vector field in action angle variables is of the form X JVH with ll BI where H is the corresponding Hamiltonian a differentiable function on M In other words the equation it Xu with u 11 can be written as q VpH 139 diH 15 The corresponding ow will be denoted by DH Some basic facts and notation H is invariant under the ow The maps I are symplectic in the sense that they preserve the symplectic form quj dpj If U is any symplectic diffeomorphism of M then the pushforward of X under U is again a Hamiltonian vector field with Hamiltonian H 0 U Furthermore U preserves the Poisson bracket g V1fVgg 7 V2f V19 Renormalization of Vector Fields 5 A change of coordinates qp Uqp is canonical if and only if the one form p dq 7 p dq is closed Locally this one form can be written as the differential of some function which we will write as p q 7 15 We will only be interested in cases where 15 is de ned globally as a function of q and 19 In this case 15 will be referred to as the generating function of U 1t satis es 1 V2 qp and p V1 qp In particular if U 1 11 uqp QqpPqp 16 then we have 620119V2 qpPqp PULP V1 9PP9P 17 Conversely given 15 not too large these two equations determine a canonical transformation of the form 16 If 15 is small say of size 6 then so are P Q and we have H o U H was i v1 052 H H as 052 18 14 KAM theory KAM theory 82511012126 is concerned with with small perturbations of integrable Hamiltonians such as KqpwpMpp 19 with w 6 Rd and M a symmetric d gtlt d matrix The dynamics for K is given by q wMp and 139 0 Notice that surfaces of constant p are invariant tori for the ow generated by K with frequency vector 1111 w Mp The goal is to construct an invariant torus for H K h with rotation vector w by iterating the following procedure Assuming that h is small say of size 6 we have KhoUKK hOe2 110 K7wV1 7h052 gt Now we try to solve 0 near 1 0 up to an error of order 52 The equation for the V th Fourier mode of 15 is iwV VhVO62 111 So among other things the average ho has to be small near 1 0 Assuming that the matrix M is nonsingular this can be achieved by a p translation and a restriction to 1 near zero Assuming in addition that w satisfies a Diophantine condition equation 111 can be solved for frequencies 1 that are not too large Finally if we also assume that h is analytic so that 11 a 0 rapidly as 111 a 00 then the equation 111 can be solved for all 1 Thus the new Hamiltonian K h 0 U is of the form K g with g of order 52 Now the procedure is iterated The KAM theorem for this situation states that under the assumptions made above the invariant torus with frequency vector w persists under small perturbations of K 6 HANS KOCH 0 025 05 075 1 Fig 1 Some orbits for the standard map with parameter value 05 61 Notice that the assumptions are also satis ed if w is replaced by cw with c 31 0 The invariant tori for different values of 0 lie on different energy surfaces The average speed of the motion varies with a but the winding numbers lim qi t 3 112 tn L105 wd do not depend on c If the goal is to construct an invariant torus with xed winding numbers but unspeci ed value of c then the nondegeneracy assumption on the matrix M can be weakened M is allowed to have rank d 7 1 as long as the range is transversal to w Such Hamiltonians are also referred to as z39soenergetz39cally nondegenemte The KAM torus for H is obtained as the limit of P7 U10U20 oU as n a 00 where U7 denotes the canonical transformation used in the n th step of the iteration described above P7 is de ned on a domain Td gtlt B7 with B7 a sequence of smaller and smaller neighborhoods of zero The limit P yields the desired conjugacy ro lt1gtor on Tdgtlt0 113 The KAM procedure has clearly a renormalization avor although some ingredients are missing most notably the scaling As we will see later by modifying this algorithm appropriately it can be made into a dynamical system Some earlier steps in this direction were also taken in 717283 15 Scales Renormalization applies mainly to systems that involve a natural progression of scales but that do not have any preferred scale The effect of a renormalization operator is to shift the scales of the system Renormalization of Vector Fields 7 The best known examples in dynamical systems may be the composition operators R F gt gt Fk modulo rescaling Here and in what follows Fk denotes the k th iterate of a map F These operators have been studied in great detail 128 after the observation of universality and scaling in one parameter families of interval maps undergoing period doubling bifurcations 1 a 2 a a 27quot1 a 2 a The k 2 version of R lifts the inverse cascade to the space of maps in the sense that F o F has a period 27quot1 whenever F has a period 2 1n problems dealing with irrational rotations the scales come from the arithmetic properties of the rotation numbers Consider eg the number a 1k 1k with k some fixed positive integer For k 1 a is the inverse golden mean lts continued fraction approximants unvn may be obtained as follows an 7 0 0 1 W T I T 1 k 114 Consider a circle C defined by a strictly monotone function 0 on R by identifying C15 with x for any real x A simple example would be 0015 x 7 1 A map on C is a function M R a R that commutes with O and a point x has rotation number uv for this map if C o M x In this formulation a renormalization operator that takes a pair F with rotation number un71vn71 to a pair with rotation number unvn is given by 0 1 R gt gt g1k modulo rescaling 115 The pair F0 with M0x x a is a fixed point of R and it clearly has rotation number a Notice that the exponents in 115 are precisely the matrix elements of T This suggests of course a generalization to problems with more than one frequency This renormalization operator R for any k has been studied in great detail 129 To give a very simple application it can be shown eg that if F is a small perturbation of F0 with rotation number a then a F0 as n a 00 This in turn can be used to establish a conjugacy between F and F0 The analogous operator can be defined also for other types of maps Such an operator was studies in connection with the breakup of invariant circles in area preserving maps of the plane 68 100 52 101 117 16 Breakup of invariant tori Let now d 2 and w 19 with 19 the golden mean E By the KAM theorem a Hamiltonian H close to an integrable Hamiltonian like 19 has a smooth invariant torus P with winding numbers wjwd The proof also shows that near this torus H is essentially integrable and so the motion is highly ordered and stable The same is true in higher dimensions In the case d 2 an invariant 2 torus has an even stronger stabilizing effect as it divides the 3 dimensional energy surface containing it into disjoint invariant regions Consider a one parameter family gt gt H13 of Hamiltonians on T2 gtlt R2 such as Ill3011 w p pf cosql 608011 7 12 116 8 HANS KOCH Which is essentially the Hamiltonian used in 47 For values of Q close to 0 this Hamiltonian has a golden invariant torus that is a smooth invariant torus With Winding number 19 1 This torus is observed to persist as Q is increased up to some value 500 Where it breaks up The breakup is also seen to promote chaotic motion in the form of hyperbolic orbits With golden mean rotation number Before the critical point ee the system has prominent periodic orbits symmetric Birkhoff orbits for each of the rotation numbers g g g associated With the continued fraction expansion of 6 1 Past the critical point these orbit With rotation number unvn turn unstable at parameter values that converge to ee in the limit 71 a 00 The convergence is observed to be asymptotically geometric With lim M 5 1 6 16279 117 This ration appears to be universal in the sense that the same values is observed Within a large class of one parameter families The critical Hamiltonian H w appears to have an invariant torus P that is non smooth Near this torus the motion for H n looks like that of H modulo a scaling of time by 19 and a scaling of space The observed eigenvalues of the spatial scaling are AT 19 Am mAZ Ag MooAT AZ 7032606 118 With m 023046 Again these values appear to be universal 0 0487 Iquot LDMW HHH HHH 39 l r FHH m vq Fig 2 Orbits for a critical Hamiltonian 2 Renormalization of Vector Fields 9 When looking for an explanation anybody that is familiar with statistical mechanics and phase transitions will immediately try to find an appropriate renormalization group transformation R acting on a space of Hamiltonians The expected picture involves the existence of a hyperbolic fixed point Hquot for R such that DRH has an expanding eigenvalue 6 16279 and no other spectrum outside the open unit disk 2 Renormalization of ows After introducing a renormalization group RG transformation for Hamiltonian ows and motivating the choices involved we generalize the construction to other vector elds on Td gtlt RI and to skew ows over Td 21 Hamiltonian systems The first use of renormalization in connection with the breakup of golden invariant tori appears to be by Escande and Doveil 48 Their transformation contained the following ingredients One is a frequency scaling H gt gt H o T TltQaPgt TQ7 T71pa where T denotes the transpose of T Here T 1J The remaining ingredients are a scaling of time and momenta and a canonical change of variables H 1 H 0 U designed to obtain a renormalized Hamiltonian of the same form as the original Hamiltonian The latter was chosen ad hoc and involved truncations since only a few selected Fourier modes were being considered An alternative approach by Kadanoff Shenker and MacKay 6810052101 uses com muting pairs of area preserving maps The breakup of invariant circles for these maps pro duces the same phenomena and numbers as described above After all one way to obtain an area preserving map it to start with a Hamiltonian on T2 gtlt R2 restrict its motion to a surface of constant energy and then consider the return map to an appropriate plane Based on the numerical results in 100 there is little doubt that the renormalization pic ture correctly describes the observed phenomena The main drawback with this approach is that commuting maps do not constitute a manifold which makes it hard to discuss some important aspects of renormalization such as the hyperbolicity of the renormalization op erator Similar problems are also encountered in one dimensional dynamics which lead to the development of alternative approaches such as the cylinder renormalization for Siegel disks 12556 The best rigorous result in this line of work is the existence a nontrivial solution of R3F F for a renormalization operator R of the type 115 acting on pairs of reversible maps 117 Whether this solution F M is a fixed point of R and whether its components 0 and M commute is not known In the Hamiltonian approach it took much longer before accurate computations be came possible 2 although some clear improvements to the original scheme were made earlier 10713 see also 16 and references therein The dif culty has to do with the 10 HANS KOCH choice for the change of variables H gt gt H 0 U In other problems Where renormaliza tion has been applied the analogue of U can be guessed from the observed scaling This scaling is typically a contraction on phase space so it may be reduced to a simple nite codimension normal form But the torus cannot be contracted As it turns out 74 it is still possible to find a RG transformation of the form 7211 H o 7 mod 9 22 Where g is some group of similarity transformations But it appears that this group needs to be infinite dimensional Among the possible similarity transformations are 0 Scaling of time or energy H gt gt n lH 7 E o Scaling of momenta H gt gt M 1HM 0 Change of variables H gt gt H 0 U With U canonical and homotopic to the identity L orbit of H under Q rtA Hamiltonians in normal form Fig 3 General form of the RC transformation R The goal is to find a suitable normal form77 for Hamiltonians and a map H gt gt GH 6 9 such that GHH o T is in normal form Whenever H is Using for CH a composition of the three similarity transformations listed above we have 1 RHHoZH H HOTH7E 23 7W Where My is a canonical change of variables TMQJ TqMT 1p 24 and E n M are normalization constants that may depend on H We Will set E 0 from now on and mostly ignore constant terms in Hamiltonians since such terms do not change Renormalization of Vector Fields the vector field The constants n and M can be determined eg by prescribing the value of two coef cients in the Taylor expansion for the torus average of RH Choosing a suitable normal form that also determines UH is more delicate This problem will be discussed in Subsections 22 and 23 We expect R to have at least one integrable fixed point describing the ow near smooth invariant tori Such xed points are in fact easy to find To be more specific let T be a matrix in SLdZ that has two eigenvectors Tw 191w and T9 1929 for two eigenvalues satisfying 191 gt 1 gt 192 Consider the integrable Hamiltonians mp my w pgt gm w 25 with m gt 0 unless specified otherwise Starting with this Hamiltonian K computing K o T and then applying a momentum and energy scaling including an angle dependent change of variables would be counter productive here we obtain 73000119n lu 1fMT 1p 7E 26 71 wapn 1m9 2g9p2E gt For K to be a fixed point for R we need an energy scaling n 1914 and E 0 lgnoring for the time being the case m 0 where the momentum scaling M is undetermined we have M fl g Notice that lMl lt 1 due to our condition 191 gt 1 gt 192 meaning that the momenta p are contracted by the scaling H gt gt M 1H M So far so good The problem arises when we try to extend R to Hamiltonians H that depend on the angle variables 1 as well Consider eg a space AP of Hamiltonians that are analytic in the domain DP defined by llmqjl lt p and lpjl lt p with p some fixed positive real number If H is analytic on Dp then H H o 7 is analytic on Tle But this new the domain is narrower than Dp in the angular direction w by a factor 1914 The question is whether this loss of analyticity can be restored by a canonical change of variables H gt gt H 0 My At first this seems unlikely since UH should be close to the identity for H close to K if we want R to be a smooth map on AP And a fixed domain loss cannot be restored by changes of variables arbitrarily close to the identity What will save the situation are cancellations 22 Resonant and nonresonant Hamiltonians Motivated by the above we start by trying to identify the good and bad terms in the Fourier Taylor series HqP Z Hmaeiqupaa pa Hpj j 27 Va I 7 where I Zd gtlt Z1 The Hamiltonian H is analytic on Dp if and only if the series 27 converges on Dp which is roughly equivalent to Hml g e pll39lp lal V a e I 28 12 HANS KOCH Here denotes the 1 norm In order to identify which of these conditions are violated for the Hamiltonian H o 7 consider its Fourier Taylor series H 0mm Z Huae T 39qMltT 1pla 29 Va I If we consider just the terms with xed degree lal then the bad terms can be identified as those for which lT1l gt A more careful analysis has to take into account that the factor Mlal in 29 improves convergence in directions where lal a 00 Alternatively we can try to identify the good modes 62quot qu that do not get ex panded under composition with Among them are the resonant modes which we now describe Assume that all eigenvalues of T other than 191 are of modulus lt 1 Then the orthogonal complement of w is contracted by T with respect to some norm on Cd that we will denote by Given real numbers 714 gt 0 to be determined later define 1VOt fllw1 SUHVH or lwwlgnlal IilI 210 The resonan 77 part llH of a Hamiltonian H and its nonresonant part ll H are now defined by restricting the sum in 27 to the index set 1 and 7 respectively In order to make precise what we mean by non expanding modes we need to introduce a norm Define AP to be the space of Hamiltonians that are analytic on the domain DP and continuous on its closure We equip this space with the norm Wit 2 lHValequot p a 211 Mada It is now straightforward to prove the following Proposition 21 If 039 M are positive and suf ciently small and if p lt p is suf ciently close to p then the restriction of H gt gt H o 7 to llAp is a compact linear operator from llA to AP with operator norm 3 1 Proof Pick 1404 in the index set 1 and consider the function EaVqp eiV39qpo Then for 7 gt 0 ME om ep T lc cpl o 212 exppHTIH 7 MM lallnlcupTI HEWHT where c is some constant depending only on T Consider first 7 p Clearly the third term in is always non positive if M gt 0 is suf ciently small lf ltd 1l S alll then the sum of the first two terms is also non positive provided that 039 gt 0 has been chosen suf ciently small This follows from the fact that wi is contracted by Tquot Alternatively if ltd 1l gt UHVH and thus lt U lnlal then we can make 3 0 by taking M gt 0 Renormalization of Vector Fields suf ciently small This shows that the term in 212 is non positive These arguments clearly extend to 7 lt p suf ciently close to p Thus if H E TAT then HHOTHHpS Z lHualllEua0llp Z lHualHEuaHrHHHr 1400614r 1400614r The assertion now follows by taking 7 lt p lt p and using the fact that the inclusion map from Ap into AT is compact QED Notice that the resonant modes which are essentially the ones that cause small de nominator problems in KAM theory are easy to deal with in this approach It should be noted also that the smallness conditions in Proposition 21 can easily be replaced by concrete inequalities To give a concrete example Using a slightly different definition of 1 an analogue of this proposition is proved in 1 for 192 0lt191 i M lt 0 lt i 191 1 20 lt 213 23 The change of variables UH Proposition 21 suggests that we take the resonant Hamiltonians as our normal form This requires that the change of variables am in equation 23 can be chosen in such a way that 1 H OHM 0 214 which makes again resonant In other words the role of My would be to eliminate nonresonant modes Now why should this equation be solvable Roughly speaking the reason is that the equation deals mainly with nonresonant functions which should avoid small denominator problems To be more precise let K0qp w p and consider a Hamiltonian H K0 h not too far from K0 Denote by h and h the resonant and nonresonant parts of h respectively and assume that 8 Hh Hp is small If U is a canonical transformation with nonresonant generating function 15 of order 8 then HoUHH O52 215 Ko hiwv1 h C62 Let us try to solve ll H 0 U 0 to first order in 5 The resulting equation for b is HT 0 which can be written as wV1 1 E h where E denotes the Hamiltonian vector field associated with a Hamiltonian 9 that is f f g The formal solution of this equation is w V1 11 L 1h 216 14 HANS KOCH with V2 w V1 V1 wVl r raw V1 1 i V211 4 V111 Here ll is a linear operator on llTAp Now by the de nition of If the two operators written as fractions in square brackets are bounded in norm by 0 1 and pnfl respec tively Thus if HVth is suf ciently small such that HllfH lt 1 then equation 216 can be solved by a Neumann series The solution 11 w V1 belongs to AP and is of order 5 The next step would be to solve equation 17 for the functions P and Q de ning the canonical transformation U generated by 15 Notice that what enters this equation is not b directly but its gradient This gradient is wV1 1V11 a function for which we have again convenient bounds By construction the new Hamiltonian Ho U has a nonresonant part of order 52 Thus we can solve equation 214 by iterating the step H gt gt H 0 U described above A generalization of this procedure will be described in detail in Subsection 74 Remarks 0 This elimination procedure shrinks domains However the domain loss tends to zero with the size of h Thus for near resonant Hamiltonians the subsequent step H gt gt HoTH more than compensates for this domain loss making R analyticity improving o It should be stressed that only the nonresonant part of h needs to be small for this procedure to work The condition HllfH lt 1 allows for Hamiltonians that are not close to being integrable For completeness let us state a concrete result about the transformation R Let w lw2 wd be a xed vector in Rd whose components span an algebraic number field of degree d We will call such vectors self similar for the following reason It can be shown 74 that there exists a matrix T E SLdZ with simple eigenvalues 19339 satisfying 191 gt 1 gt ng 2 2 Wdl such that Tw 191w Consider now a fixed matrix T with these properties Theorem 22 1 Let 0 lt p lt 05 lfp lt p is suf ciently close to p and M E C satis es 213 then there exists an open neighborhood B 6 AP of K0 such that R B a AP is well de ned analytic and compact The version of this theorem given in 1 contains additional information about the domain B 24 Other vector elds Flows q Xq on the torus Td are a special case of the above as can be seen by restricting the ow for a Hamiltonian H p Xq to the invariant torus p 0 For ows that are not described by a generating function we have to renormalize the vector field directly Let X be a vector field on a manifold M Consider a change of coordinates x Uy on M Then at DUyy So the pullback of X under H is wx DZ 1Xol 217 Renormalization of Vector Fields Consider now M Td gtlt RI and vector elds near Z w 0 In the Hamiltonian case described earlier we used a the phase space scaling 70119 TqMSp 218 with S the transposed inverse of T The same type of scaling is appropriate for other types of vector elds as well eg with S l whenever 31 d For the pullback of X X X under 7 we have 7X T1Xozp151Xoz 219 The analogue of the renormalization group RG transformation 23 is now given by RX n 1U TX 220 where LIX is a change of coordinates that eliminates nonresonant modes Here n and M are normalization constants that may depend on X This type of RC transformations has been used eg in 947079 to linearize torus ows and skew systems and in 78 to construct invariant tori for general ows near Z w 0 When dealing with specific classes of ows it is useful to choose the projections lli and the change of variables LIX in such a way that the given class is preserved under renormalization The choice given in 78 does this simultaneously for the following four classes in the case where Gqp iqp A detailed description of the corresponding elimination procedure X gt gt UX will be given in Subsection 34 l Symmetric Given a diffeomorphism G of M a vector field X on M is symmetric with respect to G if GX X The ow for X commutes with G The changes of variables LIX are generated by symmetric vector fields as well 2 Reversible If G o G I then X is time reversible with respect to G if GX 7X The ow for X satisfies G 0 gt20 G 1 Reversibility is preserved if LIX is generated by symmetric vector fields Divergence free Here trDX 0 The ow for X is volume preserving LIX is generated by divergence free vector fields H amz39ltoman This case was discussed earlier Hamiltonian vector fields are also diver gence free Some are time reversible as well eg if Hqp Hq 7p then X JVH is reversible with respect to G qp gt gt iqp AA gt5 0 VV For specific results we refer to 78 25 Skew systems Here we discuss in more detail a class of ows called skew flows These are systems of ODEs with quasiperiodic coef cients To be more specific let 5 be a Lie subgroup of GLnCC or GLn R and let Qt to be the corresponding Lie algebra Then one considers equations of the type W Ftyt y0 yo 221 with yt E 5 where F R a Qt is a quasiperiodic function with d rationally inde pendent frequencies w1wd Such systems are encountered eg in the study of the 16 HANS KOCH one dimensional Schrodinger equation with quasiperiodic potentials where 6 SL2 R The discrete analogue are products of matrices E depending quasiperiodically on the index i One of the problem is to find the spectrum of products Fm F2F1 in the limit of large m This is trivial in the case where i gt gt E is periodic We can rewrite 221 as W fltqo twyt y0 yo 222 with f a function on Td taking values in 2L This equation together with q w defines the a vector field X on the manifold M Td gtlt 6 Xqy w Hm g 6 91 try 6 M 223 The ow for X is given by 1 Qan0 10 W 12010y0 10910 6 Ma 756 R 224 where t gt gt Wage denotes the solution of 222 for yo 6 6 the identity Classical Floquet theory shows that if t gt gt qt is periodic and in particular if d 1 then the system is reducible To be more precise the vector field 223 is said to be reducible if there exists a function U Td a 6 such that 201 Uq tw6tCUQ 1 156 R q E Td 225 for some constant matrix C 6 2L lf w 6 Rd is fixed we will also refer to f as being reducible For another characterization of reducibility considering the map U M a M defined by Wm qUQy 226 The pullback of X w f under this map is given by the equation XWMI w UVMW UV U71ltf 7 DMU 227 where Dw w V Modulo smoothness assumptions 225 is equivalent to f LPG In the quasiperiodic case solving Vf E 0 leads to small divisor problems as in classical KAM theory Results based on KAM type methods have been obtained in the case where 6 SL2 R 3411140 and for compact Lie groups 8485 Another approach to the reducibility problem involves renormalization methods For discrete time cocycles over rotations by an irrational angle 04 and for 6 SU2 Rychlik introduced in 115 a renormalization scheme based on a rescaling of first return maps using the continued fractions expansion of Oz Improvements of this scheme and global non perturbative results can be found in 86876l In the context of ows renormalization techniques were used in 97 to prove a local normal form theorem for analytic skew systems with a Brjuno base ow Extensions of such RG techniques to skew systems with higher dimensional base maps or ows have become possible with the introduction in 70 of a suitable multidimensional Renormalization of Vector Fields continued fractions algorithm A renormalization scheme based on this algorithm was introduced recently in 79 It applies to Diophantine skew ows on Td gtlt 5 for arbitrary subgroups of GLnCC or GLnR and for arbitrary dimensions dn In addition the smoothness requirements are lowered from analyticity to a finite degree of differentiability depending on d and on the Diophantine exponent The RG transformations themselves are restricted to vector fields X wf with f small Among the general results are the existence of a stable codimension d manifold near f 0 of reducible skew systems Other near constant vector fields are mapped to this case by increasing the dimension of the torus In the case d 2 and 5 SL2R the stable manifold is identified with the set of skew systems having a fixed fibered rotation number see Subsection 84 The precise results and the techniques used to prove them will be described below and in Section 8 3 A single renormalization group step This section covers the more technical aspects of renormalization For skew ows this includes explicit estimates of the type needed later to deal with general Diophantine rotation vectors And for vector fields on Td gtlt R we give a complete description of the main renormalization step the elimination of nonresonant modes 31 Skew systems de nitions Skew ows are ideal for the description of a complete RG step since the analysis is quite simple estimating the action of R on resonant frequencies takes a few lines eliminat ing nonresonant frequencies involves little more than the implicit function theorem and combining the two is straightforward Consider skew ows X w f with f close to zero Given y 2 0 define f to be the Banach space of integrable functions f Td a GLn C for which the norm llwa HfoH Z HfVHQHIHW 31 Oy lEZd is finite Here fl denotes the 1 th Fourier coef cient of f Notice that f is roughly 0 The set of functions in 7 that take values in 5 or Qt will be denoted by g or A respectively We will now drop the subscript y if no confusion can arise A single RG step for skew ows is associated with a unit vector w 6 Rd and a matrix T in SLd Z which we assume now to be given Let 70w TOM 32 The pullback of X under 7 is given by 7Xqy T lw WNW 7f f o T 33 Denote by Cr the set of vectors in R that are contracted by a factor 3 7 by the matrix Tquot Choose 0 lt 039 lt 739 lt 1 if possible such that 20HTH lt 739 wL C CT2 34 18 HANS KOCH The resonant part llf of a function f E f and its nonresonant part ll f are now de ned by the equation rm Z fVeiV39q 35 VEIi where 1 is the set of integer points in CT and If its complement in Zd We will show below that it is possible to find Hf E f close to the identity such that Til1 0 36 and such that Hf E 9 whenever f E A The renormalized function NU and the renor malized vector field are now de ned by the equation NU n lTWH 7300 n lTWfX 37 where n is the norm of T lw so that the torus component of is again a unit vector 32 Skew systems estimates The resonant part of f is easy to deal with Lemma 31 ff 6 f satis es i f Ef 0 then HTfH S Tim Here Ef denotes the torus average of f The proof is one line HTVH Z HfullQHSIHWS Z HfullQTHlHWTlllflt 38 07 V I Oy ueit Notice the dependence of the contraction factor 7 on the degree of smoothness 7 Next before we can solve 36 we need to estimate some simple linear operators Given any n gtlt 11 matrix 0 define Of f0 7 Of for every function f E f Proposition 32 Assume that S 04 Then the linear operators Dw w V and D Dw O commute with l have bounded inverses when restricted to ll f and satisfy HDJWH S 0 1 HDWDAH H S 2 39 Proof Clearly Dw 6 and l7 commute with each other The first inequality in 39 follows directly from the fact that ltd 1 gt 039 whenever 1 belongs to 7 which is straight forward to check It implies llDUjlall H S 20 1HOH S 12 and the indicated bound on DWD lll l DJ16 1lli is now obtained via Neumann series QED Let us recall at this point some facts about analytic maps Let X and 3 be Banach spaces over C and let B C X be open We say that G B a 3 is analytic if it is Frechet differentiable Thus sums products and com positions of analytic maps are analytic Equivalently G is analytic if it is locally bounded Renormalization of Vector Fields and if for all continuous linear maps f C a X and h y a C the function h o G o f is analytic This shows e g that uniform limits of analytic functions are analytic Assuming that B is a ball of radius 7 and that F is bounded on B a third equivalent condition is that G has derivatives of all orders at the center of B and that the corresponding Taylor series has a radius of convergence at least 7 and agrees with G on B See eg 62 for more details Our next goal is to solve 36 Given f O h in f with O constant we seek a solution of the form U eXpD 1u where u is a function in 177 We have raw r eD luf i DQ673714 Thk rwxoheDuaDwOmhwwgt0wwo 81m The rweDur OwwwmOwuq h i u OWLH Hull 00171112 So if is suf ciently small then by the implicit function theorem the equation 1171f 0 has a SOlutiOn Uf h 0HhHZ and this solution depends analytically on f Wit a bit more work one gets an explicit bound on uf for S 06 and S 2 90 and verifies that Huff 711711 240 1Hh112 311 As a result we have Theorem 33 79 Assume that 039 and 739 satisfy 34 Let f O h with O constant and Eh 0 If lt 06 and lt 2 90 then Nltfn710 la E nghH EE 240 17111h112 312 N is analytic on the region determined by the given bounds on O and h Proof The function it in equation 312 is given by it T11h 1 2117 717 Now we can use Lemma 31 and the bound 311 In particular we have HWSHMMEWMWM an as claimed The analyticity of N follows from the analyticity of the map f gt gt uf the uniform convergence of the exponentials in 310 and the chain rule QED Notice that by construction if f belongs to A then so does Similarly if f is real valued then so is Since this theorem will be used later with very large requiring 039 to be very small by 34 it should also be noted that the domain of N is roughly of size 039 which is due to the factor 0 1 in the estimate 39 20 HANS KOCH 33 More general vector elds We start with some basic estimates for analytic vector elds on M Td gtlt Rf near Z w 0 Recall that the RG transformation for such vector elds is given by RX flagrgx 314 where LIX is a change of coordinates that eliminates nonresonant modes The de nition of resonant and nonresonant modes is analogous to the one used for Hamiltonians and the proof that X gt gt TJX is analyticity improving and thus compact when restricted to the resonant subspace is essentially the same as in the Hamiltonian case Thus we will describe here only the elimination procedure X gt gt ng But we will do this in full detail since the elimination of nonresonant modes is probably the most crucial part of R The version presented here is taken from 78 We start with some estimates on the ow generated by an analytic vector eld Y On the spaces C we use the X norm and for linear operators we use the operator norm Given p gt 0 denote by Dp the set of all vectors 11 in Cd gtlt Cl characterized by HIqu lt p and lt p If V is any complex Banach space an analytic function f Dp a V that is 27r periodic in each of the variables 1 can be written as NM Z fua6i 39qpa V q 21ij pa Phi 315 j j Va I where I Zd gtlt NZ De ne ApV to be the space of all functions 315 for which the norm Hpr Z Hfmaiiep yip a 315 Va I is nite Here M Z ilji and 04 is de ned analogously If no ambiguity can arise we will simply write AP in place of ApV The operator norm of a continuous linear map L on AP will be denoted by HLHP It is easy to check that if V is a Banach algebra then so is ApV Another basic fact about the spaces AP is the following Let 7r1qp 10 Proposition 34 Let 0 S p S p Let X E ApV and Y YY with Y 6 AP Cd and Y 6 AP 3 Then a DXW E AMV and HltDXYHp S P PVlHXHpHYiian ifp lt P b X 0 7T1 Y E AMV and HX 0 7T1 YW S HXHu ifP iiYii a iiiquotHp S P The ow by associated with a vector eld Y can be estimated eg by comparing it to the ow CIDZ for a constant real vector eld Z w 0 The following bound is obtained by applying a standard contraction mapping argument to the equation 00 0 Y 7 Z 0 3 0 1 05 ds 317 satis ed by the difference C15 I 7 CIDtZ Renormalization of Vector Fields Proposition 35 Let 739 be a positive real number and Y a vector eld in AP such that THY 7 ZHP lt 7 lt p Then the equation 317 has a unique continuous solution G 6 A9 on the interval ltl S 739 and we 7 2194 WY 7 2m 318 Next we consider the pushforward for the time 5 map CIDty If X and Y are arbitrary vector fields de ne YX YX DXY 7 DYX Proposition 36 Let 0 lt 7 lt p and t E R Let Y and X be two vector elds in AP satisfying HtYHp 3 re and HtDYHp S 86 with e S 16 Then belongs to AP an lltltIgtgtX e XIIH 36511X1196 319 ltltIgtgtX7X emclepq mumps Proof It suf ces to consider If 1 since we can rescale t Y to 1 tY Let n be a fixed positive integer By using Proposition 34 and Cauchy7s formula with contour 1 to estimate d DXY 116 X 0 1 11 320 dz n5 ZO we obtain the bound A HYXiip i39rn S 888iiXiiPa where p p This bound can be iterated n times with p decreasing by rn after each step and we find 1 An 1 WHY XII lt Hltnsgtnenuxwp 4 i n 1 322 HeSeWHXHp lteegt esHXHp In the last inequality we have used Stirling7s formula Now 00 1 A 6 651 1 n ltltIgtygt X 7XH glam X s 5 mm 323 p77 and the rst bound in 319 follows The second bound is obtained analogously with the sum in 323 starting at n 2 QED 34 A general elimination procedure Denote by A the space of vector fields on DP whose derivatives belong to AP On this space consider the norm HYH HDYHp HYHp 324 22 HANS KOCH Let ll be a xed but arbitrary projection operator de ned on all the spaces A and having norm one We also x 0 lt p lt p Let Z be any xed vector eld in A with the property that there exists a positive constant a S 1 such that M 0 Hm Zill 2 Wt 325 for all p S 7 S p and for all Y E llTAT Notice that for the choices of Z and l7 used in Subsection 22 with H N 039 and Subsection 31 the constant a is roughly equal to 039 The goal is to show that if X is suf ciently close to Z in 4 then there exists an analytic change of coordinates LIX Dp a Dp such that UX belongs to A and satis es TilQX 0 326 We start by determining an approximate solution of this equation given by the time one ow I of a vector eld Y ll Y To first order in Y the equation 326 reduces to ll X YX 0 327 Proposition 37 Let 0 lt 7 lt p Let X be a vector eld in 43 satisfying iixezH HUMP 328 Then the equation 327 has a unique solution Y E llTAp The vector eld Y satis es HYHp S lll XHp Furthermore belongs to APLT and satis es 6 T ltltIgtgtX 7XH a iwl walp 1 286T 7 2 329 ltltIgtygt X e X e KXiHH W H1 XHpHXllp Proof The rst condition in 328 implies that a HiYaX 7 Zal S 2HYHLHX ZN S ElliH 330 for every Y 6 43 As a consequence we have 1 Hi YaXlllp BM in 2m 7 Hi EX 7 2m 2 ElliHg 331 whenever Y belongs to ll Ag This shows that the linear operator l7 ll A7 a llTAp has a bounded inverse and in particular that the equation 327 has a unique solution Y E llTA7 The bound 331 also shows that this solution satis es 2 L 2 Hutsgmxt HDYHpSEHHXHp 332 Renormalization of Vector Fields The remaining claims now follow from Proposition 36 setting 8 a iHll XHp and s 7 QED Our goal is to iterate the map X gt gt described in Proposition 37 by starting with a vector eld X X0 and setting Xn1 CbthXn WXn YanD 05 333 for n 01 The expectation is that the maps U lt1gt O o c o o QM 334 converge to a solution LIX of equation 326 as 11 tends to infinity Let now 7 p 7 p Choose R 2 HZHp a and 8 2 0 subject to the constraints 8 lt 276m 8 S 2 9a26 rl r 1R 1 335 Lemma 38 78 IfX is a vector eld in A such that HX ZN S 273a HTle S 8 336 with 8 satisfying 335 then U7 converges in the af ne space 1 Ap to a function LIX that takes values in Dp The map X gt gt LIX is continuous in the region de ned by 336 analytic in the interior of this region and satis es the bounds 3 6T HUX IHp S EHH Xllp HUXX Xllp S 32REHH XHp 337 Proof Let p0 p and for m 01de ne pm1 pm72rm where rm 2 m 2r Our rst goal is to prove that 333 defines a sequence of vector fields Xm E A satisfying pm HXm e Xmilwgm rma HH XWHM We 338 If we de ne X1 Z and X0 X then these bounds hold for m 0 by 336 Assume now that 338 holds for m S n Then by summing up the bounds on Xm 7 Xm1 for m S n we obtain the rst inequality in 1 HX 7 2w lt Za Hr Xann 4 W 2arn 339 The second inequality follows from 338 by substituting the rst bound in 335 on 8 Thus Proposition 37 guarantees a unique solution to 333 and it yields the bounds e HXnJrl Xann 5T 7n1 e T 7271 2 4n g 634 Re Hll Xn1Hpn4n 7 24 Re 340 17 24 HANS KOCH Here we have used also that HXann S R which follows from the first inequality in 339 By using the second condition in 335 together with the fact that HFllgniwn S rglllFllM we now obtain 338 for m n 1 from the bounds 340 Next consider the functions bk Pg 7 I By Proposition 35 and Proposition 37 2 llmllpkil S HYkak S EH1 Xkllpk lt Tk 341 This shows that Um 1 o 11 o o 12 1 defines a function in l AP that takes Ym1 n7 values in me Here and in what follows it is assumed that 0 S m lt 11 Setting UM6 l we have the bound 7112 8 342 Z a lt gt km Wu 7 Umllp 71 1 7171 Z 453 0 Uk1m 3 Z H kllpkil S k m km p This shows that n gt gt U7 converges in l Ap to a limit LIX that takes values in D9 and that satisfies the first inequality in 337 if we set 8 Hll XHp Clearly X7 7 UXX in Ap The second inequality in 337 is now obtained by using the first bound in 340 The analyticity of the map X gt gt LIX follows from the uniform convergence of U7 7 LIX QED Under the same assumptions as in Lemma 38 is is straightforward to prove the additional bound Hugx i X i YX Oa sur xug 343 H where Y is the vector field described in Proposition 37 Here 0 is a constant that only depends on p p and HZHP 4 A nontrivial RG xed point In this section we sketch the construction 76 of a nontrivial xed point for R and describe some numerical results which indicate that there exists an analogous xed point of R12 for non twist ows 41 Observations and result As described in Subsection 16 numerical experiments with one parameter families gt gt H13 of Hamiltonians on T2 gtlt R2 whose golden invariant torus breaks up as Q is increased past some critical value ee also display sequences of bifurcations that accumulate at the critical point If denotes the parameter value where the symmetric Birkhoff orbit with rotation number ran1L7 becomes unstable where u is the n th Fibonacci number then the observation is that n 7 n1 n1 7 n converges to a universal number 6 16279 Furthermore the orbit structure of the critical Hamiltonian H w is self similar near the golden torus described by the universal constants 118 The standard explanation involves the existence of a nontrivial fixed point H for a RG transformation R In this explanation 6 is the expanding eigenvalue of Renormalization of Vector Fields The other universal constants like m and AZ are related to the re scaling of H during renormalization In particular m it the value at H of the scaling M pH that appears in the renormalization step H gt gt H 0 TH codim 2 ptmnslations smooth torus critical torus p l 01 Fig 4 Expected RC picture for the breakup of invariant tori When trying to find an RC xed point such as H it is useful to determine first possi ble subclasses of Hamiltonians that are invariant under renormalization If the phenomena under investigation is observed for a family gt gt H13 in such a class then the intersection point H w of this family with the stable manifold at H belongs to that class and thus the same should be true for the xed point H As was observed already in 74 one such class is the set of all Hamiltonians of the form H0149 wphqp 710119 Z hukCOS1quotIQ 4039 41 Vk I where w 19 and Q 71914 are the two eigenvectors of the matrix T 1 f associated with the eigenvalues 19 golden mean and 71971 Besides being even in 1 time reversibility these Hamiltonians have the property that the quantity w q evolves linearly in time We shall now describe a result 7677 concerning the existence of the fixed point H Here the resonant part lllH of a Hamiltonian H is defined by restricting the sum in 41 to pairs 1k with the property that w V 3 0 9 V or w 1 lt Hk Recall that the RG transformation 23 involves solving the equation ll H oUH 0 The solution that was described earlier yields UH as the composition Um o U 2 o of canonical transformation Um close to the identity with small nonresonant generating functions on Since we are dealing now with Hamiltonians that are far from integrable we need to include a canonical 26 HANS KOCH change of variables say Um that is not close to the identity The nonresonant generating function to for such a transformation was determined in 76 by solving numerically HioUmmHl i H10 Hn 1f1HloTH 42 As a result the remaining factor Um o U 2 o can be expected to be close to the identity The RC transformation used in 7677 can now be written as RNo oS 43 where SHqap CHHqapCH5 H 7ng o 7 o Um 44 HOUH Here CH 2h072 71 19 and UH is a canonical transformation determined by solving equation 214 Notice that the canonical transformation Um has been combined with other linear parts into a linear operator instead of incorporating it into UH ln Subsection 53 we will define a function space 8 for function h of the form 41 similar to the spaces considered earlier Here we only consider the case 6 0 The Hamiltonians considered then belong to the af ne space Hp K0 8 where K0qp w p Function in Hp that take real values for real arguments will be referred to as real Theorem 41 76 There exists an even real resonant Fourier Taylor polynomial hl an odd real Fourier Taylor polynomial to a choice of the parameters 039 H p and an open neighborhood B of K0 hl in Hp such that the following holds The transformation R is well de ned analytic and compact as a map from B to Hp It has a unique xed point H in B which is real analytic and exhibits a nontrivial scaling in the sense that 0 lt MH lt 49 Rigorous bounds on the universal constants can be found in 77 The strategy for proving Theorem 41 will be described below We note that the Hamiltonians considered here are not necessarily close to integrable However if H H with H close to the approximate fixed point H1 K0 hl then H is close to H1 And by construction H1 H1 Thus it suf ces to consider N near the Hamiltonian H1 which is resonant 42 Strategy of proof Theorem 41 is proved by converting the fixed point problem for R to a fixed point problem for a Newton like map M associated with R Mh h RH1 Mb 4 H1 Mh 45 where M is an approximate inverse of l 7 DRH1 The goal is then to show that M is a contraction near the approximate fixed point H1 This is a common strategy in computeri assisted proofs see e g 80 and references therein It usually involves explicit bounds on Renormalization of Vector Fields a nite dimensional truncation of M and error bounds for the terms that were truncated In the case at hand however working with a truncation of the full map M turned out to be computationally prohibitive This problem is solved by using the fact that M only needs to be estimated on a small open set More specifically we approximate the most complex part of M which is the map N by a much simpler af ne map N1 and then estimate the difference between the two The map N1 is defined by the equation N1H1f1H1HflHla 1a 46 where 151 is the solution of ll f1 H1 o1 0 Then N1 is an approximation of N in the sense that NH1 f1 7N1H1 f1 vanishes to first order in f1 It contains all but the first of the elimination steps described in the last subsection The map N1 can now be used to define an approximate RG transformation R1N10 OS 47 Notice that R1 is nonlinear but only due to the trivial scaling transformation 8 Define 5H1f 5H1 f 5H1 DSH1fa r r 48 RHlaf DN1H1 SH1f Then the map M can be rewritten as MW R1H1H1 a E H DR1H1Mlh b RH1 Mh R 7 R1H1 Mb 0 Denote by BO the ball in B of radius 7 centered at the origin One of the goals is to show that M maps such a ball BO into itself by verifying that a llR1H1 Hlllp 6 ltlt 7 b has an operator norm K lt l c RH1Mh R i R1H1 Mhp 073 for appropriate constants 6 K O gt 0 In order to estimate the derivative of M the bound 0 is extended to 327 Then the derivative of the nonlinear part of M is bounded by 07 on BO To be more specific we estimate Kn K 72 7720011 Mhp h 6 13W 11 12 in addition to E lt 10 and K lt 084 Then the existence of a fixed point H in BO follows by verifying that 6KTK1K ltT KK2K rltl The parameters used are a 085001 H x 004 p x 085 015 and 7 310 12 More details can be found in 76 28 HANS KOCH 43 Nontwist ows Consider Hamiltonians H T2 gtlt R2 a R of the form Hqpwphqz 201 49 where w 1941 and Q 17194 with 19 gt 0 The slope on T2 of a solution curve 15 gt gt qtpt for such a Hamiltonians is given by dQl 41 1971 azh 410 dq2 12 1 1948sz lt Assume that there exists an invariant region for the ow where this slope is bounded away from zero or infinity If H satis es the twist condition77 83h 31 0 in this region then the slope 410 is a monotone function of 2 Thus the rotation number p limtmoo q1tq2t if it exists is also a monotone function of 2 Most KAM type theorems assume such a twist or non degeneracy condition It certainly holds for Hamiltonians K177small77 and the critical Hamiltonian H most likely satisfies a twist condition near its golden invariant torus At the opposite spectrum are the shearless tori where the rotation number is a local minimum or maximum Such tori appear in many physical systems including models of the atmosphere toroidal plasma devices channel ows and others 88 29 28 7 65 12 Just like regular KAM tori a shearless torus with Diophantine rotation number persists under small perturbations of an integrable system 3254 In fact they are surprisingly stable At the point where they break up they separate two totally chaotic looking regions while in the twist case elliptic islands still dominate a non trivial fraction of phase space Apparently the breakup of such tori is also governed by universality and scaling In the case of the golden mean and related rotation numbers numerical investigations of specific two parameter families 30 31 3 4 reveal self similarity phenomena with asymptotic scaling ratios both in parameter space and phase space that seem to be independent of the family considered The self similarity transformation involves as l2 step shift in the sequence of continued fraction approximants for 19 as opposed to the ordinary l step shift describing the similarity of periodic orbits in the twist case These observations suggests that there exists a critical period 12 for an RC trans formation like R in a space of Hamiltonians that permit shearless invariant tori There is no proof for the existence of such an orbit nor are there any accurate numerical results not even from experiments on specific families 30 31 3 4 which usually yield very good values for the universal constants but only anecdotal evidence for universality The main problem is that only a few levels of self similarity can be examined since each of them involves 12 steps in the continued fraction expansion A further complication is that two parameters are needed in order to keep the rotation number of the shearless torus fixed while the nonlinearity is varied Rmunnahz wn Di Veda mas Fig 5 00mg 92 a Hamxlmman mm a neareubmalsheazless gulden cows 55 00005 0001 5 0002 0001 0 0001 0002 0 Fig 6 A mum 50921255 gulden 0205 a 30 HANS KOCH The following is a short description of a numerical RG analysis that was carried out n 55 We consider not so small perturbations of the two Hamiltonians NIH Kvqpwpv9p3 Wi 411 which are a period 2 for the RG transformation R The invariant torus p 0 for K is shearless lts rotation number 19 1 is maximal if y S 0 or minimal if y 2 0 among the rotation numbers of nearby orbits As in the twist case the analysis is restricted to a subclass of Hamiltonians H with a suitable symmetry The symmetry chosen in this case is 7HJHHOJ Jqpqp 412 Its usefulness stems from the fact that if in H then the orbit of the origin for H is invariant under J Thus an invariant torus passing through the origin divides the energy surface H H0 0 into symmetric halves and so the torus has to be shearless More specifically consider Hamiltonians HqP w 19 Maw Maw Zhukei 39kzk z 9 19 413 Vk that are odd under j and real valued for real arguments In the numerical implementa tion these Hamiltonians H and all other functions that appear in the definition of RH are truncated to M 3 N1 k 3 N2 M k 3 N3 and represented as finite arrays of Fourier Taylor coef cients lmplicit Equations like are solved by Newton7s method Af ter eliminating one of the unstable directions of R we searched for points on the stable manifold of the expected period H0 H1 H11 using a bisection algorithm along lines through K One of the first findings was an apparent additional symmetry jaanaeAiHniHmHTOHm mn6modl2 where A 1001l1 and 11 HOJa 1011 wmrpl 414 If correct then H0 H1 H5 are in fact a period 6 for the transformation N ja o R lterating N along the presumed stable manifold of N at the critical period 6 bisecting again when necessary and starting all over with different choices for the RG parameters 05 when things failed to converge an extremely tedious procedure an approximate fixed point for N 6 was found Our numerical values for the expanding eigenvalues 6 of N and for the average scaling defined by 16 MH0MH1 MH5 are 516 m 2661 5 m 1585 p x 03659 415 For comparison the numerically observed values for Si6 and lg6 computed from the data in 3 are 2678 and 1583 This supports the idea that the breakup of shearless golden tori is governed by a RG period 12 but it is clear that much more work is necessary Renormalization of Vector Fields It is also possible to give some explanation why a RC period 6 or 12 should be expected in connection with shearless invariant tori The same argument for rotation numbers in QM with 19 V112 4 n2 suggests that the corresponding RC trans formation should have a period 6 or 12 if n is odd and a period 4 or 8 if n is even More details can be found in 55 5 Invariant tori A canonical application of renormali ation for ows is the construction of invariant tori or other conjugacies The basic version discussed here applies to self similar frequency vectors and involves a single RC transformation R a fixed point for R and its stable manifold W5 However the procedure extends easily Diophantine rotation vectors once we generalize the notion of a stable manifold to sequences of RC transformations In this section we also discuss critical non differentiable and shearless invariant tori 51 Some ideas and results Consider the RC transformation for some fixed but arbitrary class of vector fields X RXn 1AX AX THOUX 51 We note that by construction UCX LIX for nonzero constants 0 Assume that X0 lies on the stable manifold of some fixed point X00 Then the sequence of vector fields X7 RWXO converges to X00 Combining the 11 RC steps connecting X7 to X0 yields Xnn1n1A3wo 77100V3X0 52 where Vk A3200 0 AXk o Agiil 53 Assuming that AXOO is one to one we have 7ltlt1 W 1Xoltgt X00 on the range of AXOO The goal now is to show that Hknknoo a c and Vk a l suf ciently fast such that VoovlooVn1gtl 54 in some appropriate topology If the limit 54 exists on a nontrivial part of phase space then PX0 0X00 55 In this case the construction yields a conjugacy to a constant vector field for every X on the stable manifold of the fixed point X00 This can be applied eg to parametrized families of vector fields gt gt X43 under conditions that guarantee that the family intersects the stable manifold ofR A procedure of this type for skew ows is described in Subsection 83 For Hamiltonian vector fields on Td gtlt Rd the limit 54 can be expected to exist only on Td gtlt In this case P is an invariant torus for H Concerning uniqueness recall 32 HANS KOCH that invariant tori With given rotation numbers typically come in one parameter families indexed eg by the energy The tori P constructed here are characterized by the property that the integrals d Am 3 2mg 56 quot j1 vanish along each closed curve y on the torus Tori With this property Will be referred to as being centered at p 0 This property is invariant under symplectic maps 16 obtained from generating functions 17 Thus since P is a limit of such maps With the p domain shrinking to the point zero P is centered at p 0 If H does not lie on the stable manifold but satisfies a suitable nondegeneracy condi tion then the procedure sketched above may still be used to construct an invariant torus for H To be more specific consider the RG transformation R for analytic Hamiltonians Where 1 O RltHgt WH AH lt5 7 With AH TH oily and H H o 7 As can be computed explicitly see Subsection 61 the expanding subspace for DRK0 at the trivial fixed point K0qp w p consist of all functions huqp 1 19 With U in the contracting subspace V of the matrix T Consider now an analytic Hamiltonian KqpwppMpfp fp0lpl3 58 Where M is a real symmetric d gtlt d matrix such that the quadratic form 1 gt gt p Mp is non degenerate When restricted to the d 7 1 dimensional contracting subspace of T We may assume that h is small since this can always be achieved With a scaling K gt gt s 1K 5 Then the family 1 gt gt Kqp v indexed by V is transversal to the local stable manifold W5 of R Thus if H is an analytic Hamiltonian close to K then the family 1 gt gt Hqpv intersects W5 at exactly one point say for v 0 Hgt gtHv W8 codim d 7 1 all ptranslates of H if M in nondegenerate Fig 7 A Hamiltonian and its ptranslates Renormalization of Vector Fields Clearly if H 0 has an invariant torus with rotation vector cw then so does H One of the issues with an RC based construction of invariant tori is the loss of regu larity that occurs in the process This concerns mostly regularity in the angular variable 1 To simplify the discussion consider again the RG transformation 57 near the trivial fixed point K0qp w p Then the equation 53 becomes Vk T W 0 AH 0 7154 59 Notice that T has d7 1 contracting directions Thus the composition with 74 shrinks the domain of analyticity in the contracting directions of T down to zero in the limit k a 00 However the non integrable part of H which determines the size of UH 71 tends to zero even more rapidly This fact can be used to control some fixed number of q derivatives of Vk and to prove that the torus P has that many continuous derivatives This result is far from optimal though These tori are known to be analytic Fortunately the analyticity of P PH for near integrable Hamiltonians H can be obtained in a different way And this works not only for Hamiltonian ows Consider torus translations Juqp q 1 019 First one verifies that for real 1 RojujT1UoR jUHHoJU 510 This is an explicit computation using that lli commutes with 7 and that ju is an isometry on the spaces 79 The equation 510 shows in particular that the local stable manifold of R at K0 is invariant under Z Here one uses the fact that the trivial fixed point is invariant under translations on Td Assume now that H belongs to this local stable manifold Then the second step is to verify that the construction of P PH yields 1THon JqflOPHOJu 511 for real translations v This identity implies that PHltQgt EqPH E0 PH 0 Jq JqEOPHqu 512 where Eq denotes the evaluation functional f gt gt fq Now it suf ces to prove that the map H gt gt PH is analytic in an open neighborhood of K0 in Hp K0 79 Then if H is suf ciently close to K0 in a space Hp with p3 gt m then 1 gt gt H o Jq is an analytic family in the domain of H gt gt PH Thus the right hand side of equation 511 depends analytically on 1 defining an analytic continuation of the left hand side It should be mentioned that the analyticity of H gt gt PH is a consequence of the explicit construction of PH via uniformly converging limits of analytic maps A construction along these lines of analytic invariant tori with self similar frequency vectors can be found in 74 In the case d 2 the procedure was generalized by Kocic 81 to frequency vectors w that satisfy a Diophantine condition with restrictions on the Diophantine exponent using a sequence of RC transformation 72 associated with the continued fraction expansion of the rotation number wlwg Recent work by Marklov Lopes Dias and Khanin 70 extends this construction to arbitrary Diophantine vectors w 34 HANS KOCH in dimensions d 2 2 after introducing an appropriate multidimensional continued fraction expansion This expansion and a RG analysis of skew ows 79 that is based on it will be described in Sections 7 and 8 It should be noted that for Hamiltonians Hqp that are linear in p there is no domain loss in the construction of T In this case P is a proper change of variables that reduces H to a constant multiple of K0 This can be applied to vector fields on Td by considering the associated Hamiltonian Hqp p A RG analysis of ows on T2 with Brjuno rotation numbers was given in 96 A RG analysis of one parameter families of Hamiltonians with a pair of golden invariant tori bifurcating from a shearless invariant torus was carried out by Gaidashev in 54 A brief description of this work which involves two different fixed points of R will be given in Subsection 55 In all these cases invariant tori were constructed for vector fields that are close to constant ie the RG fixed point used is a linear vector field or in the Hamiltonian case one of the integrable Hamiltonians K0 or K discussed earlier lnvariant tori for critical Hamiltonians near the nontrivial xed point H are discussed below This includes a description of the proof of Theorem 51 77 In some open neighborhood of H00 every Hamiltonian that lies on the local strong stable manifold of R has a non differentiable golden invariant torus Numerical results indicate that R is hyperbolic at H00 with a single relevant ex panding direction but this has not yet been proved As mentioned in Subsection 16 the Hamiltonians on the non KAM side of the stable manifold are expected to have hyperbolic orbits with golden mean rotation number A possible approach for proving a local version of this conjecture is described in 27 52 Renormalization of invariant tori As was shown above no generality is lost by using spaces like CT for the construction of the invariant torus T In most of the work that have been mentioned the differential version 13 of the torus equation rHoxrxtltrgtorH lltq0qtw0 513 was used This requires at least 7 2 1 Some recent work 778178 follows a different approach based entirely on semi conjugacies where it is natural to use equation 513 directly So there is no need to work with differentiable functions We will now describe the construction given in 77 of invariant tori for Hamiltonians on the local stable manifold of the nontrivial fixed point H described in Theorem 41 In this case the argument that was used above to show that a continuous torus P has to be analytic fails since the fixed point H is not invariant under torus translations ju In fact one of our goals is to show that the Hamiltonians on the local stable manifold of R at H are critical in the sense that their golden invariant tori are not differentiable This argument uses specific properties of the fixed point H and of its scaling map AH The other steps in the construction which we discuss first can also be adapted to the study of near integrable Hamiltonians 81 Notice that the RG Renormalization of Vector Fields transformation used near H is also of the form 57 but with AH 7 o Um oily and H H o 7 o Um The basic idea is to relate the equation 513 to the analogous equation for the renormalized Hamiltonian H For the two ows we have AH o EISt Wit 0 AH 514 with 7 depending on H in general This identity follows from the fact that T o Um oUH is symplectic and from an explicit computation showing that 514 holds in the case where Hqp is taken to be n lp 1HqMp Now consider Hamiltonians of the form Hqp w p hq 9 19 with Q w 0 Then the time renormalization is n 19 1 where 19 is the eigenvalue of T for the eigenvector w De ne MHfHof07 1 515 for any map f from Td gtlt 0 to the domain of AH Assume that T is an invariant torus for H taking values in the domain of AH and set P MH Then by using the identity 514 together with the fact that To It 1 o T we obtain Po lltAHoToTilo lJtAHoTo l rltoTil AHo iltoToTil toAHofoT 1 toP This shows that P is an invariant torus for H It also suggest that an invariant torus for a xed point H of R is a xed point of MH Such a torus P satis es the equation F o T AH o P which together with purely topological arguments can be used to prove the following Lemma 52 77 Let H be a xed point ONE and let P be an invariant torus for H that is continuous and homotopic to T If P is a xed point for MH then AH maps rangeP invertibly onto itself Let now H00 be a xed point of R with a well de ned local strong stable manifold W5 Examples are the integrable xed points 25 and K0 and for d 2 the nontrivial xed point H Given any Hamiltonian H0 6 W5 de ne H7 R H0 In order to simplify notation we will write A7 and M7 in place of AHn and MHH respectively Our first goal is to show that MH is a contraction on some appropriate space of maps whenever H is suf ciently close to H00 If we knew eg that each of the Hamiltonians H7 had a unique invariant torus P7 centered at p 0 with frequency vector w then these tori should satisfy PnMnrn1Anor 1oT1 n012 516 This argument can in fact be turned around We will use the contraction property of the maps M7 to construct via inverse limit a sequence of maps P7 satisfying 516 Then we will show that R is indeed an invariant torus for H7 36 HANS KOCH 53 Existence In what follows we will restrict our analysis to a special case Let w 1941 and Q l 71971 with 19 the golden mean Let M cw and 2 cw such that wwu 962 1 We consider only Hamiltonians of the form Hqp w p hqz where z 9 19 As a result only coordinate changes of the form U l u uqp qu uyw uZQ 517 need to be considered Just like the ow for H they are time preserving77 in the sense that 739 w q is invariant lmposing further that Mg 2 be even in q we can restrict use to be odd in q and uz even This will be referred to as the parity preserving77 property In addition uqp only needs to depend on 1 and 2 but not on y w p As a result uy enters trivially in any composition of such transformations Thus in order to simplify notation we will identify it with use uz For the same reason it and U will be identified with functions of the variables 12 The same applies to the functions F l y that appear in the construction of invariant tori In addition these functions do not depend on 2 A connection between the transformation R for time preserving Hamiltonian ows and the RG transformation for commuting maps can be found in 75 Next we introduce some functions spaces All of the functions that need to be considered here are either even or odd in q The coef cients in the cosine sine series of an even odd function f fq will be denoted by fl For functions like ym and yz we will use the space 80 if even or C0 if odd of functions f D a C with finite norm HfHo Zlfule lw39quot 1l9vlf 518 Here 67 lt l are small positive numbers to be chosen later The space C0 gtlt BO equipped with the norm H02 fZH0 maXHme02HfZH0 will be denoted by A0 Consider now the transformation MH defined earlier MHrAHoroT1 519 Concerning the composition on the right by 771 notice that f gt gt f o T 1 is bounded on 70 with operator norm close to 1 if 7 gt 0 is small For the composition of the right by AH consider first H K0 Then AH acts on y by multiplying ym by 719 1 and yz by 719 Both of these numbers are less than 23 in modulus Thus MH is a contraction on A0 in this case if 7 gt 0 is suf ciently small This clearly remains true for analytic Hamiltonians near K0 The proposition below implies that the same holds for Hamiltonians near the fixed point H Given p pmpz with positive components define Dp to be the complex neighborhood of D T2 gtlt 0 characterized by llm rl lt 6 llmacl lt pm and lt pz Here x 2 1 Let B be the space of analytic functions h Dp a C that are even in q and have finite norm thlp Z lhukl65 39quotl COSMPmQ WP 520 Vk Renormalization of Vector Fields where hmk are the coef cients in the cosine Taylor expansion of h The af ne space K0Bp will be denoted by Hp In what follows we use the same domain and norm parameters as in 76 namely p 085015 and 7 1041 Let p pmpZ2 Proposition 53 77 If6 gt 0 is chosen suf ciently small then there exists a bounded open neighborhood V of H00 in HQ such that the following holds The map H gt gt AH is analytic on V and there exists a positive a lt 1 an open ball B in A0 and a concentric closed ball B0 C B such that for every H E V the transformation M H is well de ned on B maps B into B0 and contracts distances by a factor 3 a The range of any function P E B is contained in Dp Furthermore H P gt gt MM is analytic on V gtlt B The proof in is based on computeriassisted estimates for the scaling map AH After determining numerically an approximate fixed point f for MIL the stated properties were obtained by estimating the norm of MHf 7 f and of the derivative of Mm near f Notice that Proposition 53 the existence of a fixed point Poo for the map Mm An analogous proposition for the trivial fixed point K0 follows from our earlier dis cussion together with the fact that R is analytic near K0 Now we are ready to construct a sequence of functions P7 that satisfy equation 516 Consider either H00 K0 or H00 71 Let V C HQ and B C A0 be as in Propo sition 53 and let F0F1 be arbitrary maps in B Given a sequence of Hamiltonians H0H1 in V define Pnm MnOMn1OOMm71Fm AnoAnHooAm1oFmoT mn 0 nltm Theorem 54 77 There is a constant C gt 0 such that the following holds Let H0 H1 be Hamiltonians in V Then the limits P7 limmnoo PM exist in A0 are time and parity preserving do not depend on the choice of the maps Fm and satisfy the bounds Hl nil ooHOSOsiip HHmiHOOHQ n01 522 If H7 RWHO for all n gt 0 then 516 holds If in addition V has been chosen suf ciently small and H0 belongs to W5 then D is a golden invariant torus for H7 for each n 2 0 The main ingredients in the proof of this theorem are the following Proposition 53 shows that if n lt m lt k then the difference l mk 7 1777 is bounded in norm by da where d is the diameter of B Thus the sequence m gt gt 1777 converges in A0 to a limit P7 and this limit is independent of the choice of the maps Fm The estimate 522 is obtained by using the analyticity of the map H0 gt gt P0 Assume now H7 RWHO for all n gt 0 Then 516 is obtained by taking Fm Pm for all m and using that the maps M7 are continuous In order to prove that R is an invariant torus for H7 we can use the identity 3 0 17mm 0 llit A7 0 An1 o o Am1 0 5714 0 P00 0 Q tm o 7771 523 MnoMn1ooJlm1 ol ooo llitm q gt 38 HANS KOCH where tk 194 and where le denotes the ow for Hk Here we have used that T o IDS 1195 o 7 Notice that the map in equation 523 is time preserving since the two ows change 739 by opposite amounts Thus equation 523 is an identity between maps in A0 Consider first the case where H7 H00 for all n 2 0 Then the two ows in tend to the identity as m a 00 Thus takes values in Dp for large m As m a 00 both sides converge in A0 to P7 But convergence in A0 implies pointwise convergence and since 11 and ltlgtt are both continuous and invertible we conclude that 11 0 P7 0 lift P7 This shows that R is a golden invariant torus for H7 For general H0 6 W5 near H00 we can now use that ofgm 0 r00 0 xrrtmen ofgm o 1th 0 PM 524 Again the map in equation 523 takes values in Dp for large m So by the same argument as above P7 is a golden invariant torus for H 54 Critical invariant tori The invariant torus Poo for a xed point H00 of R satis es the equation AMOPOOPOOOT 525 This shows eg that 01900 P00 0 is a xed point of the scaling map A00 lf P00 is smooth and one to one and near this xed point then the semi conjugacy 525 suggests that the eigenvalues of T are also eigenvalues of DAoo0poo This is trivially the case for the xed point K0 whose scaling map is TH with M 19 3 The eigenvalues of 7 are 1 719 1 1941 and 719 2 These are the trivial values of the scaling constants AT Am Ag and AZ whose critical values we shall now describe First we note that if U is parity preserving then the line 1 0 is invariant under U Thus the maps AH are of the form AH02 0442 526 where XH is some function of one variable The following proposition concern the xed point H described in Theorem 41 lts proof is computeriassisted Proposition 55 77 The analytic function 11 maps the interval egg2 QZQ into its interior has a globally attracting xed point 200 and the derivative of 11 at this xed point is AZ 70326063 Furthermore MH Moo 0230460196 One of the conclusions is that AH has a xed point qp 01900 with Qpoo zoo and that its derivative at this xed point has AZ as one of its eigenvalues We note that the corresponding eigenvector is not 09 as equation 526 might suggest but it has a nonzero component in the y direction that has been suppressed in 526 The remaining three eigenvalues of DAH0poo are obtained by using that UH is a time and parity preserving symplectic map they are A7 19 Am MooAZ 70706795 and Renormalization of Vector Fields xy MooA7 01424322345 The corresponding eigenvectors are the direction of the ow at 0100 and the vectors 90 and 0w respectively Consider now the RG fixed point H x H The following result implies Theorem 51 It uses the fact that Am is different from the corresponding eigenvalue 719 1 70618033 of TH Theorem 56 77 There exists an open neighborhood V of H in HQ such that if H0 belongs to W5 V then the torus P0 de ned in Theorem 54 is not differentiable Proof By using Lemma 52 and the analyticity of AHOO we can finds sets K C S and K C S C K all subsets of Dp whose interiors contain 9 graphPoo with 56 open and K K compact such that for H H00 the scaling map AH defines an analytic diffeomorphism from S to an open neighborhood of K Since H gt gt AH is analytic this extends to all Hamiltonians H in some open neighborhood of H00 in Hp Now choose V suf ciently small such that our previous results apply and the set U RangePn is contained in K This is possible by 522 and by the fact that W5 is the graph of an analytic function Thus we have P7AgilooAfloAaloPoo n 527 for all n 2 0 where the inverse scalings are defined in an unambiguous way In what follows we lift A7 and D from T2 to R2 restrict to 739 0 and use the variables 55 Since A7 is parity preserving it can be written in the form Anx5 fnx5x n5 gn5x2 528 It suf ces to consider points in S where we have a bound lt b lt QZQ By Proposi tion 55 there exists a real analytic function 0 on 7b b with non vanishing derivative such that g0 oo5 Azg05 Consider now the coordinates 5 05 and 5 Ugo5 Since UH is symplectic the functions 00 and foo in these new coordinates when restricted to 5 0 are simply multiplication by AZ and Am respectively The inverse of A7 is also parity preserving and thus admits a representation analogous to 528 In the coordinates 5 and 5 we have K1ia nia iawni7 a with on 117 analytic in SC Since all derivatives of A7 are bounded on K uniformly in n we have n55 a ooi5 uniformly on SC Furthermore mi5 a A uniformly in 5 Thus if V has been chosen suf ciently small we can find positive real numbers a and 5 lt 1 such that l ni l 3 t for all n 2 0 whenever lt 04 Here we have used the crucial fact that the eigenvalue Am of DAM at Poo0 is larger in modulus than the corresponding eigenvalue 719 1 of 71 By equation 527 the points P7 5 0 expressed in the coordinates 5 and 27 are given by asi O fsoKi1 ooKf1 ngl of0so s 594 530 40 HANS KOCH Assume now for contradiction that To is differentiable at 0 Then there exists a constant c gt 0 such that for any given 5 E R the angular component of P0sn 0 is bounded in modulus by cls l for large n Thus by using the above mentioned bound on the functions 157 we find that the angular component of Tns 0 is bounded in modulus by c nlsl lt 04 if n is suf ciently large By using now that R a P00 in A0 and the fact that evaluation is continuous on A0 we conclude that the angular component of Poos 0 is zero for all s E R Thus since the curve 5 gt gt 59 is dense in T2 the angular component of P00 is identically zero But this is impossible since P00 is continuous and homotopic to I This shows that To cannot be differentiable QED Fig 8 A critical golden invariant torus 55 Shearless tori The construction of invariant tori for Hamiltonians on the stable manifold of an RC xed point does not involve any nondegeneracy assumptions Thus it applies equally well to shearless tori Consider again the golden mean case so the vectors w and Q are the same as above We will call an invariant torus shearless if in some coordinates where the torus is at p 0 139 w M91029 GUNS 139 Oltlpl3 531 A simple Hamiltonian with such a torus is F0 where Faqpwpwv23 29p 532 with y E R nonzero This torus bifurcates as the parameter 04 is varied More specifically we have 139 0 and q w a 3722 so the Hamiltonian F404 has an invariant torus Renormalization of Vector Fields at z t a constant not time and the rotation number of this torus is 19 1 a 3yt2 533 17 19 1a 3yt2 9 Thus we can distinguish three cases a lt 0 F a has two real golden tori with shear a 0 F a has one real golden torus shearless a gt 0 F a has no real golden torus but two complex tori This type of behavior is observed and expected to persist under small perturbations of the family For near integrable area preserving maps a theorem confirming this observation was proved in 32 for arbitrary Diophantine rotation numbers using a KAM type method The following theorem focuses on golden invariant tori for near integrable Hamiltonian ows Theorem 57 54 If F is real and suf ciently close to F in some space of one parameter families of Hamiltonians then the following holds Each F a in this family has two golden invariant tori Pi and P2 and no other golden invariant tori nearby There exists a unique parameter value 04F where Pi P2 and the function F gt gt 04F is real analytic The tori P are real if and only if a S 04F If the Hamiltonians Fa are time preserving a technical assumption to simplify the analysis then the given torus for a 04F is shearless The maps a gt gt P are continuous and real analytic away from 04F This theorem is proved by using renormalization We will describe some of the ideas below with slight modifications to synchronize the definitions with those used in previous sections Presumably the method can be generalized to Diophantine frequencies using sequences of RC transformations as in 9581 The analysis involves two RG transformations with different fixed points and normal izations One of these transformations is 19 R0HHoZH HHoTH 534 with M M0 a fixed positive real number less than 19 1 As usual the canonical trans formation UH is determined in such a way that ll RO 0 This transformation R0 is considered in a neighborhood of its fixed point K0qp w p and y 31 0 is xed modulo sign such that F0 belongs to this neighborhood Define JtH Hth Jtqp qptQ 535 with 2 CQ normalized in such a way that 2 Q 1 In particular if H F a h then Hqu w p a 3vt2lz 3vt22 v23 700119 536 modulo a constant term that we can ignore Then we can choose 6 gt 0 such that the following holds Consider first the case h E 0 Then for a gt 6 all orbits for H in the 42 HANS KOCH domain Dp considered have a rotation number strictly larger than 19 1 And for 04 lt 76 the family 25 gt gt JtH intersects the stable manifold of R0 at K0 at exactly two points and thus H has two golden invariant tori Furthermore this intersection is transversal so the above generalizes to h 31 0 suf ciently close to zero This argument is used in the proof of Theorem 57 to construct the tori Pf when 04 is bounded away from zero Consider now the Hamiltonians Kqp w p yzg With the proper normalization M 19 2 these two Hamiltonians are a period 2 for the RG transformation R For Hamiltonians near K we choose the RG transformation 19 RHHoZH HHOTHojt 537 where M m 19 2 and t m 0 are determined in such a way that the coef cients of 22 and 23 in the Fourier Taylor expansion of 7 K are zero We note that the RG transfor mations R0 and R2 are hyperbolic near their trivial fixed points with a one dimensional unstable direction given by the linear function qp gt gt 9 19 In the case of R the unstable manifold at K is in fact given by the family F Thus if F is suf ciently close to F then F intersects the stable manifold of R at precisely one point The corresponding parameter value is denoted by 04F Using the procedure described in the last section we can now construct an invariant torus for FaF and if h Mg 2 then it can be shown that this torus is shearless One technical complication here is the presence of the translation is in the definition 537 of R However after the first RG step t tH is of the order of l 7 EH where E denotes averaging over the torus T2 And ll 7EH tends to zero very rapidly so that the translations is do not cause any domain problems This leaves the problem of constructing invariant tori for Fa with 04 31 04F close to 04F This is done by iterating the RG transformation R until the coef cient of z for R Foz is larger than 6 Then the RG transformation R0 can be used to construct two golden invariant tori as described above This part of the analysis requires good control over the dependence of RFa on the parameter 04 Thus it is convenient to introduce an RC transformation for families a gt gt Fa In what follows we identify the parameter space C with the one dimensional expanding subspace of DRK0 Denoting by ll the canonical projection onto this subspace we define ma MUM 538 In particular YEW 719204 which re ects the fact that F is the unstable manifold of R at K0 and that the expanding eigenvalue is 7192 The map YF is now used to re parametrize the family F after renormalization More specifically we set mFRoFoi1 539 By construction F is a fixed point of ER and it is straightforward to check that all eigenvalues of DERF are of modulus less than one Thus we expect that a F for large n This property which was proved in 1 makes it possible to get the necessary control over R Foz for 04 31 04F very close to 04F and for n correspondingly large Renormalization of Vector Fields This procedure known as the graph transform method is a common method for constructing the unstable manifold of a map R see eg 64 We Will describe a gener alization to sequences of maps of this method in Subsection 74 W8 UF0 FOJt UFaojt 9 19 E Fig 9 Renormalization of families with shearless invariant tori 6 Scaling According to the general RG picture the properties of a hyperbolic RG xed point H00 With an m dimensional unstable manifold describes universal behavior for m parameter families gt gt H13 The universal constants are eigenvalues of either the linearized RG transformation DRHOO or the linearized scaling map DAHOO 1004900 The latter describe self similarity properties of the orbit structure for Hamiltonians near the critical surface the stable manifold of R at H00 The reason is pretty clear If G is an orbit for RH With rotation vector w then AHG is an orbit for H With rotation vector parallel to Tw This fact was used eg in the analysis of invariant tori Here we discuss properties that relate to eigenvalues of the linearized RG transformation and more specifically the rate at Which periodic orbits accumulate at an invariant torus 61 Spectrum of the linearized RG transformation If H is a purely resonant Hamiltonian so that am is the identity then DRH can be obtained explicitly since it suf ces for this purpose to eliminate nonresonant modes to first order in the perturbation In this case the derivative of N H gt gt UH is given by the 44 HANS KOCH equation DAMEH7HU 3HT W r70 an This simplifies to DNH ll if Hqp does not depend on the angle variable 1 In order to simplify things further consider now R with fixed scaling parameters 7 19 and 0 lt M lt 19d Here and in what follows 19k denotes the k th largest eigenvalue in modulus of the matrix T and wk denotes the corresponding eigenvector In particular 191 19 and wl w The Hamiltonian K0qp wp is a xed point of this transformation R If we write fqp EV flpeiquot39q then DRKofqp Z uT 1pexpi1TQM 62 Assume now that f is an analytic eigenfunction of DRK0 with eigenvalue A 31 0 Then we obtain 19 n me own V e W 63 first for n l and then for any n E N by iteration Consider 1 31 0 Then T 1 19 for large n so by analyticity the left hand side of 63 is bounded in modulus by exp7m9 for some 0 gt 0 But the right hand side can tend to zero no faster than exponentially unless fl 0 This shows that the eigenvectors of DRK0 are functions of 1 only Finding these is easy On the invariant subspace of functions fqp f0p the homogeneous polynomials qp gt gt wk 19 and their products are a basis of eigenvectors In what follows we will always choose 0ltMltW y mm Then DRK0 has exactly d eigenvalues of modulus gt 1 One of them is A0 19M associated with constant Hamiltonians and the other d 7 l eigenvalue eigenvector pairs are Ak19119ka fkqPwk39P k2aquot ad39 65 Since the value of M has been fixed DRK0 also has an eigenvalue 1 with eigenvector f1qp w p But we can and will reduce this eigenvalue to 0 by choosing a scaling M MH in such a way that the torus average of w VpH is a fixed positive constant Then all eigenvalues of DRK0 other than the ones listed in 65 are of modulus lt 1 It should be noted that the expanding eigenvalues including A0 could be mapped to 0 as well by including a translation qp gt gt qpv in the definition of R with v vH chosen appropriately In this sense DRK0 has no relevan 77 expanding eigenvalues By contrast the linearized RG transformation at the critical fixed point H00 described in Theorem 41 is believed to have an expanding eigenvalue 6 16279 that is not related to coordinate changes 62 Accumulation of periodic orbits The expanding eigenvalues of the linearized RG transformation describe asymptotic rela tions between different members of the family gt gt H13 as Q approaches the value ee Renormalization of Vector Fields where the family intersects the stable manifold W5 of the given RG fixed point H00 To give an example let 73 be a collection of properties that can be assigned to closed orbits such that if has an orbit of type 73 then the corresponding orbit of H is also of type 73 Given some fixed w E RZd close to w consider the set of Hamiltonians in the domain of R that have a closed orbit of type 73 with rotation vector parallel to w Assume that this set contains a codimension m manifold 20 that is transversal to the local unstable manifold W of R Now consider the sets En R Eo They consist of Hamiltonians that have closed orbits of type 73 with rotation vectors parallel to an By the A lemma 113 these sets are codimension m manifolds that accumulate at W5 at an asymptotic rate given by the largest eigenvalue 6 of DRHOO Or more precisely if Q gt gt H13 is a m parameter family that intersects W5 transversally then for large n this family intersects 27 at exactly one point say H and n1 7 n n 7 n71 a 6 1 In particular if one of the properties characterizing 20 is marginal stability then we obtain the observed accumulation of bifurcation points described earlier where m 1 after elimination of the unstable directions related to p translations A stronger property that is expected to hold is the convergence of RI H mk to the intersection point of 27 with W This describes a joint scaling in parameter space and phase space gm H o R R 20 gm R R R H 1 El H 2 E2 23 H W Wu gt gt H Fig 10 Expected RC picture explaining universality Our goal here is to prove that such a picture is correct near the trivial fixed point K0 The RC transformation R is as described in Theorem 22 and we use the same space AP of analytic Hamiltonians as in Subsection 22 In this analysis we will keep track of constant terms in Hamiltonians so the unstable manifold W is of dimension d As a result the invariant torus PH for a Hamiltonian H on the stable manifold W5 is not only centered 46 HANS KOCH at zero but also lies on the surface of zero energy H o H 0 The families gt gt H13 considered are generated by a single isoenergetically nondegenerate Hamiltonian H via p translations H HOJ3 Jaqpqp 66 with Q 6 Rd Thus the periodic orbits described above are all orbits of the same Hamil tonian H The rate of accumulation of the parameter values describes the rate of accumulation of periodic orbits at the invariant torus In order to ensure that the family gt gt H intersects the stable manifold W5 transver sally we will assume that H lies near an isoenergetically nondegenerate integrable Hamil tonian K of the form 58 Still considering such near integrable Hamiltonians poses a problem If K has a closed orbit y with rotation vector w then it has infinitely many of them the entire torus p pw containing y consists of such orbits But a Hamiltonian H close to K has in general only finitely many w orbits closed orbits with rotation vector parallel to w The q translates of H are equally close to K but their w orbits vary by an amount of order one Thus there is no continuous map near K or near any integrable Hamiltonian that would associate with a Hamiltonian one of its w orbits That is unless we prevent the q translation symmetry of K from acting on our space of Hamiltonians We will do this by restricting the analysis to the subspace 8 of Hamiltonians Hqp that are even functions of 1 Then it is possible to find closed orbits of the form W05 11 62057 P0PtTa 57 where Q is an odd and P an even 27r periodic function with average zero The rotation vector in belongs to RZd and 739 7w is defined as the smallest positive real number if such that tw belongs to Zd We will refer to such an orbit as a symmetric w orbz t for H To be more precise we will focus on orbits on the energy surface H 0 Then we cannot prescribe the frequencies 11 but only the winding numbers 111 wd Define hwqp w p and d Am i f ijdqj 68 quot j1 Theorem 61 1 Let R gt p and let K 6 Br be a Hamiltonian of the form 58 with M as described after 58 If K suf ciently close to hw and w E RZd suf ciently close to w then there exists an open neighborhood B of K in ER and a positive integer N such that for every Hamiltonian H E B and every n 2 N some nonzero constant multiple of H has a symmetric orbit 7 with frequency vector wn 19 1T w lying on the energy surface H 10 and having AW 0 The sequence of orbit yn satisfy 71n07r0 lnlAgl 0 69 where P is the invariant torus for H obtained via translation from the torus PH for the Hamiltonian H H Renormalization of Vector Fields The proof of this theorem will be described in the next two subsections The estimate 69 is obtained from an analogous estimate 1 7114676 1n1A21O 610 for the parameter values 67 de ning the intersection of the family 6 gt gt H13 with the manifolds E7 63 Choice of the manifold 20 We assume that w 6 Rd is chosen suf ciently close to 66 such that hw belongs to the domain of R and such that v w l where v is the expanding eigenvector of Tquot As part of the construction of 00 we consider dl parameter families of Hamiltonians One of the parameters 5 is used to determine which multiple of H lies on 00 and the other d parameters uE can be related to p translations Given 5 E C and u 6 Cd satisfying 11 11 0 define an 5 7 lw 511 Let H hw h Our goal is to determine the values of 5 u and E such that the Hamiltonian F5HhuEhw5hhmE 611 has a symmetric w orbit y with the desired properties Besides the two properties F07 0 and A y 0 that are invariant under renormalization we also impose that P0 kv for some k E C Roughly speaking the conditions F o y 0 and A y 0 determine E and 16 respectively while P0 kv determines the parameter 11 In what follows we will treat 16 5 7 lw 511 as an independent parameter and treat 5111 as a function of x We start by solving the orbit equation quoty JVF O y The q component of this equation can be rewritten as TADQ V25hhm 075V2h0 y 612 The torus average denoted by E of the right hand side has to vanish so at SE 55 75EV2h o 7 613 The corresponding equation for remaining zero average part is Q ii i 751746 e Egt1ltV2hgt o 11 614 Similarly the p component of the orbit equation can be rewritten as T lDP 75V1ho y or since both sides of this equation are odd functions of q as 1313 1375D 1V1hoy 615 Next consider the condition A y 0 By using equation 612 we have 1 1 27r739 71 Awig wqigO kvPgtltwT Dodt 1 27r739 5 27r739 k PD dtk P h dt T27TT0 Q T27r0 v2 O y 48 HANS KOCH Thus the condition AW 0 is equivalent to iN Ni 5 27r739 kik k7727w 0 PVghoydt 616 Finally consider the condition F o y 0 By using that 1 27r739 5 27r739 FO y FO ydtE hw th thuloydt 27139739 0 27139739 0 5 27r739 Ek hoydt 27139739 0 we obtain the following equation N N 5 27r739 EE E h7P h dt 617 2M 0 lt V2Ov lt gt The equations 613 617 define a fixed point problem for the map 11 that takes xEkQP to i 13 For the components Q and P consider the space of Id valued analytic functions on the strip llmzl lt 7 equipped with the sup norm where 7 p3T And on the space of quintuples q EkQP consider as norm the largest component norm Then it is easy to see that 11 is a contraction near 1 0 with 11q As a result we find that for each Hamiltonian H in some open neighborhood Bw of hw in 8 the equation Fq 1 has a unique solution near 1 0 This solution yields a set of parameters uE and a w orbit y for the corresponding Hamiltonian F satisfying 15 i 11 W 1E1 lkl HQHT HPHT S OHH hwllp 618 for some constant C gt 0 Furthermore all these quantities depend analytically on H Now define so that H F Consider this function b on X Bw where X is the subspace of 8 consisting of functions of the form chw f with f haVing zero Fourier Taylor coef cients fma whenever 1 0 and 1041 S 1 The range of b is contained in the d dimensional expanding subspace W of DRULW Clearly X is transversal to W Thus the graph of b which we shall denote by 2011 is transversal to the unstable manifold W of R at the fixed point hw Denote by Ew the set of Hamiltonians H in the domain of R with the property that a constant multiple of H has a symmetric w orbit y on the energy surface H 0 satisfying AW 0 By construction 2011 is a subset of 011 Renormalization of Vector Fields 64 The manifolds Zn and orbits 0 Now consider the sets 27011 R Ew By the A lemma 113 these sets are codi mension d manifolds that are transversal to W and that accumulate at W5 as indicated by equation 610 Here the assumption is that the given family F is transversal to the stable manifold W5 and close to the unstable manifold For our families 6 gt gt Hf the transversality condition is guaranteed by the assumptions on K R and B in Theorem 61 But in order to get suf ciently close to the family F parametrizing the unstable manifold we first have to renormalize F gt gt H13 a few times An appropriate RG transformation R for families is given by WF ROFOYF 1 165 WF 620 as described in Subsection 55 see also Subsection 74 Here 11 denotes the canonical projection onto the d dimensional unstable subspace W of DRhw which we identify with the parameter space Cd An explicit computation yields DltFgtflt gt 7017 IPgtDRFltBgtfltBgt B YF lt6 621 This shows eg that the largest eigenvalue in modulus of DERF is the largest con tracting eigenvalue of DRhw which is 2912932 In what follows we assume that the scaling M has been chosen in such a way that m916f lt e lt Ag1 622 Consider now Fk ER F with F Hf and H satisfying the assumptions of Theo rem 61 Since Fk 7 F as k 7 00 the intersection parameters for Fk satisfy 610 if k is suf ciently large But then the same holds for F as well The transformation R can also be used to obtain the necessary bounds 610 on the parameter values 67 More speci cally since F7 7 F we have 6 Y5 o o YF16 Fb e 2w 623 for some unique parameter value bn if n is suf ciently large Notice that YR is simply the restriction of DRhw to the unstable subspace Y Thus the largest eigenvalue in modulus of Yil is A31 and 610 obviously holds for the family F In order to prove the same for other families like 6 gt gt Hf one uses that F 7 F and an 7 YR and b7 7 boo are of the order 06 which is small compared to Mgr Our next goal is to construct the orbits 7 described in Theorem 61 and to esti mate the values n0 Given any Hamiltonian in the domain of R the renormalized Hamiltonian H is a constant multiple of H o where 17H VOH o 0 Hair 12H 7 6qu 67716 624 If H is suf ciently close to the integrable Hamiltonian K then H5 lies on the manifold 2w and thus some nonzero constant multiple of H5 has a symmetric w orbit gn Modulo domain questions this yields a symmetric wn orbit G7 17nH noTH ognoe t 625 625 50 HANS KOCH for a constant multiple of H13 By construction the size of the domain of the transfor mations decreases exponentially with H but at a rate that is independent of M By comparison the nonlinear part of 97 is bounded by a constant times the norm of H5 7 hw as was shown in 618 and this norm decreases like 06 Thus if M is chosen suf ciently small then the maps in equation 625 can be composed as indicated yielding the desired orbits G7 By using that 77 J13 G7 and P J13 PH we have 7710 N0 n ee 0710 PHON 525 Thus it suf ces to show that the term is small in modulus compared to 67 7 600 To this end write Gnlt0gt e PHlt0gtl lvnltH gtTltgnlt0gtgt vnltH ngtlt0gtl IVnH3 0 VnH0l anH0 PH0 Since gn0 06 as mentioned earlier the first term on the right hand side of 627 is bounded by Cbe for some constants b c gt 0 The third term admits a similar bound since it describes the convergence of VnH a PH for H 6 W5 which is governed by the largest contracting eigenvalue of DRhw In order to estimate the second term we can use the analyticity of f gt gt Vnf to obtain a bound of the form 627 WMHBMO 17nH0 S Cl2l llH Kllp 628 The details leading to these estimates can be found in Putting it all together we find that 69 holds for large 11 provided that M is chosen suf ciently small and H suf ciently close to K It should be noted that the smallness condition on M only depends on the matrix T 7 Sequences of RG transformations So far we have considered only self similar rotation vectors w 6 Rd that are eigenvectors of some integer matrix T with determinant i1 Extending renormalization to more general rotation vectors involves using sequences T7 of such matrices coming from a continued fraction expansion for w see also references 717283 At this point having kept track of how the bounds on R depend on the matrix T the main ingredients in this extension are a multidimensional continued fraction expansion with bounds on the matrices T7 and a stable manifold theorem that can be applied to sequences of RC transformations Both will be discussed in this section 71 Diophantine and Brjuno numbers In the case d 2 a self similar rotation vector is a vectors w u0vo whose winding number 040 vouo has a continued fraction expansion for n 0 1 an anan1an1 a 1 71 an1 an2m Renormalization of Vector Fields that is periodic Here we have assumed that 040 gt 0 The positive integers an in the continued fractions expansion of an irrational 040 gt 0 are obtained inductively anfanla an1 a 051727 Ha where a gt gt 04 is the oor integer part function on R Alternatively the fractional parts 7 an 7 1 can be obtained by iterating 7 Czn where G is the Gauss map taking at to the fractional part of z l The relation 72 between an and 047 can also be written in the form 1 i 1 1 i 0 1 11 w 1 Thus if we start eg with tag 1040 and define w 1 an1Tglwn for n 01 then tun 1047 A renormalization procedures for Hamiltonians on T2 gtlt R2 that zooms in77 on orbits with rotation number 040 involves a sequence of RC transformations 7 associated with the sequence of matrices T7 1 RAH H OUH a Hqm manqun lp v 74 The translation parameter 1 vH has been included here optionally in order to allow elimination of the trivial unstable direction associated with p translations Instead of working near a RG fixed point one has to consider a sequence of Hamiltonians like m KnltQapwn39p3Q 39p25 07152 H with the property that K7 Define 7a 72771 0 0 R0 One of the goals isjo show that if H0 is close to K0 then H7 7 K7 tends to zero as n a 00 where H7 RnH0 This cannot be expected to work for every irrational rotation number 040 gt 0 A technical reason is the following Consider eg the effect of the step H 1 HTn onto a resonant mode whose Fourier index 1 satisfies wn 1 S MHzH The norm of such a mode changes by a factor using p l f m expmvu 7 1111 76 which we would like to be less than one Now m 1l and in order to get a negative exponent in 76 we need UWHTWH 71 lt 1 This yields g Given that 2 l we have g T771 and this leads to the bound f7 g exp7l rn Next recall that the elimination procedure H gt gt H oZH requires that ll H be bounded in norm by a constant times an Taking this as the size of the domain for Rn we need UOfOfl 39fn71 Una 172739H7 52 HANS KOCH if we wish to have nontrivial Hamiltonians that are in nitely renormalizable Assume now that we choose 71 to be a small positive constant independent of n Then the factors f7 are bounded by some fixed positive constant as well By using that an HT H 1 x 171 we see that 77 can be guaranteed if the sequence 71 gt gt an grows no faster than exponential This can be improved by choosing 72 a 0 This is satisfied if 040 is Diophantine A nonzero vector w 6 Rd is Diophantine of class Q gt 0 if there exists a constant C gt 0 such that lwvl ZOllIHl d vezw 78 This set of vectors will be denoted by DOW It is well known that DOW has full Lebesgue measure in Rd see eg 10 A renormalization group analysis of the type described above was carried out in 81 It is shown that if tag is Diophantine with Q lt 7 1 then H7 7 K7 a 0 in a space of analytic Hamiltonians provided that H0 is suf ciently close to K0 Furthermore H0 has an analytic invariant torus with frequency vector parallel to two A larger class of irrational numbers is the set of Brjuno numbers 040 is called a Brjuno number if the denominators 1 in the rational approximants pnqn 10 a1 an of a0 satisfy Zq11nqn1 lt 00 These frequencies were used in an RC analysis of near constant torus ows 96 and skew ows 97 For ows on T2 and some other single frequency problems it is known that the set of Brjuno numbers is exactly the set of frequencies for which one can guarantee linearization 72 Multidimensional continued fractions Here we give a brief description of a multidimensional continued fractions expansion by Khanin Lopes Dias and Marklov 70 which is based on the work in 8973 on geodesic ows on homogeneous spaces Consider the one parameter subgroup of G SLd R generated by the matrices Et diage t e t Cd l 156 R 79 Let F be a fundamental domain for the left action of P SLd Z on G Then for every matrix W0 6 G and for every time t E R there exists a unique matrix Pt E P such that Wt PtW0Et belongs to F A useful invariant under the left action of P on G is 6 W 39 f W 710 lt gt 0736 HV H lt gt Now consider a special choice of W0 Let w 1111 be a fixed but arbitrary vector in DOW and define W0 to be the matrix obtained from the d gtlt d identity matrix by replacing its last column vector by w W0 1 w 711 Renormalization of Vector Fields In what follows we assume that the fundamental domain F has been chosen in such a way that it contains W0 The Diophantine property of w can be used to prove the following bound 6W0Et 2 c0592 t2 0 712 where 6 d Q This is one of the main ingredients in the proof of Theorem 71 70 There exist constants 01 and 02 depending only on the Diophantine constants 0 and on the norm ofw such that for all t gt 0 WM c1 we met 713 WWW 1H S C2exp9t The information that W Wt belongs to F is used as follows Consider the lwasawa decomposition of W UAK of W where U is an upper triangular matrix with diagonal entries 1 A diaga1a2 ad and K E SOd It is possible to associate with the fundamental domain F a Siegel set77 8 containing F and being contained in a finite number of P translates of F which can be characterized by some simple conditions on the matrices U A K More speci cally U belongs to a compact fundamental domain for the left action of P on the group of upper triangular matrices with diagonal entries 1 and 0 lt aj 23 12aj1forjgtl The crucial step is to prove that ladl is bounded from above and below by a constant times Since the norms of W and W 1 are of the order of fall lag adl 1 ladl w l and ladl l respectively the estimates in Theorem 71 now follow from 712 To any cutting sequence77 of times 0 to lt 151 lt 152 lt we associate the sequences of matrices Wn W05 and PW P05 In addition define wo w and T 1374131 A HPntuOH w Aglpnwo 714 for n l2 Let t tn 7 1514 The following is a straightforward consequence of Theorem 71 Corollary 72 70 There are constants 03 06 gt 0 depending only on the Diophantine constants 0 and on the norm ofw such that for all n gt 0 HPnH S C4exp HPilH S asexp HTnH S C5exp HTilH S Caexp 19 176 d716tn d i117 6W d tn 17 6w d tn 715 AAAA An important aspect for renormalization is that the matrices Tn are hyperbolic when ever tn a 00 suf ciently fast as the following lemma implies Lemma 73 70 For all n gt 0 and for all unit vectors 5 that are perpendicular to w 1 HTKH C1026Xp1 6025 derail 716 54 HANS KOCH Proof By de nition T25 P1P5 W1Et WoP15 n 71 t t Wu E lE 1WoPn1 lf 5 is perpendicular to w 1 then 3715 is perpendicular to tag and since the last row of W3 is tag the last component of 5 is zero Thus 5 is an eigenvector of Etn with eigenvalue e tn Since W571 we have T5 ftn W1W15 718 The estimate 716 now follows from Theorem 71 QED A typical choice for the cutting sequence is 15 01 a with c a gt 0 chosen in such a way that the exponential factor in 716 is very small see eg Subsection 82 73 Composing different RG transformations With each of the vectors w 1 and matrices T7 we can now associate a RG transformation 72 for vector elds on Td gtlt R 7200 milT iX 719 Here 7 is a time re normalization factor that can either be xed at the cost of in troducing irrelevant expanding directions or determined by a normalization condition on For simplicity we will choose 7 AgilAn independently of the vector eld X The trivial RG orbit is given by the constant vector elds Zk wk0 which satisfy Z7 7Z7Zn1 for all n gt 0 The first task is to show that these RG transformations can be composed properly To be more precise denote by D7 the domain of R7 De nition 74 Let BO C Do be a xed open neighborhood of Z0 and set 771 R4 For n 12 de ne B7 to be the set of vector elds X E B71 such that X7 7 belongs to D7 and set n1X Rn1Xn Then we de ne the stable manifold for the sequence to be the set W 120 3 In the next subsection we will describe the construction of such a stable manifold in a relatively general setting The hypotheses can be veri ed for the action of on general vector elds on Td gtlt R We only give a rough descriptionmotivation here and refer to 78 for details Skew ows will be covered more thoroughly in Section 8 In order to determine the relative sizes of different quantities it is convenient to split 72 into a linear part 7 and a nonlinear part N7 nX milT iX MAX MWJWS 417331 720 Renormalization of Vector Fields Denote by E the torus averaging operator We expect ll 7 EJXW to decrease very rapidly as a function of 11 As a result terms involving N7 or ll 7 E n should be extremely small for large 11 So the hyperbolicity properties of 727 should be determined essentially by EL 1 1 M17 11 T1E 721 Furthermore we expect A1 to be a negligible factor in the discussion of these properties Recall that Tnqp an MnS p where 5 is either the transposed inverse of Tn if X d and if the class of Hamiltonian vector fields is to be invariant under renormalization or the identity matrix Other choices are possible as well but they seem less natural For the scaling Mn there is no natural choice either unless we wish to impose a nondegeneracy condition on X0 and to preserve it under renormalization In 78 we take M7 7 0 at a rate HTWH V with y suf ciently large depending on the Diophantine constants for tag As a result if X XX with Xqp O pl and X qp O plZ then X is strongly contracted by the operator 721 This can be seen from the fact that 7 X Tn lXoTM151X oT 722 The complementary subspace is spanned by vector fields of the form Xqp u vMy with u a vector in Rd 1 a vector in RI and M a X gtlt matrix The canonical projection onto this unstable subspace will be denoted by ll Although the nonlinearity N7 contributes very little it determines the domain of the RG transformation R7 Assume for simplicity that this domain D7 is a ball of radius p7 centered at ZWA Consider the rescaled RG transformations R7 1271 0 R7 0 V7171 where Vk Zk ka The following summarizes the expected properties of the RG transformations 7 in a way that relates to the assumptions made in the next subsection where 197 replaces very small and an replaces tiny R7 0 ll is linear and its restriction L7 to the range of ll has a small inverse EDan 7 R7 lP Nn and Nnz is tiny compared to l 7 ll7llDLn L7 ll7lP is very small so ll7llDRn is very small compared to ll7llDz ll 7 EQLW Lnll 7 En1 is tiny so ll 7 is tiny compared to l 7 74 An invariant manifold theorem In this subsection we describe an invariant manifold theorem for sequences of maps of the type encountered in renormalization The existence of such a manifold makes the given sequence of maps for many purposes similar to a single dynamical system In particular the construction of invariant tori with general Diophantine rotation vectors is essentially the same as the construction described in Section 5 for self similar rotation vectors see also the comments at the end of this subsection For every integer n 2 0 let X7 be a complex Banach space and let En lan be contin uous linear projections on X satisfying lanlEn Enlan lan and Hll 7 1 For each n gt 0 let R7 be a bounded analytic map from an open neighborhood D71 of the origin in 2674 to X with the following properties Rnlan1 is linear and the 56 HANS KOCH restriction Ln of this linear operator to lan1Xn1 is invertible Furthermore there exist real numbers n S 19 lt l and 8n 3 8 17 194 such that for all x E Dn1 EnRnH S EnW E 1lla 7 S 1971MltH7 Puf lla 7 23 Mill S 19 Consider now the composed maps I n Rn o Rn1 o 0 R1 The domain of E1 is taken to be 50 D0 and for n 12 the domain En of n is defined inductively as the subset of Dn1 that is mapped into Dn by I n We Will assume that the domain Dn1 of Rn is given by conditions Mala Mermaid Huwmxldm 724 Where 6k is a sequence of positive real numbers such that 6k 2 ekdk1 for all k gt 0 Theorem 75 78 Let R1 R2 be a sequence of maps With the properties described above Then W0 120 Dn is the graph of an analytic function W0 llillDOD0 a llDODO satisfying W00 0 For every at 6 W0 HEWH S 190 600M017 Fowl 725 H01 EnRnH S 600W Ewell gt Where 190 19119219n and 57 5152 5n Furthermore if the third condition in 723 is strengthened to HEDWRW 7 annil ll S 5071 7 En71ll25 73926 With gondn1 S 8 then DWO0 0 and HPn nWMl W u P0ll2 727 The proof of this theorem follows the traditional graph transform method For each n 2 0 we consider a space fn of analytic families F bn a Xn Where bn denotes the open unit ball in lanXn Then for n gt 0 and F E fn1 define RnF Rn o F o Ynjg an rnmn o F 728 By construction lanfRn In Where In is the inclusion map of bn into Xn Furthermore RnIn1 In by the second condition in 723 If F0 is any family in the domain of 5R1 define Yn Ynerl and Fn EKnFn1 for n 12 It can be shown that the orbit approaches In at a rate 190 or Renormalization of Vector Fields faster for any F0 in the domain of 5R1 The re parametrization maps Y7 approach L7 at the same rate which allows us to de ne 27 znF0 lim Y l o Yn712 o o Yg10 729 Notice that RnFn1zn1 for all n gt 0 and in particular F020 is in nitely renormalizable The function W0 described in Theorem 75 is now de ned by setting W0ac 205 gt gt x s for every x 6 ll 7 P0D0 Fig 11 The reparametrization maps Y7L More details and an application of this theorem to the construction of invariant d tori for analytic ows on Td gtlt Rf can be found in 78 Following the construction described in Subsection 52 every vector field X on the local stable manifold W at Z0 can be shown to have a analytic invariant torus with frequency vector wo assumed to be Diophantine This translates immediately into a result for m parameter families where m is the codimension of W If the vector fields are restricted to an invariant subclass such as the ones described in subsection 24 then the number of required parameter reduces to the number of unstable directions in this subclass Notice that a trivial reduction by 1 can always be achieved via a time scaling X gt gt TX A further reduction is possible via translations or scaling in R if the analysis is restricted to vector fields satisfying a nondegeneracy assumption In the Hamiltonian case the number of necessary parameters can be reduced to zero in this way as was explained in Subsection 51 58 HANS KOCH 8 Reduction of skew ows Here we apply the framework developed in Section 7 to a concrete case the reducibility problem for near linear skew ows In addition we will describe a way of dealing with vector elds near w A with A constant but not necessarily equal to zero The analysis of the case SL2 R also involves a reduction of the number of unstable directions that is not of the type mentioned above The work described in this section is from reference 79 81 A general result The first result describes a general class of vector fields X w f that are reducible to the trivial vector field w0 See Subsections 25 and 31 for a definition of reducibility and for notation In what follows the constants and O in the Diophantine condition 78 are consid ered xed Define WOW d 1 25 2M1 571d gtl 71 81 Theorem 81 79 Given y 2 72 gt WOW there exists an open neighborhood B of the origin in f and for each Diophantine unit vector w satisfying 78 a manifold W in B such that the following holds W is the graph of an analytic map W ll 7 EB 7 EB which vanishes together with its derivative at the origin and which takes values in A when restricted to A Every function f on W is reducible to zero The corresponding change of coordinates V belongs to 75 and depends analytically on f where 8 y 7 72 If in addition f E A then V belongs to gs and if f is the restriction to Td of an analytic function then so is V Here a function 11 defined on W is said to be analytic if 1 o W is analytic on the domain of W This theorem can also be applied to vector fields Y wg with g is close to a constant matrix A but not necessarily small Assume that A 6 Qt admits a spectral decomposition A H J H1J1 Hng where the J are linearly independent mutually commuting matrices in Qt with the property that t gt gt exptJj has fundamental period 27r The vector H 6 RI will be referred to as the frequency vector of A In order to see how Theorem 81 can be applied to g m A we start with a skew system Y 1119 on T gtlt 5 and then take d m f Consider the associated function fq e T39JgxeT39J 7 H J 1 any E T gtlt Tl 82 Notice that f w 0 If Y is regarded as a vector field on M by identifying w and x with 1110 and x 0 respectively then the above relation between 9 and f can be written as f 99 Way qequotquoty 83 If w w H is Diophantine of type 78 and 9 belongs to the manifold 9W then the ow for X w f can be trivialized with a change of coordinates V as described in Renormalization of Vector Fields Theorem 81 The same now holds for 9 However the corresponding change of variables V9q Vqe T39J is not of the desired form since it still depends on the coordinates 7 But as we will show later 11 Vgac tw 1V9x Vx tw 1etCVx 84 for some matrix C 6 Qt with frequency vector H provided that V is differentiable What remains to be shown in specific cases is that the space of functions of the type 82 has a reasonable intersection with the manifold W This procedure can be characterized as transforming some quasiperiodic motion on 5 into motion on an extended torus This makes it possible to treat all frequencies of the system in a unified way In the case 5 SL2R which will be discussed in Subsection 84 it also has the advantage that the analysis of near constant skew ows Y w 9 can be reduced to a purely local analysis near f E 0 82 The stable manifold With each of the vectors w 1 and matrices T7 obtained in Subsection 72 we can associate a RG transformation for skew ows Nnf nngg Zlf where 7 AgilAn Let us reformulate Theorem 33 for each of these RG transformations The condition 34 reads T71 QUnHTnH lt Tn HTn H S 3W 5 6 win 85 and Theorem 33 becomes Theorem 82 Assume that 01 and 71 satisfy 85 Let f O h with O constant and Eh 0 If lt In6 and lt 2 90 then MM n10 5 Illel S WW Efz g 24U7Y173HhH2 86 N7 is analytic on the region determined by the given bounds on O and h The goal now is to compose these RG transformations The domain 3 of N7 is defined to be the ball of radius 2 90 in A centered at zero Ignoring the constant component C of f the range of N7 is by a factor lt 2717 smaller than its domain By multiplying up these factors we get the following condition for the composability of the transformations N7 A1 H 4771 01 3 an 87 j1 Lemma 83 Given 71 gt 705 there eXist two sequences 71 gt gt 01 and n gt gt 72 ofpositive real numbers less than one both converging to zero such that the following holds If tag is a unit vector in R satisfying the Diophantine condition 78 then there eXists a sequence 60 HANS KOCH n gt gt P7 of unimodular integer matrices such that with T7 and A7 as de ned in 714 the conditions 85 and 87 hold for n 12 It is straightforward to verify these estimates by using the results of 70 described in Subsection 72 with a cutting sequence of the form 15 01 or and with the choice 01 Joe dtf 71 7390 exp7d 71W 1t d tn 88 For the factor A1 in 87 it suf ces to use the trivial bound A1 3 HP1 Consider now the composed RG transformations N7 N7 0 0 N1 with domains defined as usual Let r0 2 1101 By Theorem 82 and Lemma 83 the question whether the function fn1 NWAUO belongs to the domain of N7 only depends on its average 3an To be more precise given an open set B C B1 containing zero define B0 B and E f e 1Ef1ult rm 7 Agl4 7rxlr0 for n 12 Then B7 is contained in the domain of A77 Theorem 84 79 If y gt 71 then there eXists a non empty open neighborhood B of the origin in f such that W 120 B7 is the graph of an analytic function W ll 7 EB a EB Both W and its derivative vanish at the origin This is now essentially a corollary of Theorem 75 if we set Rnf 7 an7 1 The projections used are are simply lan E7 E and the parameters are 6 19 r and 6 2 83 Conjugacy to a linear ow In this subsection we outline a proof for the remaining parts of Theorem 81 concerning the reducibility of vector fields on thelnanifold W Consider f0 6 W and let f7 N71ltf0 In order to simplify notation the transfor mation Hf de ned in Subsection 32 and the ow wmfn will be denoted by U7 and 11 respectively For m gt n 2 0 define meltqgt Um1Ti1 T111q U1T11qUnltqgt 89 Then the ows for fm and f7 are related by the equation MM meltq twn 1 1 nm quot tT1 TJ11qlemlt 810 We would like to show that the three terms on the right hand side of this equation each have limits as m a 00 with the 1 term in the middle converging to the identity This would yield the desired conjugacy Due to the matrices T1 involved these limits will have to be considered in go using that Hf o Tn lllo WHO 3 The time t map 11 associated with a vector field X w f can be estimated as usual by applying the contraction mapping principle to the corresponding integral equation and Renormalization of Vector Fields then using the group property The resulting bound is 7 111 g glltfllv 7 1 And for changes of variables we can use the bound 1R0Ailef exp30 H14Evaei1 811 which is obtained from the results in Subsection 32 and holds for f in the domain of N These estimates can now be combined with the bound 07711123117 E TWKVIIIlefon 812 obtained from Theorem 82 and Lemma 83 to prove the following Lemma 85 If f0 6 W then the limit V7 limmmoo Vm exist in go and satis es HWSW W WMM em Furthermore the maps f0 gt gt V7 are analytic and 1 mm tun1v q 256 R 814 This lemma will now be applied for y 72 with 72 gt 71 fixed If f has some excess regularity that is if f belongs to A with y gt 72 then this excess regularity is inherited by V0 The mechanism is analogous to the one described in Subsection 51 To be more specific consider the torus translations jpf 1 fqp It is straightforward to check that the manifold W is invariant under 7 and that the map f0 gt gt V0 commutes with jp Denote by H the map that associates to each f in the domain of W via f0 f 1 Wf the value V00 Then VoltPgt HUN 19 6 Ti 815 Given that H is analytic this identity links the regularity of V0 to the regularity Ofp gt gt jpf as a family in 772 More precisely we have Lemma 86 Let y 2 w gt 71 and 8 y 7 72 If f0 6 W then the function V0 described in Lemma 85 belongs to gs and has a directional derivative DWOVO in 75 As elements of f5 both V0 and DWOVO depend analytically on f0 Furthermore if f0 is the restriction to 11d of an analytic function then so is V0 Lemma 85 and Lemma 86 imply the statements in Theorem 81 concerning the re ducibility of functions f E W and the regularity of the corresponding change of coordinates The fact that V takes values in 5 if 5 is a proper Lie subgroup of GLnC or a Lie subgroup of GLn R whenever f takes values in the corresponding algebra 21 follows by construction see also the remark at the end of Subsection 32 As was described in Subsection 81 Theorem 81 can also be applied to vector fields Y 1119 with 9 close to a nonzero constant matrix A H J However a direct appli cation reduces Y as a skew system over 11d instead of 11 quot which involves extra frequencies The following result shows that these frequencies can be factored out 62 HANS KOCH Lemma 87 Let y 2 72 1 with 72 gt 71 Assume that f E A is of the form 82 and belongs to W Let V V0 corresponding to f0 f Then the ow for Y wg is given by equation 84 for some 0 6 2L The corresponding map f gt gt C is analytic Proof The first equality in 84 follows from Lemma 85 and the de nition 83 Let tac Vx MINI rVac 1 Vx twetAVac tw 1 816 fort E R and x E 11 quot Notice that t is the ow for a skew system Z w h on 11 gtlt 5 and since V E 91 by Lemma 86 the function h belongs to A0 Consider now an arbitrary sequence 15 such that tan a 0 on the torus Tl asj a 00 Then expth J a 1 Furthermore disttjwtjw a 0 on the torus Td and since V is of class C1 we have ttj a tac uniformly in at if t 0 By the cocycle identity for the ow t the same holds for any If E R and the convergence is uniform in t This implies see eg 53 that the function 25 gt gt tac is periodic or quasiperiodic with frequencies in FH 51 m As a result he nu tltmgt tltzgt4 817 is also periodic or quasiperiodic in t with frequencies in But the frequency module oft gt gt hx tw is clearly a subset of Fw 1111 wm and since Fw FH is empty h has to be constant Setting 0 h we obtain tx etc and the identity 84 now follows from 816 A computation of from the equations 817 and 816 yields 0 VAV l 7 DEVV 1 evaluated at x This identity between matrices if x is fixed together with Lemma 86 shows that 0 depends analytically on f QED 84 The special case Q SL2R Consider now the group 5 SL2R and the corresponding Lie algebra Qt of traceless 2 gtlt 2 matrices In this case Theorem 81 yields a stable manifold of codimension 3 On the other hand it is known at least in the analytic setting that reducibility is a codimension 1 phenomenon governed by the so called bered rotation number The goal here is to describe how this fits in the framework of renormalization Consider first the ow for Y w 9 on the product of Td l with R2 0 it 9100 twvt 110 110 818 Denote by at the angle between vt and some xed unit vector no and let a0 040 Then the lift of this angle to R evolves according to the equation 0215 7e atJJgx0 tweat u0 uo 040 040 819 where denotes the standard inner product on R2 Here and in the remaining part of this section J 7 Assuming that the components of w are rationally independent we can define the fibered rotation number of Y gY lim E 820 Renormalization of Vector Fields This limit exists and is independent of the initial conditions 0 and a0 66 If w is fixed we will also write 99 in place of 9Y From the definition of G we see that 9Y H if and only if gX 0 Thus we may restrict our analysis to skew ows with fibered rotation number zero Theorem 81 deals with precisely such ows However the functions 82 are of a particular type and more can be said in this case Denote by A9 the subspace of functions 9 E A with the property that gq 910 for all 1 zr in Td l gtlt T1 Theorem 88 79 Given y 2 72 gt WOW and a gt 0 the following holds for some R gt 0 Consider a constant skew system w A on Td l gtlt 5 for a matrix A 6 Qt that has purely imaginary eigenvalues say iHi Assume that w w H satis es the Diophantine condition 78 and that S alHleH Then there exists an open neighborhood BO of the constant function x gt gt A in A9 containing a ball of radius R centered at this function such that for any 9 6 BO the one parameter family A gt gt g AA contains a unique member in BO say 9 whose associated skew ow has a bered rotation number H If y 7 72 8 2 1 then 9 is reducible to a constant C 6 Qt as described by equation 84 via a change of coordinates V E gs Furthermore the function 9 and ifs 2 l the quantities O and V depend real analytically on g This theorem is proved by first performing a change of coordinates g gt gt L lgL with L E 5 such that L lAL HJ followed by a constant scaling Y gt gt CY of the resulting skew system which converts wH to a unit vector After that the task is reduced via the map 6 to the study of vector fields X w f with f of the type 82 Thus in view of Theorem 81 and Lemma 87 it suf ces to prove besides real analyticity that the family A gt gt f AJ intersects the manifold M in exactly one point characterized by gf AJ 0 A sketch of the proof will be given at the end of the next subsection The main dif culty with this approach is that the subspace A1 of functions f E A of the form 82 is not invariant under renormalization Notice that this subspace can also be characterized by the identity fq 07 e fqe Thus that the torus average of f E A1 is necessarily a constant multiple of J This may seem to explain the statement about one parameter families in Theorem 88 However this property is neither invariant under renormalization nor does is guarantee that the ow for X w f remains bounded 85 Excluding hyperbolicity The three unstable directions under renormalization correspond to elliptic parabolic and hyperbolic behavior of the ow The goal is to show that a vector field X w f with f E A1 close to zero can only generate an elliptic ow by excluding eg the possibility that the renormalized functions f7 Nnf0 have the following property De nition 89 Let 51 be the set of unit vectors in R2 We say that a vector eld X wf has the ezrpandz39ng cone property if for every 1 E Td there exists an open cone Cq in R2 not intersecting its negative with vertex at zero and a unit vector uq in this cone such that the following holds The map q gt gt 51 Cq de nes two continuous functions from Td to 1 The function q gt gt uq is continuous as well and homotopic to a 64 HANS KOCH constant Furthermore for every 1 E Td the cone IJ qCq is contained in Cq tw for all t gt 0 and the length of tends to in nity as t a 00 The usefulness of the expanding cone property stems from the fact that it is invariant under coordinate changes of the form 32 or 226 with V continuous and homotopic to the identity A simple condition that implies this property is the following Proposition 810 Assume that f Td a Qt is continuous and of the form f O h with O 6 Qt symmetric and lt for all 1 E Td Then X wf has the expanding cone property This proposition is proved first for h 0 where it is trivial and then a perturbative argument is used for lt Lemma 811 If f belongs to A1 then X wf cannot have the expanding cone property Proof Consider first an arbitrary f E A such that X wf has the expanding cone property Let 1 E Td be fixed Using the notation of Definition 89 denote by Aq the set of all nonzero UO 6 R2 such that vt IJqvo belongs to Cq tw for some and thus each suf ciently large positive 15 This set is clearly open Notice that if UO is any nonzero vector in R2 with the property that vt IJ qv0 tends to infinity as t a 00 then UO belongs to either Aq or 7Aq This follows from the fact that is area preserving so the angle between vt and has to approach zero and that the opening angles of our cones are bounded away from zero Thus given that the two disjoint open sets iAq cannot cover all of R2 0 it is not possible that lvtl a 00 as t a 00 for every nonzero v0 6 R2 Assume now for contradiction that f belongs to A1 Define zrz e uq with u as described in Definition 89 Then 11 6Tt 7 lquq tends to infinity as t a 00 But as 7 increases from 0 to 27r the vectors 2Tz cover all of 51 since u is homotopic to a constant function This implies that 11 zvo tends to infinity in length for each nonzero UO 6 R2 which was shown above to be impossible QED Now we are ready to renormalize and to continue our sketch of the proof of Theo rem 88 Denote by 3 the one dimensional subspace of Qt consisting of real multiples of the matrix J The idea is to consider the family Fs s l 7 Ef associated with a function f E A1 close to zero where s E A and to show that the parameter value 5 20 where this family intersects the stable manifold W belongs to 3 This is proved by contradiction If the intersection takes place at a point 5 outside 3 then by renormalizing the family F as in the proof of the stable manifold theorem see subsection 74 we can find 11 gt 0 and so 6 3 such that Fns nFso is much closer to its average value 5 than this value is to 3 By using that the re parametrization maps Y7 for skew ows are very close to multiples of the identity we can in fact choose so in such a way that 5 is symmetric Then F7 37 has the expanding cone property The same is true for Fso since this property is preserved under coordinate changes of the form 32 or 226 But Fso belongs to A1 Renormalization of Vector Fields 65 and we get a contradiction with Lemma 811 This shows that 20 E 3 And Lemma 85 implies that gF20 0 Finally assume for contradiction that gFsO 0 for some so 6 3 different from 20 By renormaliZing F we can achieve 8 7 gt 22 which implies using again that the maps Y are close to multiples of the identity that Fs is close enough to an antisymmetric matrix to have a strictly positive determinant But then F 3 cannot have a vanishing fibered rotation number and the same is true for Fso This shows that the family f gt gt f AJ intersects W at a unique point characterized by gf AJ 0 For details we refer to the proof of Lemma 75 in 79 Acknowledgments This review grew out of lecture notes for a mini course given at the Fields Institute in Toronto during the Fall 2006 thematic program on Renormalization and Universality in Mathematics and Mathematical Physics The author would like to thank the organizers of this program especially Mikhail Lyubich and Michael Yampolsky as well as the Fields Institute and its staff for their generous support and hospitality References 1 JJ Abad H Koch Renormalization and periodic orbits for Hamiltonian flows Commun Math Phys 212 3717394 2000 JJ Abad H Koch and P Wittwer A Renormalization Group for Hamiltonians Numerical Results Nonlinearity 11 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121671228 1987 r Sullivan Bounds quadratic quotft 39 39 and renor quot 39 J Publications II Mathematics into Twenty rst Century 4177466 1992 WI Thirring A Course in Mathematical Physics I Classical Dynamical Systems SpringerVerlag Berlin New York Wien 1978 Sr Tompaidis Approximation of Invariant Surfaces by Periodic Orbits in HighDimensional Maps some Rigorous Results Experimental Math 5 1977209 1996 J Wilbrink New Fixed Point of the Renormalisation Operator Associated with the Recurrence of Invariant Circles in Generic Hamiltonian Maps Nonlinearity 3 5677584 1990 Mr Yampolsky Complex bounds for renormalization of critical circle maps Ergodic Theorr Dynr Systr 19 2277257 1999 Mr Yampolsky Hyperbolicity of renormalization of critical circle maps Publr Mathr lnstr Hautes Etudes Sci 96 1741 2002 Mr Yampolsky Global renormalization horseshoe for critical cirole maps Commun Math Phys 240 75796 2003 M On the 39 39 o In Proc ln AMS Centennial quot operator Nonlinearity 16 156571571 2003 l r r fa mum quot References 18 19 20 21 22 33 36 37 38 43 42 45 44 49 50 63 92 91 90 98 99 105 109 119120 m References 3 4 24 25 23 33 31 30 29 28 35 46 52 63 67 68 69 93 98 101 100 103 104 105106 108112 114117123124126 127125 m A Property of the Identity Function An Exercise in Induction ayadev Misra 91699 Let f be a function from naturals to naturalsi It is given that Property P1 Vn lt fn Prove that f is the identity function We will actually prove that the same result holds under the more general condition given be ow Property Q2 Vn 3i i 2 2 z lt fn l have heard that this problem appeared in a mathematical olympiadi The problem was shown to me y van de Snepscheut on 121389 who received it from Richard Birdi This is a belated recording of my response to van de Snepscheut though the generalization is newt Henceforth all variables are naturals Lemma 1 f is increasing iiei Vn n S Proofzz There seems to be no direct proof of this result by induction We will show instead that R1 Wm Pun where Rin is Vt n S The de sired result follows by setting t to 0 in each Rini The proof of R is by induction on n R10 Vt 0 S This follows because f is a function from naturals to naturalsi Rin R1 n 1 We prove n 1 S 1 t for arbitrary t assuming that Rin holds true Induction hypothesis V31 nS fns n S fnt t 7 n 2 0 from above Set 8 to t 7 n in the rst term nS fnfnt in Rewriting n S f2n t From property P f2n t S 1t n lt 1 t arithmetic n1Sfn1t D Lemma 2 f is monotone iiei Vm t m S n S Proofzz true Set n to in Lemma 1 S f 201 From property P f2 lt 1 f n lt f 71 1 Induction on naturals D Corollary lt n lt m7 by taking contrapositive of Lemma 2 Theorem 1 f is the identity function7 ie7 n7 for all n roo true Property P fwm ltfmw Corollary of Lemma 2 lt n 1 Lemma 1 nS n S lt n 1 Arithmetic n f n A Generalization We show that if property Q Vn 3i i 2 2 lt fn holds then f is an identity function Note that ifi 0 for all n then the property is a tautology7 n lt n 1 For i l the conclusion is incorrect the successor function satis es the property Lemma 3 f is increasing7 ie7 Vn n S Proof Let S Vn Sn Where Sn Vt n S fn We prove S is by induction on n S0 Vt 0 S Follows trivially Sn Sn 1 By induction hypothesis assume that A Vs n S fmgt Claim For all natural lat we have n S fk n t Proof is by induction on k 0 n S n t Follows trivially k 1 true Assumption A n S 3 lnduction hypothesis n S t Set 8 to fknt 7 n note 8 2 0 n S fn fknt 7 n arithmetic n S fk1n t D Now we show that n 1 S 1 t7 for any t For the given mt let j be such that fj n t lt 1 t such a j exists from Property true Claim above n S fj n t given that t lt fn1t n lt 1 t arithmetic n1Sfn1t D it it it Corollary For any natural 16 n7 we have n S fk Proof is by induction on k Lemma 4 f is monotone ie7 m S n S Proof Let m be an arbitrary natural Let i be such that lt 1 such an 239 exists from Property true Let n7 k 71 in Corollary to Lemma 3 NoteiZQ kZO f0 3 fl 1fm Given lt fm1 fm lt fm1 The result follows by induction on natural numbers D Corollary 1 lt n lt m Corollary 2 For any 16 k 2 07 and all rmn7 we have fk lt fk n lt m Proof is by induction on In Theorem 2 n7 for all n Pick an arbitrary n and let lt 1 true assumption 1 lt fn1 1n corollary to Lemma 3 let n7 k1 1i7 2 NoteiZZ kZO I Mn lt fn 1 A fn 1 S fl 2fn 1 arithmetic I 1 lt fl 2fn 1 Rewrite above I 710600 lt 161401 1 Corollary 2 of Lemma 4 with k7n7mi717 n 1 High Performance Control of an Intelligent Wheelchair Using Visual Information Shilpa Gulati 382007 Motivation Safe navigation Without constant control from driver Reduce stress Increase mobility Gracefully perform tasks such as Pass through a door Go up and down ramps Turn sharp corners Problem De nition Pose is Location Orientation Start from a pose such that Location Within circle of radius R from door Both door edges clearly Visible Move gracefully and passthroughdoor Stop on other side of door Approximately at distance R from door Facing away from door PassThroughDoor V One possible Velocity pro le Graceful Motion Velocity and acceleration are within comfortable limits Change smoothly No backingup maneuvers Passenger always facing forward in direction of motion Sufficient clearance from door edges and walls for passenger comfort No collisions One Approach Motion Planning then Feedback Control Most common approach Plan desired path with motion planner open Zoop control Make robot follow desired path with feedback controller closed loop control Motion Planner 1 Optimal Control 2 Vector Fields 3 Many more Desired PathTraj ectory y qd Error Actual Desired Feedback Controller Kinematics and Dynamics of Wheelchair Control I A Commands uV Let s Jump into Feedback Control Now Kinematic Model of Wheelchair l i voosH y vsinH 0 w Nonlinear System Unicycle Model gtltv v is linear velocity along CTP w is angular velocity about C Nonholonomic Constraints if sin6 cos6 0 y 0 9 Constraint among velocities that cannot be integrated Physically means that Wheelchair cannot move sideways Reduces feasible paths makes motion planning dif cult 2 inputsvw 3 outputs x39 y39 9 Controllability and Stabilizability Does there exist piecewise smooth input that takes the system from a point to another in its neighbourhood Yes 7 Can the system be exponentially stabilized Cannot be stabilized to a pose with smooth time invarying feedback Can be stabilized to a trajectory Good for us Types of Control Problems Path Following Follow a cartesian path Linear velocity v assumed constant Only parameter to be controlled 8 Use a to control 8 Easy Path Following Path Following Control Law Follow a Wall Follow wall so that y yd constant e 2 y ya We want the system to behave like a damped spring Damped Spring gt kl 1626 0 System converges tOyZJd vcos66k1vsin6k2620 792 w k1tan9 m6 Types of Control Problems Trajectory Tracking Follow a time varying trajectog Y L10 x ly l 6dtT Two parameters to be controlled explicitly ext eye e30 controlled implicitly due to nonholonomic constraints X0 ya Two control inputs Possible Trajectory Tracking Two Maj or Approaches for Trajectory Tracking Feedback Linearization Novel et al 1 Oriolo et al 8 De Luca et al 6 and many more Sliding Mode Control Yang and Kim 9 Is path following sufficient Why should we look at trajectory tracking We want the wheelchair to start from rest move and then come to rest again Velocity is time varying Just following a path may not be enough Or maybe it will be Let s look at trajectory tracking Trajectory Tracking Feedback Linearization I Exact linearization based on state transformations Different from approximate linearization around an operatlng p01nt We want to control Z 1101 X3 yT We want find a relationship between input u u coT and output z x yT Differentiate output to get i E6u Trajectory Tracking Feedback Linearization II We got i E6u If E6 is invertible we can nd new inputs v 121 v2 T such that u E611 So we get one input per output 21 V1 Now write control laws lt gt lt gt for these two decoupled z 1 2 2 linear systems Trajectory Tracking Sliding ControlI Convert problem of tracking qdt to that of staying on a surface st for all t gt O s 0 is solution of differential equation with qt 1010 Design surface so all trajectories point towards it Once on surface trajectory stays on surface Control Law Change u based on which side of the surface system is on Sliding Surface Which is better Feedback Linearization or Sliding Control I Sliding Control suffers from chatter System is constantly pushed from one side to the other Can overcome this by specifying a boundarylayer around surface to get smooth control laws M ny other methods to remove chatter Boundary Layer Sliding Surface Which is better Feedback Linearization or Sliding Control II Sliding Control does not need exact knowledge of system parameters Qualitative description in the form of bounded functions is sufficient Let control estimate based on qualitative model be 72 Design control u to be u a signs signs 1 8 gt 0 signs 1 s lt 0 Feedback linearization lets us use a familiar linear control method Can we do without motion planning for specialized tasks I Possibly Alter system dynamics using External forces Control inputs So that system naturally follows the trajectories we want Can we do without motion planning for specialized tasks Divide into regions separated by sliding surfaces Different control law in each region Blend the control laws for smoothness Determine sliding surfaces by gradient lines in a vector field Still guring out the details Suggestions Ideas REGION 2 REGION 3 REGION 3 I I l REGION 1 Now let s talk about using Visual information for control We need to estimate error 6 between actual and desired trajectory Use sensory information Vision is good depth color texture Challenge Identify landmarks that are clearly Visible despite motion Will a xed camera do Or will we need to actively track landmarks with a moving camera Vision Based Control PoseBased approach Convert image data to robot s state space to compute control inputs Eberst et al 2 door navigation ImageBased approach Compute control inputs directly from sensor inputs Ma Kosecka and Sastry 7 curve properties Patel et al 10 multiple sensors for door navigation So What neWI To the best of my knowledge Most research has focused on Tracking specific trajectories such as curves Without regard for the extent of the robot and comfort of passenger Vision based navigation navigating safely but not necessarily comfortably No one has used information from a body mounted camera alone for high performance feedback control So What s new 11 Results in all the research show either of these Backup maneuvers chattering spikes in angular velocity Will not do for our application Research Plan On a straight line trajectory Evaluate feedback linearization and sliding mode control Assume point mass and perfect odometry Eliminate backup maneuvers Use visual information Assume noiseless sensors Introduce noise and uncertainty Evaluate controller on more complex paths References 1 B dAndrea Novel G Campion and G Bastin Control of nonholonomic wheeled mobile robots by state feedback linearization The International Journal of Robotics Research 146543 559 1995 2 C Eberst M Andersson and H Christensen Visionbased doortraversal for autonomous mobile robots Proc IEEERSJ International Conference on Intelligent Robots and Systems 1620 625 2000 3 R Fierro and F L Lewis Control of a nonholonomic mobile robot Backstepping kinematics into dynamics Journal of Robotic Systems 143149 163 1997 4 B Kuipers Control tutorial 2004 5 JP Laumond S Sekhayat and F LamirauX Robot Motion Planning and Control chapter Guidelines in Nonholonomic Motion Planning for Mobile Robots pages 1 54 SpringerVerlag1998 6 A De Luca and M D Di Benedetto Control of nonholonomic systems Via dynamic compensation Kybernetica 29593 605 1993 References 7 Y Ma Kosecka J and S Sastry Visionbased doortraversal for autonomous mobile robots Proc IEEERSI International Conference on Intelligent Robots and Systems 1620 625 2000 8 G Oriolo A De Luca and M Vendittelli WMR control Via dynamic feedback linearization Design implementation and experimental validation IEEE Trans on Control Systems Technology 106835 852 2002 9 IM Yang and J H Kim Feedback control of a nonholonomic wheeled cart in cartesian space Proc IEEE International Conference on Robotics and Automation 15578 587 1999 10 S Patel S H Jung J P Ostrowski R Rao and C J Taylor Sensor based door naVigation for a nonholonomic vehicle Proc IEEE International Conference on Robotics and Automation 33081 3086 2002 A Consensus Protocol in a Prison ayadev Misra 222004 The following problem was posed at a recent conference by Leslie Lamport who had heard that it was posed in a popular radio program Car Talk in the National Public Radio in US I solved a special case of this problem assuming a certain initial state Anup Rao a student in my graduate class solved the general case in an hour Problem A set of prisoners iassume there are at least 27 are asked to play the following game by the warden There is a room in the prison which has two switches initially the switches are in arbitrary positions The warden will bring one prisoner at a time to the room and the prisoner must ip one of the switches The prisoners do not know the order in which they will be taken to the room but they know that every prisoner will visit the room over and over until the end of the game The game ends when some prisoner announces every prisoner has been in this room at least once h If the announcement is correct all prisoners go free if incorrect they all are executed The game continues until the announcement is made The prisoners are allowed to confer and decide on a protocol prior to the start of the game Once the game starts they are not allowed to communicate nor can they nd out who is being taken to the rooml The problem is to devise a protocol for the prisoners Solution special case In the original formulation there are two switches A and Bi Simplify the problem by assuming that there is a single switch 5 which a prisoner may or may not ipl This is a special case of the original formulation whenever S is ipped ip A and whenever S is not ipped ip Bi Henceforth flip denotes ipping switch 5 Let variable 8 denote the state of the switch 0 for off and l for on so flip changes the value of 8 The prisoners choose a leader who will make the announcement all other prosoners are followers The number of followers N is at least 1 Each follower will vote by ipping the switch appropriately and the leader will count the number of votes The im follower has a variable vi which is 1 if he has voted and 0 otherwise Clearly vi 1 implies that the ith follower has visited the rooml The leader has a variable 5 for maintaining the count lnitially no follower has voted and c is 0 Assume for the moment that the switch is initially off so 8 0 The protocol is described by the steps taken by the leader and the followers initially for all followers vi 0 c 0 s 0 Follower if v 0 not voted and s 0 switch is off then flip s 1 vi vil Leader if s 1 switch is on then flip s 0 c 01 if c N then Announce The proof obligations are 0 Safety Whenever the leader announces all vis are l 0 Progress Eventually the leader announces Proof of Safety Let 1 denote the sum of all the vis Then 0 S v S N It is easy to see that 0 8 v is an invariant it holds initially and after each step by a follower or the leader Since 5 N s 0 is a precondition of Announce we conclude from 08 1 that v N holds prior to Announce Hence all vis are 1 Proof of Progress We show that 5 increases eventually if it is below N There fore eventually c Since 5 never decreases c will remain true and the leader will Announce lf 5 lt N s 1 then on its next visit to the room the leader will increase c lf 5 lt N s 0 some follower has not voted yet so 8 will become 1 either by the voting of this or some other follower Then 5 lt N s 1 holds and from the previous argument 5 will be increased Solution general case We have assumed that initially s 0 Now we remove this assumption Then we have the invariant c s S v 1 It is no longer true that c N 1 N 1 could be N 1 However 5 can be off by no more than 1 from v This discrepancy can be handled if every follower votes twice Now the precondition of voting is changed to vi lt 2 and of Announce to c 2N Proof of Safety A precondition of Announce is c 2N s 0 From the in variant cs S vl we get 2N S vl as a precondition of Announce We claim that then every prisoner has visited the room at least once because if Nl pris oners have visited the room and each has voted twice 1 S 2N2 or vl lt 2N Proof of Progress Similar to the previous case

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