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# Introduction to Sociology SOC 100

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Date Created: 09/19/15

NOTES ON THE DYNAMICS OF LINEAR OPERATORS JOEL Hi SHAPIRO ABSTRACTi These are notes for a series of lectures that illustrate how continuous linear transformations of in nite dimensional topological vector spaces can have interesting dy namical properties the study of which forges new links between the theories of dynamical systems linear operators and analytic functions 1 INTRODUCTION TO TRANSITIVITY Our story begins in a separable complete metric space X on which acts a not always continuous mapping T We are interested in the behavior of the sequence I7 T7 T27 T37quot of iterates of T where T denotes the composition of T with itself n times and we will be particularly interested in knowing when there exists a point z E X for which the T orbit orb T a orb zTxT2x of z under T is dense in X When this happens we call T topologically transitive and refer to x as a transitive point for T Before long we will be considering T to be a linear transformation on a metrizable topological vector space in which case the word hypercyclid7 will be used instead of transitive more on the reason for this terminology shift later Note that if z is a transitive point for T then the same is true of every point in orb T x so once there is one transitive point there is a dense set of them Even for simple metric spaces like the unit interval or the unit circle the study of transitive maps can be fascinating We begin with some elementary examples that illustrate this point 11 Irrational translation modulo one The metric space here is the closed unit interval I 01 Fix an irrational real number 04 and de ne and the mapping T I a I by Ta z 04 mod 1 That is Ta is whats left when you subtract off the integer part of z a Now T is not a continuous map of I but if this causes you problems then identify Date May 27 2001 2 JOEL H SHAPIRO the endpoints 0 and 1 to make I into a circle and regard T to be rotation of the this circle through the angel 27m a continuous map More precisely let e27 which maps I continuously onto the unit circle T Let R T 7 T be the mapping of rotation through the angle 27m ie T2 e2quotmz Then R is a continuous mapping of T onto itself and E o T R o E which because E is almost one to one77 exhibits the action of T on I as being essentially the same as77 that of R on T In particular to show that T is transitive it7s enough to do this for R In fact well do more 12 Proposition Every orbit ofR hence every orbit of T is dense Proof Let u ezmu and note that the irrationality of Oz guarantees that v is not a root of unity Thus orbR1 1 v if is an in nite subset of T and therefore because T is compact has a limit point in T In particular there is a strictly increasing sequence of positive integers such that 0 lilrnlwnk1 7 WM lilrn lwnktl m 71 lilrn anktl M 7 1 hence 1 is a limit point of its own R orbit Thus given 8 gt 0 there is a point C E orb R 1 such that the arc between 1 and C has length lt 8 The successive powers of C therefore partition the unit circle into non overlapping arcs of length lt 8 hence every point of the circle lies within 8 of one of these powers all of which belong to orb R 1 Thus orb R 1 is dense in T establishing the transitivity of B To see that every R orbit is dense it7s enough to notice that for C E T Orb R7 C 39 orbR717 and to observe that the map 2 7 C2 is a homeomorphism of T which therefore takes dense sets to dense sets and in particular transfers the density of orb R 1 to orb R C I Why transitivity Let7s return to the general situation where X is a metric space on which acts a map T and lets say that a subset E of X is T mvarz39arzt if TE C E1 For x E X the closure of orb T y is the smallest closed T invariant subset of X that contains x Thus for maps like irrational translation mod one or equivalently rotation of the unit circle through an 1 Warning This terminology is standard in operator theory but not universal LINEAR DYNAMICS 3 angle that is an irrational multiple of 7139 density of every orbit is equivalent to the fact that the only closed invariant sets are the empty set and the whole space Such maps are called minimal The more modest concept of transitivity merely asserts that open invariant sets are either empty or dense Perhaps its time to give an example of a transitive mapping that is not minimal 13 The baker map This is the map E I a I de ned by Bz 2x mod 1 The name comes from a strategy for kneading dough take a strip of unit length roll it out to double the length cut it in half translate the right half on top of the left half knead the two halves together into one strip of unit length and repeat the process Note that the baker map is not minimalithe origin is a xed point More generally for each dyadic rational z 12 q and 71 positive integers with q S 2 Bmz 0 for all m 2 n so there is a dense set of points with nite orbits Nevertheless 14 Proposition The baker map is transitive Proof Represent each point z of I by a binary expansion ie z 0102 a 0 or 1 means that z 221 ak2k Binary rationals 12 have two such expansions one that is nitely nonzero and the other that is not For these choose the nite one The binary expansion of every other point of I is unique For x represented as above Bz 11203 ie B performs a backward shift on binary expansions Enumerate the countable dense set of binary rationals in I as b1b2 Form the point z E I as follows begin with the nite recall binary expansion of b1 follow it with a zero after which you copy in the expansion of b1 again then two zeros then the expansion of b2 then three zeros then back to b1 four zeros b2 ve zeros b3 six zeros etc The idea is to get the nite expansion of each binary rational copied into that of z in nitely often each time followed by successively more zeros This produces a point z E I with each binary rational a limit point of orb B x hence z is a transitive point for B I Characterizing transitiVity Suppose the mapping T of the separable metric space X is transitive Let trans T denote the necessarily dense set of transitive points of T To better describe trans T x a countable basis B7 of open sets for the topology of X For 4 JOEL H SHAPIRO example this could be the collection of open balls of rational radius with centers in some xed countable dense subset of X Then its easy to check that 1 trans T U T B j n Since trans T is dense in X so is the union on the right for each positive integer j and this means that for every basic open set B and every nonempty open set V there is a non negative integer n such that T Bj V 31 0 or equivalently Bj T V 31 0 Now every nonempty open subset of X contains a B so we have just proved 15 Proposition IfT is a transitive map of a necessarily separable metric space X then for every pair UV of nonempty open subsets ofX there is a non negative integer n such that T U V 31 0 or equivalently U T V 31 0 Corollary If a map is transitive then the orbit of every nonvoid open set is dense In case T is a continuous transitive map on X the description 1 shows that the set of transitive points ofT is a dense G5 Suppose in addition that X is complete then Baire7s Theorem asserts that the intersection of every countable collection of dense open sets is again dense so every countable intersection of dense G57s is another dense G5 In summary 16 Proposition Every transitive continuous mapping of a complete metric space X has a dense G5 set of transitive points Every countable collection of such maps on X has a dense G5 set of common transitive points If a complete metric space has no isolated points then every dense G5 set is uncountable see 32 513 page 103 for example so continuous mappings of such spaces are either non transitive or have an uncountable dense set of transitive points It might appear that these remarks do not apply to the baker map which is not continuous on I but by the same trick used for irrational translation mod one we also can think of B as acting on the unit circle lndeed letting E I a T denote the exponential map of 11 e we have E o B E2 which allows the transitivity of B to be transferred to the squaring map77 2 a 22 of T Since the squaring map is continuous it has a dense G5 and therefore uncountable set of transitive points and from this its easy to see that this property transfers back to the non continuous baker map LINEAR DYNAMICS 5 We close this section with one more consequence of the characterization 1 of the set of transitive points where now T is continuous and the separable metric space X is complete Suppose that such a T obeys the conclusion of Proposition 1 above ie that for each pair U V of nonempty open subsets of U there is a non negative integer n such that T U V 31 0 Then the union on the right hand side of 1 is a G5 that intersects every nonempty open set ie it is a dense G5 The set of transitive points of T is therefore a countable intersection of dense G57s so it is nonempty by Baire7s Theorem 32 56 page 97 Thus for continuous maps of complete separable metric spaces the converse of of Proposition 1 holds These observations form the content of 17 Birkhoffls Transitivity Theorem A continuous map T of a complete separable metric space X is transitive if and only iffor every pair U V of nonempty open subsets ofX there is a non negative integern such that T U V 31 Q or equivalently U T V 31 9 Corollary A homeomorphism of a complete separable metric space onto itself is transitive if and only if its inverse is transitive So far our only invertible example is translation modulo one of the unit interval by an irrational number In this case transitivity of the inverse is clear it is just translation modulo one by the negative of that irrational number Later we will encounter many other interesting examples of invertible transitive maps Transference of transitivity We have used the exponential map E z a e27 to connect the behavior of some discontin uous maps T ofthe unit interval irrational translation modulo one and the baker map with continuous maps R of the unit circle irrational rotation and the squaring map respectively via the intertwining equation77 E o T S o E This is a special case of something quite general 18 De nition Suppose T X a X and S Y a Y are mappings of metric spaces and E X a Y is a continuous map ofX onto Y for which V o T S 0 V In this case we call S a factor of T and T an extension of S If VX is just dense in Y we ll say S is a quasi factor of S and T a quasi extension of T IfV is a homeomorphism ofX onto Y we say S and T are conjugate 6 JOEL H SHAPIRO Clearly conjugacy is an equivalence relation Going back to our examples Rotation of the circle through an irrational multiple of 7139 is both a factor of irrational translation rnod one and a quasi extension ofthat map via the quasi conjugacy Ve2quot z 0 S x lt 1 which maps T continuously onto the dense subset 01 of the closed unit interval Similarly the squaring map on the circle is both a factor and a quasi extension of the baker map of the unit interval 19 Proposition IfT X a X is transitive theri so is every quasi factor of T Proof Suppose V Y a Y is a continuous map of the metric space Y onto a dense subset and VoT SoV We establish the desired result by showing that Vtrans C trans To this end suppose that z is a transitive point of T An induction shows that V o T S 0 V for every non negative integer n hence Vorb T orb SVx Since V is continuous with image dense in Y the image of any dense subset of X is dense in Y in particular this is true of the S orbit of Vx which was just revealed as the image under S of the dense T orbit of x I 110 A shift map In proving transitivity for the baker map B we used the fact that it acted as a shift on binary expansions of numbers in the unit interval In fact what we were really doing was proving transitivity for a certain extension of B Let E be the space of all sequences of zeros and ones and let 6 denote the backward shift on E that is if xxn1 nltoo62then The metric d de ned on X by CHM Z 2 1960 7 WM 9571 E E is complete on o and d convergenc1is the same as coordinatewise convergence Thus the map V E a I de ned by 00 Va Z 27727 x E E is continuous with VE I Clearly1V o B B 0 V which establishes the claim that the baker map E is a factor of 6 In our proof of transitivity for 6 what we really did was construct a transitive vector for B and interpret it via V as a point of I LINEAR DYNAMICS 7 Here are two other well known maps of the unit interval 1 leave it mostly as exercises to show that they are conjugate to each other7 and that both are factors of the squaring map 111 The quadratic map This is the mapping Q I 7 I de ned by Q 4M1 7 Exercise Show that Q is afaetor of the squaring map Outline of proof De ne V1 T 7 7171 by V1e Re em cos x So V is the orthogonal projection of T onto 7171 except for the points i1 it is two to one Some common trigonometric identities show that if z E T then V1Qz 2Vjz2 7 17 so if 0 denotes the squaring map of the circle and Q1 2x2 71 on 71717 then V1 00 Q1 0 V1 Thus Q1 is a factor of U7 and so inherits its transitivity Next7 map 7171 homeomorphically onto I 071 in the simplest way 12 The map Q2 V2 0 Q1 0 V271 is therefore conjugate to Q7 hence is a factor of o A little arithmetic shows that Q2x 2x 7 12 Finally the homeomorphism 1 7 z of I onto itself yields V3 0 Q2 0 V34 Q7 hence Q is conjugate to Q27 and so is a factor of U7 hence a transitive map Upon composing the maps properly7 you can see that the map V T 7 I that comes out of all this and exhibits the quadratic map as a factor of the squaring map Voo QoV is Ve sin2 x 112 The Tent Map This is the map T I 7 I de ned by Tz 2x if 0 S x S 127 and T 21 7a if 12 S x S 1 The name comes from the shape of the graph7 which is an inverted V77 based on I The shape of the graph is qualitatively like that of the quadratic map7 so one would conjecture that the two maps are conjugate This would imply transitivity for the tent map7 also a reasonable conjecture if one interprets the maps action in terms of mixing dough its just like the baker map7 except that instead of cutting and translating the stretched out dough7 you just double it over at its midpoint 113 Exercise Show that if Vz sin2x so that V is a homeomorphism ofI onto itself then V o T Q 0 V hence the tent and quadratic maps are conjugate Chaotic maps 8 JOEL H SHAPIRO The following discussion comes almost entirely from 3 a beautiful short paper which dis cusses the question What does it mean for a map to be chaotic There seems to be no consensus on this issue but most authors seem to agree that chaotic maps should at least be transitive However transitivity by itself does not seem to capture the essence of chaos Consider for example the map of irrational translation mod one acting on the unit interval Although it is transitive orbits that start close together stay close together so at least in this respect the map is too regular to be considered truly chaotic To exclude such examples it seems desirable to also require a form of instability which asserts that each point z E X should have points arbitrarily close to it whose orbits in some uniform sense do not stay close to orb T More precisely 114 De nition A mapping T of a metric space X depends sensitively on initial conditions or has sensitive dependence if There epists a number 6 gt 0 such that for every 8 gt 0 and every x E X there is a point y E Bz8 such that dT pT y gt 6 for some non negative integer n The number 6 in this de nition is is called a sensitivity constant for T Some authors take chaotic to mean transitive plus sensitively dependent on initial conditions77 see for example 30 However sensitive dependence has a aw it is not in general preserved by conjugacy To see this let T be the map of multiplication by two77 on the positive real axis S the map of translation by ln 277 on R and V 0 00 a R the natural logarithm Then V o T S 0 V so T and S are conjugate but T does not have sensitive dependence whereas S does2 In his classic text 15 Devaney proposed a third property that chaotic maps should have dense sets of periodic points A point z E X is periodic for T if there is a positive integer n such that T z x The least such n is called the period of x in particular xed points are precisely those of period one 2However as pointed out in 3 it is not dif cult to show that if a map T of a compact metric space X has sensitive dependence then so does any map conjugate to T Indeed suppose V o T 0 V where V X A Y is a homeomorphism onto Yr Let 6 be a sensitivity constant for T and let D6 denote the set of pairs zz E X X X with dzz 2 6 This is a compact subset of X X X so its Vimage is a compact subset of Y X Y that is disjoint from the diagonal of that space Since this diagonal is also compact these two compact sets lie some positive distance by apart and one checks easily that 6y is a sensitivity constant for S i LINEAR DYNAMICS 9 Let per T denote the set of all the periodic points of T The same argument that showed quasi factors inherit transitivity now shows that they also inherit denseness of periodic points lndeed7 if V o T S 0 V7 then Vper C per Thus if V is continuous and has dense range7 and per T is dense7 then so is per ln particular7 the property of having a dense set of periodic points is preserved by conjugacy Reasoning that chaotic maps should have pervasive elements of both randomness and predictability7 Devaney de ned a map to be chaotic if it has all three of these properties transitivity7 sensitive dependence7 and a dense set of periodic points 157 Page 52 It might appear that Devaney7s de nition7 relying as it does on sensitive dependence7 is not in general preserved by conjugacy7 but in 37 Banks et al showed otherwise 115 Theorem Suppose a continuous mapping T of a metric space X is transitive and also has a dense set of periodic points Then T depends sensitively on initial conditions Proof The rst step is to observe that each point of z lies uniformly far from some periodic orbit More precisely There epists 60 gt 0 such that for each x E X there is a periodic point q with dist x7 orb gt 6 To see why this is so7 note rst that two periodic points for T either have the same orbit or have disjoint orbits Since per T is assumed dense in X7 there must be two periodic points with disjoint orbits Let 260 be the distance between these orbits Then any point of x has to lie at distance 2 60 away from at least one of these orbits I claim now that 6 604 is a sensitivity constant for T To prove this7 x z E X and 0 lt 8 lt 6 Our goal is to show that there is a point y E B78 and a positive integer in such that dme7Tmy gt 6 Since T is transitive7 it has a dense set of transitive points7 so one of these7 call it t lies in B78 lf dTmt7Tmp gt 6 for some m7 we are done7 with y t So assume that each point of orb t lies within 6 of the corresponding point of orb p7 ie that the orbit of the transitive point t is dragging the orbit of z through the space77 In this case well see that for any periodic point p E Bz7 67 some point of orb is more than 6 away from the corresponding point of orb p7 and this will complete the proof with y p 10 JOEL H SHAPIRO So x such a periodic point p and denote its period by n Note that the distance from p to orbq is 2 46 7 6 36 Because T is continuous there is a ball U centered at q with T7U C BT7q6 for j 012 n 7 1 Because t is a transitive point for T there is a positive integer k such that Tkt E U hence for 0 S j lt n we have TkHt within 6 of orb q and so Tk x lies within 26 of orb Now there7s a unique integer j between 0 and n 71 such that h j is a multiple of n Fix this j and set in h j We7ve already observed that Tmp p lies more than 36 distant from orb q and have just seen that me is at most 26 distant from that same orbit Thus dTmp me 2 dist Tmporb 7 dist Tmorb gt 36 7 26 6 as desired Cl 116 De nition We say a mapping T of a metric space X is chaotic if it is transitive and has a dense set of periodic points Thus chaotic maps have sensitive dependence and chaotic ness is preserved by conjugacy indeed it is inherited by quasi factors Let7s try this concept out on our examples We7ve already seen that irrational translation rnodulo one and also its factor irrational rotation of the circle although transitive is not sensitively dependent hence not chaotic Of the remaining rnaps all of which are transitive recall that the tent map is conjugate to the quadratic map which is a factor of the squaring map which is a factor of the baker map which is a factor of the backward shift 6 on the metric space E of all sequences of zeros and ones So if we can prove B is chaotic well know all of these factors are chaotic too We already know 6 is transitive so its enough to nd a dense set of periodic points But this is obvious the periodic points of B are just the periodic sequences of zeros and ones these correspond to binary expansions of binary rationals in the unit interval and any sequence x E E is approximated to within 2 by the periodic sequence you get by repeating ad in nitum the rst n coordinates of x Thus we have proved 117 Theorem The backward shift the squaring map the tent map and the quadratic map are all chaotic LINEAR DYNAMICS 11 2 HYPERCYCLICITY BASIC EXAMPLES In this section we consider complete metric spaces that are also vector spaces over the complex numbers and for which the vector operations are jointly continuous That is vector addition viewed as a map X gtlt X a X and scalar multiplication viewed as a map C gtlt X a X are both continuous Such spaces are called F spaces The most common examples of F spaces are Hilbert spaces and more generally Banach spaces However there are others for example a The Lebesgue spaces LPQL where M is a measure and 0 lt p lt 1 are F spaces with the metric de ned by dw lfgl d 129 6 Wu b If G is an open subset of the plane then the space CG of continuous complex valued functions on G can be metrized so that a sequence convergence in this metric if and only if it converges uniformly on compact subsets of G The resulting space is thus an F space One way to obtain such a metric is to exhaust G by an increasing sequence of open subsets Gn where the closure of each Gn is compact and contained in Gn1 Then let dnfg be the supremum of 7 as 2 ranges over Gn nite because the closure of Gn is a compact subset of G and set dug Zan fgfg g 6 0a c The collection HG of functions that are holomorphic on G is a closed subspace of CG and therefore an F space in its own right In fact it was in just such a space that hypercyclicity aka transitivity was rst observed for a linear operator The result due to G D Birkhoff dates back to 1929 and asserts that for each a E C the operator T HC a HCC of translation by a77 de ned by 2 Taf2 f2 a z E C f E is hypercyclic Note that T is invertible with inverse Tw which makes good on an earlier promise to provide more examples of transitive invertible maps The rst example of hypercyclicity for an operator on a Banach space was exhibited by Rolewicz in 1969 The setting is the sequence space 1 for 1 S p lt 00 and the operator is 12 JOEL H SHAPIRO constructed from the backward shift B de ned on p in exactly the same way it was de ned on the sequence space E in the last section 3 Bzz27z37 wheres z17z27 66 B itself is a contraction on p S for each x E 6 so there7s no hope of its being hypercyclic7 however Rolewicz proved that if you multiply B by any scalar of rnodulus gt 1 the resulting operator is hypercyclic We begin with a suf cient condition that provides a uni ed proof of hypercyclicity for both these operators7 and for many others as well This result was discovered by Carol Kitai in her Toronto thesis 227 but she never published it7 and it was rediscovered later by Gethner and Shapiro 17 Dont be fooled by the seemingly cornplicated statement as well see shortly7 its proof is easy7 and the result is often quite easy to use 21 Suf cient condition for hypercyclicity Suppose a continuous linear transformation T on an F space X satis es these conditions a There eists a dense subset Y ofX on which Tn converges to zero pointwise b There eists a dense subset Z ofX and a mapping S Z a X not necessarily either continuous or linear such that i TS is the identity map on Z ii 5 converges to zero pointwise on Z Then T is hypercyclic on X Proof Fix two nonernpty open subsets U and V of X Using the density of Y and Z7 choose y E U Yandz E V Z ThenT y Oand SnZ O Thuszny5 quot2 y7 hence xn E U for all suf ciently large n Now even though S and T need not cornrnute7 the fact that TS I on Z means that TnS I on Z also Thus by the linearity of T used for the rst and only time here7 Tnxn Tnyn z a 27 hence T z E V for all suf ciently large n Thus T U V 31 Q for all suf ciently large n7 which by Birkhoff7s Transitivity Theorern Theorem 17 is more than enough to imply that T is transitive III I For our rst application7 recall the backward shift B de ned on p by LINEAR DYNAMICS 13 22 Rolewicz7s Theorem 31 For every scalar A of modulus gt 1 the operator AB is hypercyclic on p for each 0 lt p lt 00 Proof Fix a scalar A with W 1 We will apply Theorem 21 to T AB To this end let U denote the forward shift on 17 Ur 012 where z x1x2 E 6 and set S A lU Since BU I on 17 we also have TS I Let Z p and note that a 0 as n a 00 let Y be the collection of nitely nonzero sequences in 17 a dense subspace of 1 because p lt 00 Then for each x E Y we have End eventually zero so the same is true of Snd hence the hypothesis of Theorem 21 are ful lled That7s all there is to it I We can think of AB as a special weighted backward shift lf w wn is a bounded sequence of non zero complex numbers de ne the operator Bw on p by 4 Bwx z1w1 x2w2 x 1 x2 E 17 Then Bw is a continuous linear operator on 17 which converges pointwise to zero on the set Y of nitely non zero sequences employed in the previous proof Suppose further that lim sup lwnl gt 1 Since none of the weights wn are zero we can de ne the right inverse operator Sw on Z 1 by Swz 0 1w1 2w2 Once again all hypotheses of Theorem 21 are satis ed so Bw is hypercyclic without any help from scalar multiplication To summarize 23 Theorem 17 4 Ifw is a bounded sequence of compler numbers with no element zero and lim sup lwnl gt 1 then the weighted backward shift Bw is hypercyclic on p for euery0ltpltoo Let7s turn now to some examples from analytic function theory We begin with a natural complement to Birkhoff7s translation theorem that was surprisingly not proved until much later 24 MacLane7s Differentiation Theorem 25 1952 The operator of di erentiation is hypercyclic on 14 JOEL H SHAPIRO Proof Let Y Z denote the dense subspace of all polynomials in Let D denote the differentiation operator and S the operator of integration from 0 to z N 5102 ma 12 1 where 102 2770 anz Then for each polynomial p DSp p7 D p is eventually zero7 and S p a 0 uniformly on compact subsets of C Thus the hypotheses of Theorem 21 are satis ed7 so D is hypercyclic on C I In fact7 this argument works just as well for HG where G is any simply connected plane domain All that is needed is that the polynomials be dense in HG7 and this is provided by Runge7s Theorem This suggests the question of whether or not simple connectivity of a plane domain G is characterized by hypercyclicity of the differentiation operator on This is in fact trueisee 35 for the details7 and for further variations on this theme For our nal application we prove Birkho s result on the translation operator mentioned at the top of this section 25 Birkhost Translation Theorem For each eompler number a the operator Ta given by is hypercyclic or Proof Instead of working with polynomials we consider exponentials EA de ned by EA2 e V A72 6 C The key is that EA is an eigenvector for Ta with eigenvalue e 7 so in particular if Re aA lt 0 then TSEA exp nRe aAE a 0 in HCC n a 00 Thus if Y is the linear span of the exponentials EA for Re aA lt 0 a half plane H of points A whose boundary is the line through the origin orthogonal to a7 we see that T Tm a 0 pointwise on Y Similarly7 let Z denote the linear span of the Eys for Re aA gt 0 Then TglEA a 07 so if we can show that Y and Z are dense in HC then well have the hypotheses of Theorem 21 satis ed with S Ta In fact much more is true LINEAR DYNAMICS 15 26 Density Lemma Suppose A is any subset ofCC with a limit point in C and let EA be the linear span of the functions EA with A E A Then EA is dense in Proof Suppose A is a continuous linear functional on HCC that takes the value zero on each exponential function EA for A E G By the Hahn Banach Theorem it is enough to prove that A E 0 on For each R gt 0 and each f E HCC let HfHR maxlfzl S R H HR is a norm on HC and the open balls for each of these norms forms a basis for the topology of Thus the inverse image of the unit disc under A contains an 39 llRball centered a the origin for some R gt 0 In other words A is a bounded linear functional relative to the norm HR so by the Hahn Banach theorem it extends to a bounded linear functional on S By the Riesz Representation Theorem there is a nite Borel measure M on the closed disc S R such that A is represented by integration against M in particular Mofa oeHw Since the support of M is compact the function F de ned on C by FwaaWaw is entire and D F0 z dnz n 012 But our hypothesis is that F vanishes on A and since A has a nite limit point the identity theorem for holomorphic functions insures that F vanishes on C hence the same is true of each of its derivatives Thus f2 dnz 0 for every non negative integer and therefore f fdn 0 for every holomorphic polynomial f and so for every entire function f every entire function is the limit in HC of the partial sums of its MacLaurin series Thus A vanishes on HC so by the Hahn Banach Theorem EA is dense in This completes the proof of the Lemma and with it the proof of Birkhoff7s Translation Theorem I Exercise Use the same idea to show that for any nonconstant polynomial p the operator pD is hypercyclic on HC and even on HG for any simply connected plane domain G In fact the result of this exercise extends to any continuous linear operator L on HG that commutes with D and is not a constant multiple of the identity see 18 5 for the details 16 JOEL H SHAPIRO Examples of nonhypercyclicity Why do most people nd it surprising that linear operators can be hypercyclic7 or mixing7 or chaotic Perhaps its because the most common onesithe nite dimensional operators7 are not This is most easily seen by rst considering a more general situation Given an F space X7 denote the dual space of X the space of continuous linear functionals on X by X Now if X is not locally convex we may have X 0 eg7 X L1 07 1 with 0 lt p lt 17 but this does not affect our arguments If T is a continuous linear operator on X7 de ne its adjoint T X a X by TA A CT for A E X so that T is a linear transformation of X 27 Theorem Suppose T is a continuous linear operator on an F space X If the adjoint operator T has an eigenvalue then T is not hypercyclic Proof We are saying that there is a continuous linear functional A on X that is not identically zero7 and a complex number 04 such that TA 04A7 so for every positive integer n we have TWA anA Thus if z is any vector in X then AT x T Ap oLAz n 071727 hence Aorb T7 oz Ap3 Now the set on the right hand side of this identity is never dense in C whereas if orb Ta were dense in X7 then its A image would be dense in C by the continuity of A the image of a dense set under a continuous map is dense in the image of the whole space This contradiction proves the theorem El Corollary There are no hypercyclic operators on nite dimensional F spaces Proof Each nite dimensional F space is isomorphic to C for some positive integer n Therefore for each linear operator on such a space7 the adjoint can also be viewed as an operator on C ie7 as an n gtlt n complex matrix So T has an eigenvalue7 and therefore T is not hypercyclic El Exercise No compact operator on a Banach space can be hypercyclic LINEAR DYNAMICS 17 3 MIXING TRANSFORMATIONS In this section we return to the general setting of continuous maps of complete metric spaces This is motivated by the fact that the proof of our suf cient condition for hyper cyclicity Theorem 21 of the last section actually yields much more than was promised In the rst place the conclusion itself is formally stronger than what7s needed to be able to apply Birkhoff7s Transitivity Theorem the conclusion shows that for every pair UV of nonempty open subsets of X there is a non negative integer N such that T U V 31 Q for all n 2 N Birkhoff7s Theorem only demands one n for which the intersection is empty Whenever this stronger property is true of a mapping of a metric space we call that map ping topologically miping Thus for example all the linear operators we have proved to be transitive are actually mixing Next the proof of Theorem 21 never made full use of linearity it works just as well if X is merely a complete metrizable topological group with the group identity taking the place of the vector space zero element and T is a continuous homomorphism of X into itself The group need not even be abelianl For example the space E of 01 sequences introduced in 110 is a group where addition is done coordinatewise modulo two and the backward shift 6 on that space which we proved to be transitive by constructing a point in E with dense orbit actually obeys the hypotheses of our re ned suf cient condition Thus 6 isn7t just transitive it too is mixing 31 Exercise Every quasi factor of a miping map is miccing Clearly the maps of irrational translations mod one though transitive are not mixing However all the other maps we discussed in 1 are factors of the backward shift on 2 so they are mixing 32 Proposition IfX is a complete separable metric space and T a continuous mapping on X that is miccing then for every strictly increasing sequence of positive integers there is a dense G5 subset of points z E X for which T wz h 2 0 is dense in X Proof Fix a countable basis for the topology of X and enumerate the pairs of basis sets obtaining a sequence UV f with each of the basis sets occurring in each coordinate 18 JOEL H SHAPIRO in nitely often Because T is mixing 3 V1 6 N such that n 2 V1 gt T U1 V1 31 0 In particular 3 k1 e N with T k1U1 m V1 7s 0 Similarly 3 V2 6 N such that n 2 V2 gt T U2 V2 31 0 from which it follows that 3 kg 6 N with k2 gt kg and T k1U1 V1 7s 0 Continuing in this manner you get a strictly increasing sequence such that T k739Uj 31 0 In particular for any nonvoid open subsets U and V of X there ex ists an index k such that T kU V 31 0 The proof of Birkhoff7s TransitiVity Theorem repeated almost word for word now shows that there is a dense G5 set of points z E X such that is dense in X I As a special case for each xed positive integer n we may take nk kn in the result above thus obtaining 33 Corollary IfT is a continuous miping transformation of a complete metric space X then T is transitive for every positive integer n By contrast 34 Proposition Not every transitive map has transitive powers Proof To make examples of this phenomenon let T be any continuous mixing map of a complete metric space X and form a new X as follows 55 X gtlt 12 and Tz1Tz2 Tim Tx1 for 95 e X So X is the disjoint union of subsets X gtlt where j 1 2 and T maps X onto X in the same way T would map X onto itself and it similarly interchanges X and Perhaps one should think of each of these subsets as a copy of X with the rst one colored red and the second blue Then T has the action of T except it takes each red point to the same point but now colored blue and Vice versa LINEAR DYNAMICS 19 Now X is the product of two metric spaces7 X7 with its metric d and the two point space 17 27 with the discrete metric 6 Therefore the metric on X de ned by Wait 1170 i Way 5U k puts the product topology on X7 thereby making it into a complete metric space on which T acts continuously Clearly T2 is not transitive since it takes into itself for eachj 17 2 On the other hand T itself transitive lndeed7 because T is assumed to be mixing7 there is a dense G5 subset of points z E X for which Tznx is dense in X7 and a similar set of z7s for which Tzn x is dense It follows that there are points z in X for which both sequences are dense Thus Tznx7 1 is dense in X while T2 171 is dense in X hence orb T7 7 1 is dense in X7 ie7 T is transitive I By re ning this construction just slightly we can create examples where the underlying space X is connected For this7 suppose X is connected and T X a X is continuous and mixing7 but with a xed point p E X For example7 the tent and quadratic maps on 071 have these properties both x the points 0 and 1 Let X be X with p71 and p72 identi ed to a point well call 13 De ne T on X in the obvious way its just T on Xp7 and it xes the point 13 The resulting T is continuous on X7 and if V X a X is the identity on Xp7 17 p7 2 and takes p7j to 137 then V is also continuous and V o T T 0 V7 ie7 T is a factor of T7 and therefore is also transitive But just as before7 T2 is not transitive7 since for j 172 it still takes each of the disjoint sets into itself I For a more concrete example7 it might be instructive to show that the map you get by this construction from the tent map on the unit interval is conjugate to the double tent map77 de ned on 7171 by Tx 1 for 71 x S 0 and 7T for 0 S x S 1 35 Product maps The construction given above for T is actually a special case a much more far reaching idea Let be a nite or countable collection of metric spaces On each Xi we can assume without loss of generality that the metric dl is bounded by one otherwise7 replace it by di1 Then the cartesian product X of the spaces Xi is a metric space in the metric d 2 idi7 and a sequence of points converges in this product metric if and 2O JOEL H SHAPIRO only if it converges in each coordinate Thus X d has the product topology it is compact if each Xid is compact and complete if each Xd is complete Suppose T a mapping of X for each i Then the product map77 T is de ned on the product space X by letting T act in each coordinate If each T is continuous X then T is continuous on X and conversely In particular the map T constructed above is an example ofjust such a product construction Both set theoretically and topologically X X gtlt 1 2 and T T gtlt S where now T X a X is the original map on X and S 12 a 12 is the permutation that interchanges 1 and 2 36 Proposition IfX is the product of a nite or countable collection of metric spaces and T is a miccing transformation ole for each i then the product map T is miping on the product space X Proof Suppose U and V are basic open sets in X ie that U U with U open in X and U X for all i 2 some n1 Similarly V V with V open in X and equal to X for all i 2 some possibly different index n2 Let n be the larger of n1 and n2 Because each T is mixing we may choose a positive integer N such that kN TkUimiH oforeach1971 hence for each such k T kU m V H TkU m V gtlt H X 7 o i1 i2n1 which shows that T is mixing CI 37 De nition A mapping T of a metric space X is called weakly mixing ifT gtlt T is transitive on X gtlt X We have just seen that every mixing transformation is weakly mixing and it is easy to check that weakly mixing transformations are transitive more generally if a product mapping is transitive then the same is true of each coordinate mapping However the converse is not true there are transitive maps T that are not weakly mixing The next result implies that the the examples constructed in the proof of Proposition 34 with T is transitive but T2 not all have this property LINEAR DYNAMICS 21 38 Proposition Suppose T is a continuous map of a complete metric space X IfT is weakly mipiag then T2 is transitive Proof We use Birkhoff7s characterization of transitivity for this setting Fix U and V open in X and nonempty It is enough to nd an even integer in such that T mU O V 31 0 Use the transitivity of T gtlt T to choose 71 gt 1 such that T gtlt Tr U gtlt U m V gtlt T 1V 0 But the left hand side of this expression is just T U O V gtlt T U T 1V hence T U O V and T 1U V are both nonempty Since either 71 or n 71 is even it follows that T2 is transitive El 4 HYPERCYCLIC COMPOSITION OPERATORS Historically the notion of transitivity was not foremost on the minds of operator theorists an understandable oversight since when dealing with linear operators one thinks about in variant subspaees but not so much about invariant sets Suppose X is a vector space T a linear transformation on X and z a vector in X Corresponding to the fact that orb T p is the smallest T invariant set containing x the linear span of this orbit is the smallest T invariant subspace containing x and the closure of this linear span is the smallest closed T invariant subspace of X containing x If span orb T p is dense in X we say T is cyclic and call x a cyclic vector for T Thus noncyclic vectors generate proper closed invariant subspaces and if there are no cyclic vectors then there are no closed invariant subspaces except for 0 and X The rst example of a Banach space operator having only trivial closed invariant subspaces was constructed by En o 16 in the 1980s and his work was later simpli ed by Read who eventually showed that it is even possible for an operator on a Banach space to have no hypercyclic ie transitive vector 29 and hence no closed invariant subsets other than the zero vector and the whole space However none of the examples produced so far is set in a Hilbert space and for this special case its a famous open question to decide if every bounded operator has a closed invariant subspace or a closed invariant subset 31 0 or the whole Hilbert space 22 JOEL H SHAPIRO So it makes sense to think of transitivity of a linear operator as a very strong form of cyclicity This is why operator theorists use the term 77hypercyclic77 instead of transitive7 In this section we7ll nd more examples to illustrate the point that hypercyclicity occurs surprisingly often Then we7ll explore a few of the ways in which hypercyclicity is a more robust77 concept than cyclicity Composition operators Recall that if G is an open subset of the plane7 then the space HG of all complex valued functions holomorphic on G can be made into an F space by a complete metric for which a sequence fn in HG converges to f E HG if and only if fn a f uniformly on every compact subset of G If G1 and G2 are open subsets of C and p G1 a G2 a holomorphic map not necessarily one to one or onto7 then p induces a composition operator CW HG2 a HG1 de ned by Oof f 0 r7 f E HG2 lf G17 G27 and G3 are all open subsets of C with p G1 a G2 and 7 G2 a G3 holomorphic maps7 then wa GW 0 Cw We will focus primarily on holomorphic self maps p of plane domains G7 for which the composition formula above yields the iteration formula G G W W 710717 77 where7 to avoid confusion with the n fold pointwise product7 pn denotes the composition of p with itself 71 times This simple observation suggests that there should be intriguing connections between the dynamical behavior of a composition operator CW with that of its inducing map p In particular7 which composition operators are hypercyclic ori In order the operators foremost7 we restrict attention to the simplest setting G U7 the open unit disc Note7 however7 that the Riemann Mapping Theorem will allow us to transfer dynamical results about composition operators on HlU to HG where G is any simply connected plane domain 31 C lndeed7 Riemann7s theorem guarantees that there is a univalent holomorphic map a taking lU onto G7 hence the corresponding composition operator CO is an isomorphism one to one7 onto7 linear7 bi continuous of HG onto 1 If p G a G is a holomorphic self map of G7 then 7 o o p o o is a holomorphic self map of U that is holomorphically conjugate to p As for the corresponding operators7 LINEAR DYNAMICS 23 Cw CU 1C CU so Cw HG a HG is similar ie linearly conjugate to 0 HlU a Henceforth p will always denote a holomorphic self map of U and we will abbreviate holomorphic and one to one77 by univalent Our rst result severely limits the kinds of maps that can produce hypercyclic behavior 41 Proposition If 0 is hypercyclic on HlU then p is univalent and has no red point in U Proof Suppose p has a xed point p E U Then for f E HlU any function in orb CW f must have value fp at p hence the same must be true of any function in the closure of this orbit Thus no CW orbit is dense hence 0 is not hypercyclic Suppose p is not univalent so there exist distinct points p q E U with p q Then if f E HlU each function in orb wa takes the same value at p as at q and again this property gets passed on to functions in the closure of the orbit Once again no orbit can be dense CI Note that this proof did not make any use of the special properties of the unit disc so the result is valid for any open set G In a sense to be made precise later univalent self maps of the unit disc are modelled by linear fractional transformations so we consider this class of maps rst We need to know how these maps are classi ed in terms of their xed points then next few paragraphs review this matter 42 Linear fractional transformations Recall that a linear fractional transformation henceforth a LFT is a mapping of the form a wa The condition A 31 0 is necessary and suf cient for p to be nonconstant as you see when where A ad 7 be 31 0 c 31 0 by checking the formula a A 1 W 2 3 When 0 0 then the condition A 31 0 implies that neither d nor a is zero hence p is a nonconstant af ne mapping of C We extend p to a mapping of the extended complex plane 24 JOEL H SHAPIRO C CUoo onto itself by de ning ltpoo ac and ltp7dc 00 if c 31 07 and ltpoo 00 if c 0 the af ne case If E is then identi ed with the Riemann Sphere 2 via stereographic projection7 then p becomes a homeomorphism of 2 onto itself We employ the classi cation of LFT7s in terms of their xed points Each LFT has one or two xed points in lf there7s just one xed point the map is called parabolic If a parabolic map p has its xed point at 007 then its easy to check that p is a translation ltpz z 739 for some complex number 739 lf7 however7 its xed point is p 6 7 then the LFT 042 12 7p takes p to 007 and so a o p o of1 is an LFT that xes 007 and so is a translation Thus the parabolic LFT s are precisely the ones that are linear fractionally conjugate to translations Note also that 00 is an attractive xed point for any translation7 in the sense that the sequence of iterates converges to 00 uniformly on compact subsets of C Thus if p is any parabolic LFT with xed point p E C then the sequence of iterates gal converges to p uniformly on compact subsets of The other possibility is that p has two distinct xed points7 say p and Q If these are 0 and 00 then p is a complex dilation ltpz n2 for some H E C In the general case7 one xes an LFT 04 that maps p to 00 and q to zero7 eg7 042 z i Q in which case oil 0 p 0 oz xes 0 and 007 hence is a Is dilation7 as above Now Is is not uniquely determined the maps 2 a n2 and z a 1Hz are linear fractionally conjugate to each other But this is as bad is things can getiif one conjugation of p to an LFT with xed points 0 and 00 gives you a multiplier of K then any other one will give either H or 1H see 347 Chapter 0 for more details If H is positive we say p is hyperbolic7 otherwise p is lopodrornic Note that in the hyperbolic or loxodromic cases one of the xed points is attracting and the other repelling The only other possibility is W 1 in which case p is called elliptic Here the xed points are neither attracting nor repelling 43 Linear fractional maps of the unit disc How does our classi cation of linear frac tional maps fare if we additionally require that the unit disc be taken into itself Let LFTlU denote this class of maps7 and Aut U denote the subclass of conformal automorphisms of LINEAR DYNAMICS 25 U ie linear fractional maps that take U onto itself Since we aim to study hypercyclic composition operators we will focus on maps in LFTU that x no point of U If p E LFTU is parabolic then because the xed point is attractive it must lie on the unit circle Upon conjugating by an appropriate rotation we may assume this xed point is 1 Now the map 2 7 1 7 2 takes the unit disc onto the open right half plane P and takes 1 to 00 so this map conjugates p to a translation of l into itself ie a map of the form it 7 w 7 where Rei39 2 0 Note that p rnaps U onto itself if and only if Rer 0 So parabolic maps are LFT conjugate to translations of the right half plane into itself with the autornorphisrns corresponding to the pure irnaginary translations Similarly if p E LFTU is hyperbolic and an automorphism then both its xed points must lie on 3U as before this is clear for the attractive one and the repelling one is attractive for p l which is also an automorphism of U In this case any LFT that sends one of these xed points to the origin and the other to 00 conjugates p to a positive dilation as we have seen and takes the unit circle to a straight line with U going to one of the half planes bounded by this line Conjugation by an appropriate rotation takes this half plane to P while leaving the dilation unchanged Thus every hyperbolic automorphism of U is conjugate to a positive dilation ofllD Finally if p E LFTU is hyperbolic but not an automorphism then its attractive xed point must still lie on 3U while the repulsive one is either in U the case were not con sidering or outside the closure of U By conjugating with a rotation we may assume the attractive xed point is 1 Let q denote the repulsive one Then the re ection if of q in the unit circle lies in U so for an appropriate unirnodular constant to the U autornorphisrn z 7 wp 7 752 takes 10 to zero while xing 1 This map therefore takes p to 00 and therefore conjugates p to a map ltIgt E LFTU that xes both 1 and 00 Thus ltlgtz a2 b where lal lbl S 1 since ltlgtU C U and a b 1 since ltlgt1 1 It follows that 0 lt ab lt 1 so ltlgtz a2 17 a for some 0 lt a lt1 Note that ltIgt also belongs to LFTU and that p is LFTU conjugate to ltlgt Note further that ltIgt is a hyperbolic automorphism of the half plane H Rez lt 1 and that its attractive xed point is 1 the same as for p With these results in hand we can classify those p E LFTU that induce hypercyclic cornposition operators on The result is surprisingly unsubtle 26 JOEL H SHAPIRO 44 Theorem Every p E LFTlU with rzo red point m U induces a hypercyclic composi tion operator or Proof The idea is to reduce everything7 case by case7 to Birkhoff7s translation theorem 25 Fix p E LFTlU with no xed point in U Then p is either parabolic or hyperbolic a If p is parabolic then we know that there is a LFT a mapping lU onto l such that p 0 1 0 T7 0 07 where T7w w 739 for w 6 P7 and 739 is a complex number with non negative real part so 739 maps l into itself As we saw at the beginning of this discussion7 the composition operator CU HOP a HlU establishes a similarity in the language of previous sections a linear conjugacy between CW on U and CT on P So it is enough to prove that CT is hypercyclic on For this7 note that the restriction map R HCC a Hll gt7 de ned by Rf fl for f E HCC is a 1 1 continuous linear map from HC into HOP it can be viewed as the composition operator induced by the identity map l a CC Moreover R07 CTR7 where in the rst instance CT is acting on HCC and in the second on Finally7 the range of R7 namely H0 is dense in HCithis is a consequence of Runge7s theorem which insures that the polynomials are dense in both spaces their density in HCC is also obvious from the convergence properties of power series Thus CT on H0 is semi conjugate to CT on HC7 and therefore the former operator inherits the hypercyclicity that Birkhoff7s Translation Theorem provides for the latter one b Suppose p is a hyperbolic automorphism Then p is LFT conjugate to a dilation AT llD a R de ned by ARw rw for some xed 0 lt r lt 1 The principal branch A of the logarithm takes l univalently onto the horizontal strip S W llle lt 7T27 and the corresponding composition operator effects a similarity between CAF HOP a HO and the translation operator TMP7 now acting on Just as we saw in the parabolic case7 the map that restricts an entire function to S is a continuous linear embedding of HC into HS with dense range7 and it reveals TMP7 acting on HS as a quasi factor of the same operator acting on Hence7 as before7 the translation operator on HS is hypercyclic7 and therefore so is its conjugate7 CW on c Finally7 suppose p E LFTlU is hyperbolic7 but not an automorphism Then we have seen that p is LFTlU conjugate to the restriction to U of a hyperbolic automorphism ltIgt LINEAR DYNAMICS 27 of the half plane P Rez lt 1 So it is enough to show 04 is hypercyclic on By part b above 04 on HP is conjugate to a composition operator induced by a hyperbolic automorphism of U and so is hypercyclic The same observations we employed to prove part b show that 04 acting on HlU is a quasi factor of CW acting on HP so it too is hypercyclic This completes the proof of our theorem I 45 Chaos In 116 we de ned a map to be chaotic if it is transitive and has a dense set of periodic points Which of the linear maps we7ve proved transitive are chaotic Answer All of them To see this recall that the operators of differentiation and translation have the exponential functions EA2 6 as eigenvectors DEA AEA and TaEA eaAEA Thus if A is an n th root of unity EA is a periodic point of D with period 71 Any linear combination of such periodic points is again periodic with period equal to the least common multiple of the original periods Since the set of roots of unity is dense in the unit circle the Density Lemma shows that D has a dense set of periodic points The same argument works for T except now you start with exponentials EA such that 6 1quot is a root of unity As for composition operators on HlU induced by linear fractional maps our proof that they are all hypercyclic established that they are all quasi factors of translation operators acting on HC and therefore they inherit the chaotic behavior just proved for those oper ators El Remark You can think of conformal automorphisms with no xed point in U as non Euclidean translations of the unit disc with the attractive xed point of p which necessarily lies on 3U playing the role that 00 plays in the Euclidean case From this point of view the fact that such maps induce hypercyclic composition operators on HlU can be viewed as the non Euclidean analogue of Birkhoff7s Translation Theorem Thus it is only tting that the non Euclidean result which was proved about sixty years ago by Seidel and Walsh 37 actually follows from Birkhoff7s Beyond linearfractional Having disposed of the linear fractional case its time to ask whether or not a composition operator induced by an arbitrary univalent self map of U is hypercyclic or chaotic on 28 JOEL H SHAPIRO Given the special nature of the proofs used in the linear fractional setting one might be tempted to dismiss the question as too general to admit a de nitive solution However one would be wrong one of the landmark theorems of classical analytic function theory renders the general problem an almost trivial consequence of our analysis of the linear fractional situation This is 46 The Linear Fractional Model Theorem Ifltp is a univalent holomorphic self map of U then there epists a linear fractional map 7 and a univalent map a U a C such that o o p 7 o o Ifltp has no cced point in U then there are two possibilities a 7 can be taken to be dilation 72 r2 for some 0 lt r lt 1 and 0lU C P or b 7 can be taken to be a translation 72 z 739 for some 739 in C There is actually much more to this remarkable theorem and well talk about this in a moment Note that the two cases distinguished above correspond precisely to what happens for hyperbolic and parabolic maps in LFTUU so in some sense these maps are models77 for univalent self maps of U that x no point of U Just how one distinguishes hyperbolic from parabolic behavior for such general self maps of U is a fascinating question which we will discuss shortly But rst note that theorem above makes short work of the hypercyclicity problem for composition operators on 47 Theorem Suppose p is a univalent holomorphic self map oflU that has no cced point in U Then CW is chaotic on Proof The proof is the same one we used in the linear fractional case In case 7 is a dilation so that 0lU C P a further mapping by the principal branch A of the logarithm replaces it by a translation that takes G AolU into itself Since G is simply connected Runge7s Theorem insures that the polynomials are dense in HG hence the entire functions are dense in The rest follows just as before our translation and therefore CW itself is exhibited as a quasi factor of the same translation acting on In the parabolic case the same proof works except that now there is no need to call upon the logarithm El Remarks Recall that in the proof of Birkhoff7s Translation Theorem Theorem 25 it was the suf cient condition 21 that proved hypercyclicity for translation operators on Thus by the rst paragraph of 3 Every translation operator on HCC is miccing Since all LINEAR DYNAMICS 29 the composition operators on HU that we studied above turned out to be quasi factors of such translations Every composition induced on HU by a univalent cced point free holomorphic self map ofU is miccing So thus far all our hypercyclic examples turn out to be both chaotic and mixing This is not the case in general For example in 8 examples are given of hypercyclic translation operators T on some Hilbert spaces of entire functions for which the spectrum of T is the single point Since periodic points of linear operators are eigenvectors whose eigenvalue is a root of unity these translation operators have no hope of being chaotic We will see in the next section that thanks to an example of Salas not every hypercyclic operator is mixing Iteration and linear fractional models The Denjoy Wol Theorem In a certain sense every holomorphic self map of U has an attractive xed point if there is not one in U then there is a unique boundary point that serves the purpose This is the content of the famous Denjoy Wol Theorem which gures importantly in many aspects ofthe study of composition operators To simplify its statement let7s adopt some terminology o A point p E 3U is a boundary cced point of p if p has non tangential limit p at p o The notation 3 indicates uniform convergence on compact subsets of U o If the derivative of p has a nontangential limit at a boundary point p of U and the non tangential limit of p at p whose existence follows easily from that ofthe derivative has modulus one we say p has an angular derivative at p and denote the limit by ltp p It may seem harsh to require that p have non tangential limit of modulus one at any boundary point at which its angular derivative is to exist but when dealing with composi tion operators this is exactly what makes the concept meaningful see 34 Chapter 4 for example 48 The Denjoy Wolff Iteration Theorem Suppose p is an analytic self map ofU that is not an elliptic automorphism a Ifltp has a ped pointp E U then pn 3p and l pl lt 1 b Ifltp has no cced point in U then there is a pointp E 3U such that can i p Furthermore gtk p is a boundary ped point of p and gtk the angular derivative ofltp epists at p with 0 lt ltp p S 1 3O JOEL H SHAPIRO c Conversely ifltp has a boundary red point p at which p p S 1 then p has no cced points in U and on L p The xed point p to which the iterates of p converge is called the Denjoy Wol point of p Part a7 which is an exercise based on the Schwarz Lemma7 is not really part of the original theorem it is included here only for completeness For a proof of Theorem 487 and for further connections with the theory of composition operators7 see 347 Chapter 5 or 137 Section 24 Classi cation of linearfractional maps The Denjoy Wol Theorem suggests a linear fractional like classi cation of arbitrary holo morphic self maps of U For motivation7 let7s review how the linear fractional self maps of U fall into distinct classes determined by their xed point properties cf 34 Chapter 0 These are 0 Maps with interior cced point We didn7t concentrate much on this case previously7 but by an argument based on the Schwarz Lemma7 the interior xed point is either attractive7 or the map is an elliptic automorphism In both cases the map is conjugate to a dilation z a A2 for some complex number A with 0 lt W S 1 Hyperbolic maps with attractive cced point on all Upon chasing through the classi cation of these maps as conjugate to dilations of the right half plane7 you see that they are the self maps of U having no xed point in U7 with derivative lt 1 at the attractive boundary xed point 0 Parabolic maps These have exactly one xed point on the Riemann Sphere7 necessarily lying on all These maps are characterized by the fact that they have derivative 1 at the xed point The parabolic self maps of U fall into two subclasses7 which one distinguishes by examining their the action of the corresponding maps on the right half plane o The automorphisms These are distinguished by the property that each orbit is sepa rated in the hyperbolic metric meaning that7 for each 2 6 U7 the hyperbolic distance between successive points of the orbit stays bounded away from zero LINEAR DYNAMICS 31 o The aonautomopphisms For these7 the orbits are not hyperbolically separated7 ie7 the hyperbolic distance between successive orbit points tends to zero An elementary argument establishes these last two statements The rst just re ects the fact that automorphisms are hyperbolic isometries The second is best viewed in the context of the right half plane ll Suppose 7 is a parabolic self map of U with xed point at 17 and let Tw Z i7 and 1 Toy oTil Thus T is a linear fractional mapping of U onto l that takes 1 to 007 and one easily checks that 1w w 1quot1 It follows that ib 1 has non negative real part otherwise 1 could not map l into itself7 and since 7 is not an automorphism of U7 1b1 cannot be pure imaginary Now hyperbolic discs in l of xed radius have this property their Euclidean size is proportional to the real part of their hyperbolic center see section 47 or 357 Chapter 4 for the details Our hypothesis on the translation distance ib 1 insures that for each w E l the Kll orbit has unbounded real part7 but xed Euclidean distance W 1l between successive points Thus for all suf ciently large n7 the hyperbolic disc of radius 8 about 11nw contains Qn1w7 hence the orbit of w is not separated Motivated by the classi cation of linear fractional self maps of U7 and encouraged by the restrictions the Denjoy Wol Theorem places on the values the derivative of an arbitrary self map can take at the Denjoy Wol point7 we introduce the following general classi cation scheme 49 Classi cation of arbitrary selfmaps A holomorphic self map p of U is of o dilation type if it has a xed point in U o hyperbolic type if it has no xed point in U and has derivative lt 1 at its Denjoy Wol point 0 parabolic type if it has no xed point in U and has derivative 1 at its Denjoy Wol point As in the linear fractional case7 the maps of parabolic type fall into two subclasses o Automoiphie type Those with an orbit that7s separated in the hyperbolic metric of U o Noa automoiphie type Those for which no orbit is hyperbolically separated 32 JOEL H SHAPIRO It can be shown that either all orbits are separated or none are separated for a special case of this see 7 With these ideas in hand we can state the full strength version of the 410 LinearFractional Model Theorem Suppose p is a uriiyalerit holomorphie self map oflU Theri there epists a holomorphie uriiyalerit map a U a C arid a linear fractional map 7 such that 7lU C U 7olU C olU arid 6 o o p 7 o 039 Furthermore a 7 viewed as a self map oflU has the same type as p b Ifltp is of hyperbolic type theri 7 may be taken to be a conformal automorphism of U c Ifltp is of either hyperbolic or parabolic automorphie type theri a may be taken to be a self map of U We call the pair 7 G or equivalently 7o a linear fractional model for ltp The fact that 7 maps the sirnply connected dornain G olU into itself follows immedi ately from the functional equation This equation establishes a conjugacy between the original rnap p acting on the unit disc and the linear fractional map 7 acting on G Since the action of 7 is known the subtleties of p lie encoded in the geometry of G History of the LFM Theorem The Linear Fractional Model Theorern is the work of a number of authors whose efforts stretch over nearly a century The dilation case is due to Koenigs 23 1884 In this case equation 6 is Sehrb39der s equation 0 o p AU where necessarily A ltp 0 see 35 Chapter 6 for more details The hyperbolic case is due to Valiron If one replaces the unit disc by the right half plane sending the Denjoy Wolff point to 00 then the resulting functional equation is again Schroder7s equation but this time A is the reciprocal of the angular derivative of the original disc map at the Denjoy Wolff point 38 1931 Finally the parabolic cases were established by Baker and Pornrnerenke 28 2 1979 and independently by Carl Cowen 10 1981 Once again the situation is best viewed in the right half plane rather than the unit disc with the Denjoy Wolff point placed at 00 Then equation 6 is just 0 o p o i in the autornorphic case 28 and o o p a 1 in the nonautornorphic case In 10 Cowen uni ed the proof of the Linear P ractional Model Theorern by means of a Riemann surface construction that disposes of all LINEAR DYNAMICS 33 the cases in one stroke see also 13 Theorem 253 He later introduced linear fractional models into the study of composition operators using them to investigate spectra 11 These models have also gured prominently in previously mentioned work on subnormality 12 and compactness 36 Distinguishing the parabolic models The problem of distinguishing the two parabolic cases of the Linear Fractional Model Theorem is in general quite delicate lt7s shown in 7 4l that if p has enough differentiability at the Denjoy Wolff point then cases are distinguished by the second derivative of p at that point There is however some subtlety here its shown in 7 6l that for example 02 differentiability at the Denjoy Wolff point is not enough to allow the second derivative to distinguish the two cases Necessity of Univalence Although we have stated the Linear Fractional Model Theorem only for univalent maps p the result is true even if p is not univalent provided we are willing to give up the conclusion of univalence for the intertwining map a In case p is of dilation type with xed point p E II we must also assume that ltp p 31 0 5 WHY HYPERCYCLICITY IS INTERESTING We7ve observed hypercyclic phenomena in some interesting classes of operatorsiweighted shifts and composition operators There is much more to say about composition operators see 6 but right now Id like to shift gears and discuss some of the functional analytic aspects of hypercyclicity Suppose for example that you are an operator theorist interested in invariant subspaces Then you are interested in cyclicz39ty so why bother with hypercyclicity except that it is a formally stronger concept than the one you want to study One answer is that if an operator is hypercyclic then it will in general have a far greater proliferation of cyclic vectors than one that is merely cyclic Indeed we have already noted that for a hypercyclic operator the collection of hypercyclic vectors is a dense G5 set By contrast the cyclic vectors for a cyclic operator need not be dense Here is an example On HlU let M2 denote the operator of multiplication by the independent variable 2 ie szzzfz fEHlU andzElU 34 JOEL H SHAPIRO admittedly7 there is some abuse of notation here 51 Proposition f E HlU is a cyclic vector for M2 if arid only iff has no zero in U Before proving this result7 lets note that by Hurwitz7s Theorem an immediate consequence of the argument principle any limit in HlU of a sequence of never vanishing holomorphic functions is either identically zero or never vanishing itself Thus Proposition 51 has this CO nseque HOS 52 Corollary The operator M2 on HlU is cyclic but its collection of cyclic vectors is riot derise Proof of Proposition 51 If f vanishes at some point of U then so does everything in orb MZ7 f7 hence so does anything in the closure of this orbit So the orbit closure cant be dense7 ie7 f cant be cyclic note that this argument works for any plane domain Conversely7 if f vanishes nowhere on U then 1f E HlU7 and so by power series conver gence7 there is a sequence of polynomials pn that converges in HlU ie7 uniformly on com pact subsets of U to 1f Thus the sequence pnf7 which is contained in span orb MZ7 f7 converges in HlU to 17 and so 1 belongs to the closure of the linear span of orb MZ7 f Now the aforementioned linear span is MZ invariant7 hence so is its closure7 and so every polynomial belongs to this closure In other words7 the linear span of orb MZ7 f is dense in HlU7 ie7 f is cyclic for M1 CI Note that the argument of the last paragraph works for any simply connected plane domainiall that7s needed is the polynomials approximating 1f7 and in this generality the power series convergence argument gives way to Runge7s theorem In fact this character ization of cyclic vectors holds in my plane domain7 but now one needs to use the full strength of the fact that HG is a topological algebra with identity I leave this as an exercise for the interested reader7 noting only that the key to success is the fact that a proper closed ideal of HG is maximal if and only if it consists of all functions that vanish at a pre assigned point of G 247 Proposition 1397 page 110 Recall that all of our hypercyclic examples so far have been chaotic7 and therefore7 by Theorem 115 these maps have sensitive dependence on initial conditions In fact7 for linear operators7 hypercyclicity itself implies sensitive dependence LINEAR DYNAMICS 35 53 Proposition Every hypercyclic operator on an F space has sensitive dependence on initial conditions Proof Suppose T is hypercyclic on X and let HOT denote the collection of hypercyclic vectors for T Fix z E X and note that HOT z is a dense G5 subset of X because HOT has this property and translation by z is a homeomorphism of X Thus any neighborhood of z contains a point y of HOT x Since z 7 y is a hypercyclic vector for T the orbit of z exhibits the appropriate divergence from the orbit of y El Exercise If a continuous linear operator on an F space X is hypercyclic then every vector in X is a sum of two hypercyclic vectors So far it has been the suf cient condition 21 that has yielded hypercyclicity in all our examples In the rst two paragraphs of 3 we noted two ways in which this condition is more powerful than rst advertised a When properly rephrased it works for continu ous homomorphisms of complete metrizable topological groups b It provides not just hypercyclicity but a stronger property miccing Question Does every miping operator on an F space obey the hypotheses of Theorem 21 I dont know the answer to this one but by the end of this section I hope you7ll agree that the question is a reasonable one There is a corresponding question for hypercyclicity whose formulation depends on the observation that the proof of Theorem 21 our suf cient condition for hypercyclicity works just as well under much weakened hypotheses Here is the theorem that this proof actually gives 54 The Hypercyclicity Criterion Suppose T is a continuous linear transformation an F space X and that for some subsequence of positive integers oo a There epists a dense subset Y ofX on which TTLW converges to zero pointwise b There epists a dense subset Z ofX and a sequence Sk of mappings S Z a X not necessarily either continuous or linear such that i T llek converges pointwise on Z to the identity map on Z ii Sk converges to zero pointwise on Z Then T is hypercyclic on X 36 JOEL H SHAPIRO Note that if T satis es the hypercyclicity criterion then so does T 69 T7 ie7 in the termi nology of 377 T is weakly mixing It is an open question whether or not every hypercyclic operator on an F space satis es the hypotheses of the hypercyclicity criterion However in this direction we have a striking recent result of Res and Peris 4 55 Theorem Suppose T is a continuous linear transformation of a separable F spaee X Then T is weakly miping on X ie TEBT is hypercyclic on XEBX if and only ifT satis es the hypotheses of the hypercyclicity criterion Proof Its easy to see that if T satis es the hypotheses of the hypercyclicity criterion on X7 then the same is true of T 69 T on X 69 X7 hence T is weakly mixing Its the converse that requires some work Suppose7 then7 that T 69 T is hypercyclic on X 69 X Fix a vector Ly E X that is hypercyclic for T 69 T We will verify the hypotheses of the hypercyclicity criterion for T with Y Z orb T7 p dense in X because z is a hypercyclic vector for T the trick is to nd the subsequence and the approximate right inverses Sk Since T is hypercyclic7 its range is dense in X7 hence the range of TN is also dense for every positive integer N From this it is easy to check that for each N the vector LTNy is hypercyclic for T 69 T ln particular7 for each zero neighborhood U in X there is a vector u E U such that Lu is hypercyclic for T EDT Let7s denote the distance from a vector i E X to the origin by this is just for notational convenience unless X is a Banach space7 will generally not be a norm By the work of the last paragraph7 we can inductively choose a strictly increasing sequence of positive integers and a sequence of vectors in X so that 7 llT kxll7 and llT kluk 7 are all lt V k E N From the second of these conditions7 Tn a 0 pointwise on Y We de ne the approximate right inverses from Z Y a X by setting SkT xT uk k071727 The de nition is well made because z is a hypercyclic vector for T7 hence the points in its orbit must be distinct if two points of an orbit coincide7 the entire orbit is eventually LINEAR DYNAMICS 37 periodic hence nite Because a 0 it follows that Sk a 0 pointwise on Y Finally for each kn E N T kSkTquotz WWW TnT kuk a T z as k a 00 where the convergence on the right is provided by the last inequality of 7 above Thus T kSk a I pointwise on Y as desired I This proof has a curious consequence If an operator on an F space satis es the hypotheses of the hypercyclicity criterion then it satis es those hypotheses with Y Z 56 Hypercyclicity vs mixing At this point its appropriate to mention a striking example due to Hector Salas 33 of a bilateral weighted shift T on 2Z which along with its adjoint is hypercyclic By way of contrast note that in the Rolewicz example AB where B is the backward shift on 2 and A a scalar of modulus gt 1 the Banach space adjoint can be identi ed with A lS where S is the forward shift so T is in this case a strict contraction hence not hypercyclic This example and others like it made the Salas example seem quite surprising Furthermore in Salas7s example the weighting coef cients of the shift T are real so with respect to the standard orthonormal basis for 2Z the matrices for both T and T have only real entries It follows from this and an intriguing unpublished result of James Deddens see below that T 69 T is not even cyclicl Now we saw in Proposition 36 that any direct sum of mixing transformations is again mixing so in the Salas example either T or T is not mixing In particular 57 Theorem There eccist hypercyclic operators on Hilbert space that are not raiding hence do not satisfy the hypotheses of our rst su cient condition Theorem 2 Because the result of Deddens is striking useful easily proved and unpublished Id like to end this section by giving it a proper statement and proof 58 Deddensls Theorem 1982 Suppose T is a bounded linear operator on a separable Hilbert space H whose matrip with respect to some orthonormal basis ofH consists entirely of real entries Then T 69 T is not cyclic 38 JOEL H SHAPIRO Proof Suppose fg E H We7ll prove the theorem by writing down a vector in H 69 H that is orthogonal to the T 69 T orbit of f 9 Let 63O be the promised orthonormal basis for H relative to which all the matrix entries ltTe emgt are real Now f and 9 have representations f Zane and g anen relative to this basis with square summable coef cient sequences an and bn respectively Therefore there exist vectors 7 and g in H de ned by 7 EEG and 2E6 I claim that 77 is orthogonal in H DH to the orbit of f 9 Indeed for each non negative integer n ltT T f797 717 ltTmf7T 97 7 gt 7ltT f7 gt ltT 7gt lt7T f gt lt9 T 7gt lt7Tmf7 gt ltTn7 ggt lt7Tmf gt ltT f gt 07 where the next to last equality we nally use the fact that all the matrix coef cients ltTe emgt are real I 6 COMPOSITION OPERATORS ON H2 So far we7ve considered composition operators only on the full space HlU of functions holomorphic on the unit disc Now Id like to shift the scene to a more subtle setting the Hardy space H2 which is a subspace of HlU that in its natural norm is a Hilbert space H2 is arguably the best place to study the interaction between the theories of linear operators and analytic functions The purpose of this section is to prepare the way for the next one in which we7ll study hypercyclicity for linear fractionally induced composition operators on H2 discovering in the process some interesting contrasts with the HlU case In this section l7ll develop some basic properties of H2 and prove that that every compo sition operator restricts to a continuous mapping of H2 into itself If you7ve already had an introduction to Hardy spaces and composition operators on them skip this section What7s here comes almost verbatim from 34 Chapter 1 61 The Hardy space H2 For f E HlU and every non negative integer 71 let f 0nl Then the series 220 is the Taylor series of f with center at the origin LINEAR DYNAMICS 39 it converges uniformly on compact subsets of U to f The Hardy space H2 is the collection of functions f E HlU with 20 lt 00 We equip H2 with the norm that is naturally associated with its de nition 00 12 8 Hfll fn2gt and note that this norm arises from the natural inner product 9 ltf ggt Z fn n f g 6 H2 n0 Let T be the Taylor transformation77 from H2 into the sequence space 2 de ned by Tf The mapping T is clearly linear7 and from the de nition of the H2 norm7 it is an isometry HTfH for every f 6 H2 62 Proposition T maps H2 onto 2 In particular H2 is a Hilbert space in the inner product Proof Because square summable sequences are bounded7 a simple geometric series estimate shows that if the complex sequence a an30 lies in 2 then the associated power series 20 anz converges uniformly on compact subsets of U to an analytic function f By the uniqueness of power series representations7 an fn for every n7 hence Tf El so TH2 2 El Thus H2 is the sequence space 2 disguised as a space of analytic functions Note in particular that 63 Corollary The sequence of monomials 2 n 0 1 2 is an orthonormal basis 7 7 739 forHZ Some properties of the functions in H2 can be easily discerned from the de nition of the space Here is one 64 Growth Estimate For every f E H2 andz E U S 7 lzlz 12 4O JOEL H SHAPIRO Proof Use successively the triangle inequality and the Cauchy Schwarz Inequality on the power series representation for f lf2l lZfn2 lZlfnllzl m p lM8 ltgt A 3 V 4m V s 7 A M8 V 37 V s 7 l WNW The exercise below shows that the exponent in the Growth Estimate is best possible Exercise For oz real let faz 17 2f Show that fa E H2 if and only ifa lt 12 Suggestion Use the Binomial theorem and Stirling7s formula to show that na l 65 Corollary Convergence in H2 implies uniform convergence on compact subsets oflU Proof Suppose fn is a sequence of functions in H27 f is a function in H27 and anifll a 0 Our goal is to show that fn a f uniformly on compact subsets of U For this7 suppose K is a compact subset of U Let r maxlzl z E Then for z 6 K7 the Growth Estimate yields 1 WW2 7 1 WWW which shows that as n a 007 M fll riaxlfnw f2l S W E ie that fn a f uniformly on K I 0 7 However some properties of H2 do not follow easily from the de nition For example7 is every bounded analytic function in H2 In order to answer this question reasonably7 we need a different description of the norm LINEAR DYNAMICS 41 66 Proposition A function f E HlU belongs to H2 if and only if lim lfreltl2d0ltoo 74117 27139 When this happens the limit of integrals oh the left is Proof The functions eme form an orthonormal set in the space L20 27d hence for each 0 S r lt 1 the integral on the right is 220 The result now follows from the monotone convergence theorem I It is now an easy matter to show that every bounded function in HlU belongs to H2 In fact we can do better Let Hquot0 denote the collection of bounded analytic functions on U and for b E Hquot0 let HbHoo suplbzl z E U The integral representation given above for the H2 norm shows immediately 67 Proposition Ifb E H ahdf E H2 theh bf E H2 and Hbe S In particular upon taking f E 1 we obtain Corollary Ifb E H then b E H2 with S 68 Multiplication operators act on H2 Proposition 67 reveals an interesting class of linear transformations on H2 For b E Hquot0 let Mb denote the operator of pointwise multiplication by b That is be bf Clearly Mb when viewed as a mapping on all of HlU is linear note that for this we dont need b to be bounded According to Proposition 67 Mb maps H2 into itself with HbeH 3 for each f 6 H2 hence Mb is a bounded linear operator on H2 with norm S We call M5 the multiplication operator induced by b The most famous of these is the one induced by the identity map bz E 2 If we identify H2 with the sequence space 2 this mapping of multiplication by Z77 gets revealed as the forward shift on 2 which appeared in previous sections as the right inverse of the backward shift Do Composition operators act on H2 This is not a trivial question Suppose you have f E H2 and want to determine if wa 6 H2 Using the de nition of H2 we would substitute 2 for z in the power series expansion of f expand the various powers of the power series of p by the binomial theorem and regroup the resulting double series to identify the new powers of 2 which are now complicated 42 JOEL H SHAPIRO numerical series involving the coef cients of f and those of the powers of p Done this way7 there seems to be no reason why wa should be in H2 A calculation using the alternate characterization of H2 provided by Proposition 66 fares just as badly7 since it raises the specter of an unpleasant7 and possibly non univalent7 change of variable in an integral After these pessimistic observations7 it is remarkable that composition operators do pre serve the space H27 and do so continuously The key to this is the following result7 proved by Littlewood and published in 1925 69 Littlewood7s Subordination Theorem Suppose p is a holomorphie self map oflU and ltp0 0 Then CW is a contraction mapping on H2 Proof The proof is helped signi cantly by the backward shift operator B7 de ned on H2 by Bro Z n 1gt2 f 6 H2 n0 The name comes from the fact that B shifts the power series coef cients of f one unit to the left7 and drops off the constant term Clearly7 HBfH S for each f 6 H27 and one might expect this fact to play an important role in the proof7 but surprisingly it does not Only the following two identities are needed7 and they hold for any f E HlU 10 f2 MD zBf2 2 6 UL lt11 B f0fn no12gt To begin the proof7 suppose rst that f is a holomorphic polynomial Then foltp is bounded on U so by the work of the last section there is no doubt that it lies in H2 the real issue is its norm We begin the norm estimate by substituting ltpz for z in 10 to obtain fltPZ f0 ltP2Bfltr7z 2 E U Let us rewrite this equation in the language of composition and multiplication operators 12 wa f0 MWOWBf At this point7 the assumption ltp0 0 makes its rst and only appearance It asserts that all the terms of the power series for p have a common factor of 27 hence the same is true LINEAR DYNAMICS 43 for the second term on the right side of equation 127 rendering it orthogonal in H2 to the constant function f0 Thus7 13 WNW lf0l2 lleOwallZ S lf0l2 llOwall27 where the last inequality follows from Proposition 67 above since HltpHoo S 1 Now succes sively substitute Bf7B2f7 for f in 13 to obtain llOwallZ S le0l2 llOwBZfllZ llOwBZfllZ lt lef0l2 HOWBE J H2 HOwBWHZ lt lB f0l2lleB 1fll2 Putting all these inequalities together7 we get V L llwaHZ E Z lka0l2 H0an1fH2 k0 for each non negative integer 71 Now recall that f is a polynomial If we choose 71 be the degree of f7 then Bn f 07 and this reduces the last inequality to llwallZ E Z lka0l2 Z lfkl2 llfll27 k0 k0 where the middle line comes from property 11 of the backward shift This shows that CW is an HZ norrn contraction7 at least on the vector space of holornorphic polynornials To nish the proof7 suppose f E H2 is not a polynomial Let fnz 220 the n th partial sum of the Taylor series of f Then fn a f in the norm of H27 so by Corollary 65 fn a f uniformly on compact subsets of U hence fn o p a f o p in the same manner It is clear that S HfH7 and we have just shown that Hf 0 pH 3 Thus for each xed 0 lt 7 lt1 we have 1 7r it 2 i 1 7r it 2 iwlfnww m 6197313 wwlfnww m d6 hmsup llfnwll hmsup HM HfH To complete the proof7 let 7 tend to 17 and appeal one last time to Proposition 66 I 44 JOEL H SHAPIRO To prove that CW is bounded even when p does not x the origin7 we need to study conformal automorphisms of U from a different point of view For each point p 6 U7 de ne the holomorphic function Dip on U by p72 04172 The map so de ned belongs to Aut U7 interchanges p with the origin7 and is its own inverse see7 for example7 327 12271267 pp 2547256 Write p 0 Then the holomor 175239 phic function 7 04p 0 p takes U into itself and xes the origin By the self inverse property of 0417 we have p 04p 0 ii and this translates into the operator equation CW CwCap We have just seen that Cw maps H2 into itself Thus7 the fact that CW does the same will follow from the rst sentence of the next result 610 Lemma For each p E U 0 is a bounded liriear operator on H2 with 1 1 W 5 Hail 7 1 W Proof Suppose rst that f is holomorphic in a neighborhood of the closed unit disc7 say in RU lt R for some R gt 1 Then the limit in formula 66 can be passed inside the integral sign7 with the result that lt14 W W W This opens the door to a simple change of variable in which the self inverse property of Dip 2d6 gures prominently 1 7r weevil lfltapequotl2d0 27139 1 W i i g lf6tl2la6 ldt 1 W 39t 2 1 lplz l MW gt n 7W 1 2 1 7r 1 7 39 W lflt6itl2dt 1 lpl fZ 1will l LINEAR DYNAMICS 45 Thus the desired inequality holds for all functions holomorphic in RU in particular it holds for polynomials It remains only to transfer the result to the rest of H2 and for this we simply repeat the argument used to nish the proof of Littlewood7s Subordination Theorem I At this point we have assembled everything we need to show that composition operators map H2 into itself 611 Theorem Suppose p is a holomorphie self map of U Then CW is a bounded linear 1lltp0l CW 3 1 MON Proof As outlined earlier we have CW CwCap where p ltp0 and 7 xes the origin operator on H2 and Since each of the operators on the right hand side of this equation sends H2 into itself the same is true of CW As for the inequality this follows from Lemmas 69 and 610 I leave the details to you D 7 HYPERCYCLIC COMPOSITION OPERATORS ON H2 Now its time to consider hypercyclicity for composition operators on H2 The argument that showed such operators can only be induced by xed point free univalent holomorphic self maps of U works again to give the same result for H2 So we begin as before with linear fractional maps once again looking initially at the automorphisms In dealing with composition operators on HlU we could cavalierly map the unit disc to other domains ultimately identifying our operators as quasi factors of translation operators on this doesnt work in H2 where the situation is a lot more rigid so we have to take our stand pretty much in the unit disc The material of this section follows very closely that of 34 71 72 71 Theorem Suppose p E Aut U pes no point of U Then CW is hypercyclie on H2 Proof As discussed in 43 the automorphism p being non elliptic has a unique attractive xed point 04 E 3U If there is another xed point B then this too must lie on the unit circle since it is the attractive xed point for the inverse of p which is again an automorphism of 46 JOEL H SHAPIRO U Suppose rst that we are dealing with the case of two xed points We will produce the cast of characters required for the hypothesis of the Theorem 21 Let Y denote the set of functions that are continuous on the closed unit disc analytic on the interior and which vanish at a We claim that C 7 0 on Y For this note that for every C E 6U we have MK 7 oz hence if f E Y then fltpnC 7 fa 0 Since f and pn are continuous we can use boundary integral representation 14 of the H2 norm the Lebesgue Bounded Convergence Theorem yields the desired result n 1 7r 139 Nomi g lfltpn6 l2d0 7 0 n 7 00gt There are several ways to see that Y is dense in H2 Here is one based on elementary Hilbert space theory Suppose f E H2 is orthogonal to Y Then for every non negative integer n the polynomial z 1 7 042 belongs to Y so it is orthogonal to f 0 lt f z 1 7 042 gt 1 7 o7 n It follows upon iterating this identity that o7 f0 for each n Since 04 has modulus one all the Taylor coef cients of f have the same modulus and since f 6 H2 these coef cients must all be zero Thus the only H2 function orthogonal to Y is the zero function Since Y is a linear subspace of H2 it must therefore be dense We note for further reference that the only property required here of 04 is that it lie outside of U the argument actually shows fa lU then the set of polynomials that vanish at 04 is dense in H2 To nish the proof let S Cf CW4 As noted above p 1 is also an automorphism of the disc with attracting xed point B the repulsive xed point of ltp So if we take Z to be the set of continuous functions on the disc that are holomorphic in the interior and vanish at B then S maps Z into itself and the previous arguments apply to show that Z is dense and S 7 0 on Z The hypotheses of Theorem 21 are therefore satis ed so CW is hypercyclic indeed even mixing on H2 The case where p has just one xed point is even easier take Y as before and set Z Y I leave the details to you I 72 The Linear Fractional Hypercyclicity Theorem Suppose that p E LFTlU has no coed point in U Then LINEAR DYNAMICS 47 a 0 is hypercyclic on H2 unlcss p is a parabolic non automorphism b Ifltp is a parabolic non automorphism then 0 fails to bc hypercyclic in a very strong sense Only constant functions can bc limit points of CW orbits Proof of a We have already proved the result for automorphisms so it remains to do it for hyperbolic non automorphisms Let p be such a map and suppose oz and B are its xed points with oz the attractive one As before we seek to nd the dense sets Y and Z and the map S that will satisfy the hypotheses of Theorem 21 The space Y is exactly the one that worked in the automorphic hypercyclicity result and it works again with no change in the argument It is the space Z that requires some care Suppose rst that the repulsive xed point 6 lies on the line through the origin and oz but is on the other side of the origin from a Let A be the disc whose boundary is the circle perpendicular to this line that passes through oz and B so now U is inside A and all is tangent to 3A at a Since p xes oz and B and preserves angles it maps the boundary of A onto itself and therefore takes A onto either itself or the exterior of 3A But p takes lU into itself so the latter possibility is ruled out Therefore p is a conformal automorphism of A Let Z be the collection of functions that are continuous on A analytic on A and which vanish at B As we noted above the fact that 6 lies outside lU insures that the polynomials that vanish at 6 form a dense subset of H2 thus Z is dense De ne the map S Z a Z by 672 fltP 12 2 E U The fact that ltp 1z is not always in U is of no importance here nor is the fact that S is neither de ned nor bounded on H2 What is important is that ltp 1A C A that S is de ned on Z and that CWS is the identity on Z all of which is obvious In addition the fact that ltp lt a B for each C 6 all yields precisely as in the last section that S a 0 on Y Thus the hypotheses of the Theorem 21 are again satis ed so 0 is hypercyclic on H2 If the repulsive xed point B is not in the required position it could even be at 00 for example then there is a conformal automorphism ya of U that xes oz and takes 6 to the desired position This is a simple exercise instead of the unit disc work in the right half plane with 00 in place of a An appropriate af ne map gives the desired automorphism of 48 JOEL H SHAPIRO lP Then p yaozbo yujl where 7 is a linear fractional self map oflU that has its xed points arranged properly so Cw is hypercyclic and 0 is similar to Cw hence also hypercyclic El Remarks Once it has been observed that p is an automorphism ofthe larger disc A a more elegant line of argument suggests itself De ne the space H2A in some obvious way show that it is a dense subspace of H2 and has a stronger topology Then use the Automorphism Theorem to conclude that 0 is hypercyclic on H2A and transfer this hypercyclicity to H2 by noting that 0 on H2 is a quasi factor of 0 on H2A It is also interesting to investigate why the proof given above for hyperbolic non automorphisms does not work for parabolic ones The point is that in the hyperbolic case the big disc A is precisely the union of the successive inverse images of U under p A U ltp U V L If on the other hand p is a parabolic non automorphism then this union turns out to be o as is easily seen by representing the map as a translation of the right half plane strictly into itself hence the set Z de ned in the proof above contains only the zero function Proof of Now we assume that p E LFTlU is a parabolic non automorphism so it has only one xed point in S and this lies on 3U Without loss of generality we may take this xed point to be 1 otherwise conjugate p by an appropriate rotation to produce a similar composition operator induced by a parabolic automorphism with xed point at 1 We return to an earlier idea seeking to understand parabolic self maps of U by mapping the unit disc to the right half plane where the parabolic map becomes a translation Since our particular map xes the point 1 we use the transformation w 1 7 2 which takes our the original map p to an LFT ltIgt that maps l into itself and xes only the point at 00 Thus ltIgt is translation of the right half plane w w a where necessarily Rea gt 0 Rea is 2 0 because ltIgtll gt C P and gt 0 because ltIgt is not an automorphism of lP Similarly the n th iterate on of p gets transformed into translation by mm ltIgtnw w no Our proof will depend on knowing how quickly the p orbits of points in U get close to each other and to the attractive xed point 1 Suppose z in U and w is the corresponding point in P so 2 1 w 7 1 and z w1 w LINEAR DYNAMICS 49 Then 2 2 4Rew 1 7 z w 1 lw 1 2 It follows7 then7 that if z 6 U7 w is the point of l corresponding to ltpz7 and we that of and lilz ltp07 then w no corresponds to pnz and we no to pn0 Thus 4Re w nRe a 17 n 2 W M w w z and 2w0 7 w W W0 7 from which follows lt15 31330710 7 WM c1 and gm nzwa e 0420 where 01 and 02 are non zero constants that depend on 2 and 1 Now x f 6 H2 Our goal is to show that if the orbit of f under CW clusters at some 9 6 H27 then 9 must be a constant function For this we need a growth estimate on dz erences of functional values that is analogous the one obtained in Growth Estimate 64 for the values themselves We begin with the derivative For 2 6 U7 2 lf 2l2 Zn mzm S lt2n222n1gt Wm Upon taking square roots on both sides of the last inequality7 we get this growth estimate on the derivative of f Z lt 2 m To get the desired estimate on differences7 suppose 27w E U and S To estimate zelU fz 7 fw we integrate f over the line segment joining z and w7 and use the inequality 50 JOEL H SHAPIRO above lf2fwl c firow w MG 3 i ll lwizl S llfll Thus for each pair of points 27 w 6 U7 lwizl 16 lf2fwl S llfllm In 16 above7 substitute pnz for 27 and pn0 for w7 and use the estimates of 15 above the result is lflt n2flt n0l WHWN min1lz171l n0l32 lt t n 2 7 const 7 cons 7 7 where the constant in each line depends on f7 27 and p7 but not on n Thus 17 11glflt n2 f0l 0 2 E U To nish the argument7 suppose g E H2 is a cluster point of the CW orbit of f Then for some sequence nk 00 we have f 0 pm a g in the norm of H27 and therefore pointwise on U By 17 this implies 92 90 lignlfwndzh fltm0l 07 hence g E 90 Thus only constant functions can be limit points of the CW orbit of an H2 function I Hypercyclicity for more general composition operators on H2 Suppose p is a holomorphic self map of U7 with linear fractional model 7156 As usual7 denote by U the Riemann map of U onto G The basic principle behind operating here is If the polynomials in o are dense in H2 then cyclic behavior for Cw on H2 can be transferred to cyclic behavior for CW For example7 suppose Cw is hypercyclic on H27 so that 7 is a holomorphic self map of U that7s either hyperbolic or a parabolic automorphism According to the Linear Fractional LINEAR DYNAMICS 51 Model Theorem7 we may take G C U7 so if we let V HlU a HlU denote Ga acting on the restrictions to G of functions in H27 we can interpret the functional equation 1 o o o o p as asserting that VGw GWV Now our hypothesis on o is that V has dense range in H27 so 0 is a quasi factor of Cw and therefore 0 inherits the hypercyclicity of CW The same holds for other concepts such as cyclicity7 chaos7 mixing One way to insure that the polynomials in o are dense in H2 is to employ 73 Walshls Theorem Suppose G is a simply connected domain whose boundary is a Jor dan curve Let the holomorphic function a map lU uniyalently onto G Then the polynomials in o are dense in H2 The result that is usually called Walsh7s Theorem actually asserts that the polynomials in z are uniformly dense in AG7 the subalgebra of CG consisting of functions holomorphic on G see7 for example7 267 Theorem 397 page 98 A theorem of Caratheodory asserts that F extends continuously and univalently to G7 so Walsh7s original result asserts7 in our situation7 that the polynomials in F are dense in AUU7 which is clearly dense in H27 and this yields Theorem 73 see 347 81 for more details The main point of the monograph 7 is to nd conditions on a univalent holomorphic self map p of U which guarantee that G is the interior of a Jordan curve The main result shows that If the Denjoy Wol point to ofltp lies on 3U and the closure of ltplU touches all only at w and p has quotsu cient di erentiability at u then the composition operator induced on H2 by p has the same hypercyclic be havior as its linear fractional model ie 0 is hypercyclic indeed chaotic and miccing ifltp is of hyperbolic or parabolic automorphic type and not hypercyclic if p is of parabolic non automorphic type Here suf cient differentiability77 varies from case to case7 but G4 works for all of them Of course the negative result about parabolic non automorphic type maps cannot be deduced from regularity of the model lnstead7 enough differentiability is assumed at the Denjoy Wolff point to allow the estimates that worked in the linear fractional case to be used for the more general one For speci c references to these results7 see Table ll on page 12 of 52 JOEL H SHAPIRO 7 To give you a feeling for how the arguments go I present below the hyperbolic case taken almost word for word from 34 83 pp 1347137 In order to simplify notation let7s agree that a Jordari domain is a simply connected plane domain whose boundary is a Jordan CUI VS 74 A Valiron type Theorem Suppose p is a uriiualerit holomorphic self map oflU of hyperbolic type with CZ smoothriess at its Derijoy Wol poirit u Suppose further that the closure ofltplU lies in UUw Theri there epists a hyperbolic 7 E LFTlU arid a holomorphic uriiualerit map a oflU onto a Jordari domain coritairied in U such that o o p 7 o a Proof We may without loss of generality suppose that the Denjoy Wolff point is 1 Write A ltp 1 let H denote the half plane Rez lt 1 and set 72 A2 1 7 A so 7 is a hyperbolic automorphism of H We7ll rst nd a mapping a with all the required properties except that its image will lie in H The theorem as stated will follow upon mapping H conformally onto U We7ll be able to copy the original Koenigs argument almost word for word if we use the map 2 7 1 7 z to map 1U onto the open disc U0 of radius one and center 1 Let7s still use the notation p for the resulting self map of U which now xes the origin at which it is assumed to be CZ smooth This means that p has the nite Taylor expansion77 18 ltpz A2 22B2 z 6 U0 where 0 lt A ltp 0 lt 1 and the function E is bounded on the closure of U0 The idea is to resurrect Koenigs7s original proof for the interior xed point case but where Koenigs used the Schwarz Lemma we will employ an estimate derived from 18 We will obtain a solution a of Schroder7s equation 0 o p AU on U0 with o a Jordari map with positive real part We proceed just as did Koenigs obtaining a as a limit of normalized iterates on 5pr Note rst that since p has positive real part so does each of its iterates and therefore so does each map on We claim that the sequence on converges uniformly on U0 For this let 6 maxlBzl z 6 U0 and let A be the intersection of U0 with the closed disc of radius 1 7 A2 centered at the origin By 18 above we have lt19 lltp2l 121 lt2 e A LINEAR DYNAMICS 53 Since 1 A2 lt 17 this last inequality insures that A C A7 so inequality 19 can be iterated for each 2 E A to yield 1 1 2 1 1 nltzgt1 1 n71lt2gt1 1 n72lt2gt1 7 121 By our de nition of A7 this last estimate shows that 1 A 1 A n lt lt20 1mm 7 23 2 zem for each non negative integer 71 Note that the origin is now playing the role of the Denjoy Wolff point for p7 so pn 7 0 uniformly on compact subsets of U0 We are assuming that p takes U0 into U0 U 07 so the closure of U0A in U0 is compact Thus pn 7 0 uniformly on U0 ln particular7 there is a positive integer N such that pnU0 C A n 2 N Following Koenigs7 we set Fz 7 z e no and note that for each 2 6 U0 the expansion 18 implies 21 11F2l llzllB2l Skil lzl Now x 2 6 U0 lfj 2 N then pgz 6 A7 so using respectively 21 and 20 above with j 7 N in place of n and pNz7 which belongs to A7 in place of z7 we obtain 7 N 117 Fwy2 Emwwzm g for each 2 6 U0 Since N is independent of the point z E U07 this last inequality shows that each term of the in nite series 11 7 is dominated by the corresponding term of a convergent geometric series Thus 2 l1 7 converges uniformly on U07 by the Weierstrass M test Since this convergence is passed on to in nite product 00 HFWAZD 2 1 11m Un27 3970 Hoe 7 the sequence an therefore converges uniformly on U0 to a function 039 that xes the origin7 is continuous on U07 is holomorphic and univalent on on7 and obeys Schro39der7s equation on U0 Furthermore7 039 has positive real part on U0 since it is non constant there and7 as we noted above7 each an has positive real part 54 JOEL H SHAPIRO Thus7 in order to show that 0U0 is a Jordan domain7 it only remains to check that o is univalent on all But this follows from Schroder7s equation and the univalence of p on U0 The argument is this If 021 022 for a pair of points 21722 E 3U then upon multiplying by A and using Schroder7s equation which we have just proved holds on the closed disc we see that oltp21 oltp22 If neither 21 nor 22 is zero7 then both p images belong to on7 on which we know a is univalent Thus 21 ltp227 so 21 22 since p is assumed to be univalent on U0 Suppose on the other hand that one ofthe original points7 say 21 is zero Then Schroder7s equa tion and the fact that 00 0 yield oltp22 0 But if 22 31 07 then 22 E on7 contradicting the fact that Rea gt 0 on U0 Thus 22 07 so a is univalent on U0 ln summary7 we have shown that there is a continuous7 univalent map a de ned on U0 that has non negative real part7 is holomorphic on on7 and satis es Schroder7s equation ooltp AU on U0 Upon transferring this result back to the unit disc by means of the map 2 gt gt 1 7 2 our accomplishment looks like this Ifltp obeys the hypotheses of the Theorem then it has a Jordan model 1150 where 7M2 A2 17 A and 0 maps lU into the half plane Rez lt 1 The only problem remaining is that G need not lie in U7 but this is easily remedied Let 739 be a linear fractional transformation that takes the half plane Rez lt 1 onto the unit disc7 and xes 1 Necessarily 739 E LFTlU De ne o 72 739 o w o 7 17 another member of LFTlU with attractive xed point at 17 0 6 739G7 a Jordan sub domain of U7 and o If 739 0 a7 a univalent mapping of U onto Since 7 is hyperbolic7 so is 757 and E o p 71 0 57 so 757 is the desired linear fractional model for p I 8 WHY HYPERCYCLICITY IS VERY INTERESTING In this section we show that hypercyclic operators have special properties not possessed by general transitive mappings LINEAR DYNAMICS 55 We observed in the rst section of these notes that nontrivial transitive mappings might have non transitive squares in fact the nontrivial permutation map of the discrete metric space 1 2 onto itself as this property as does every product of this map with a transitive one With this construction we see that even continuous group homomorphisms can be transitive without having transitive squares Carol Kitai in her 1982 dissertation 22 Remark 213 asked if this could happen in the linear setting and in 1995 Shamim Ansari gave a striking argument to show that that it cannot 81 Ansarils Theorem IfT is a hypercyclic operator on a metrizable topological vector space X theri T is hypercyclic for arty positive iriteger 71 Here is a way to think about trying to prove Ansari7s Theorem Let x be a hypercyclic vector for T and x an integer n gt 1 Then 22 orb T p orb T p U orb Tn Tx U orb Tn Tn lp Now in any topological space if a nite union of sets is dense at least one of the sets must be somewhere dense ie its closure must contain a nonempty open set To see why this is so we may assume the collection of sets with dense union is minimal ie if we remove one them then union of what remains is not dense So remove one of the sets in this minimal collection The closure of what remains misses a nonvoid open subset of the space and this open subset must therefore belong to the closure of what was removed So that removed set is somewhere dense Returning to our hypercyclic situation 22 therefore guarantees that least one of the sets orb T Tkx is somewhere dense Thus Ansari7s Theorem will be proved if we can show that for linear operators somewhere dense orbits are everywhere dense Note that our examples of nonlinear transitive maps with nontransitive squares also show that in the general situation somewhere dense orbits need not be dense Here is another open question this one posed by Domingo Herrero in 1992 20 Suppose p1z2 wN is a nite subset of an F space X and UL orb T x7 is dense in X is T hypercyclic on X Herrero7s question was just recently answered in the af rmative indepen dently by George Costakis and Alfredo Peris To get a nice statement let7s call an operator for which a nite union of orbits is dense multihypercyclic 56 JOEL H SHAPIRO 82 The CostakisPeris Theorem 9 27 Every multl hypercycllc operator is hyper cyclic Note that like Ansari7s theorem this result would follow easily if we could prove that for linear operators somewhere dense orbits are everywhere dense Peris in 27 posed this problem explicitly and within the last few months Paul Bourdon and Nathan Feldman solved it af rmatively thus providing a uni ed proof of both Ansari7s Theorem and the Costakis Peris Theorem 83 The Bourdon Feldman Theorem Suppose T is a continuous linear operator on a locally conuep F space X and z E X If orb T p is somewhere dense in X then it is dense in X Proof Actually Bourdon and Feldman prove their result for any locally convex Hausdorff topological vector space over the complex scalars You7ll see below that neither completeness nor metrizability ever plays an essential role in the argument However the Hahn Banach theorem enters at one point in the proof so local convexity is required I dont know if this result is true in topological vector spaces that are not locally convex For convenience we7ll employ the following notation throughout the proof 0 orb T n will henceforth be abbreviated just orb o The closure of orb will be denoted clorb o The interior of the clorb will be denoted clorb o 73 is the collection of holomorphic polynomials with complex coef cients If S C 73 and y E X then 8T pT p E S and STy pTy p E S In this notation to say that a vector y E X cyclic for an operator T on X is to assert that 73Ty is dense in X The proof will be broken up into ve steps Throughout we assume z E X has a some where dense orbit ie clorbO 31 Q although in Step I this will not be used STEP 1 Ify E orb then clorb clorb Proof That clorb C clorb follows from the corresponding set containment for orbits Now clorb differs from clorb by just a nite set of isolated points from which follows the reverse containment El LINEAR DYNAMICS 57 STEP H Each element of orb is a cyclic vector for T Proof Suppose y E orb By Step I orb is somewhere dense and since orb C orb C 73Tx we see that 73Tx is somewhere dense Now 73Tx is a vector space hence so is its closure Since this closure contains an open set it is the whole space Thus y is cyclic for T I The next step provides the crucial element of the argument To put it in perspective note that clorb is T invariant STEP III The complement in X of clorb ls T z39nuarz39ant Proof By Step I we may replace x by any element of its orbit without disturbing clorbo Now clorbo is a nonempty open set each point of which is a limit point of orb a so some point of orb belongs to clorbO Replace x by this point In other words Without loss of generality we may henceforth assume that z E clorbo Suppose in order to reach a contradiction that Xclorbo is not T invariant ie that for some y clorbO we have Ty E clorbO Actually We may assume that y clorb Indeed if were unlucky and y is in clorb then since its not in clorb it must be on the boundary of clorb In particular there is a point y clorb a but close enough to y that Ty is close enough to Ty to keep it in the open set clorbO we use the continuity of T here Then rename y as y We may also assume that y pT for some p E 730 This is a similar argument z is cyclic for T Step I ie 73Tx is dense in X so we can nd p E 73 so that pTx is close enough to y that it lies in the open complement of clorb a and so that its T image stays in clorb continuity of T and open ness of clorb again Note that p is not the zero polynomial since pTx 31 TpTx Now because clorb is T invariant and contains TpT clorb D T pTx pTT 1x n 012 hence clorb D pT orb Upon taking closures 23 clorb D pTclorb Ta 3 pTclorb Ta pTclorb a 58 JOEL H SHAPIRO the last equality following from Step I But z E clorbo recall that we showed earlier that there was no loss of generality in assuming this7 so by 23 above7 pTx E clorb But this is a contradiction we have chosen p so that pTx clorb I Step IV 73T has dense range for every p E 730 Proof This is the only place where we use local convexity and the fact that the scalar eld is C Fix p E 730 and factor it into linear factors 102 z 7 a1z 7 042 z 7 at where 0417 704 are complex numbers Then pT has a similar decomposition into linear factors T 7 avgI7 so it is enough to show that each such factor has dense range For this7 x 04 E C and suppose the range of T 7 04 is not dense in X Then7 thanks to local convexity7 the Hahn Banach Theorem provides a continuous linear functional A on X that annihilates ranT 7 0417 but is not identically zero on X Now A o T 7 04 07 so A o T 04A We are saying7 of course7 that 04 is an eigenvalue of T with eigenvector A Now just as in the proof of Theorem 27 we have 24 Aorb a Ax n 07 17 27 In this case Aclorb is a nonvoid open subset of C continuous linear functionals are open maps7 hence Aorb is somewhere dense in C But its a simple exercise to check that oz Ap is nowhere dense in C7 regardless of the value of Ax7 so thanks to 24 we have arrived at a contradiction This all began with the assumption that ran T7 04 is not dense in X7 so that couldnt have been correct I STEP V Completion of the proof We are assuming that clorb 31 Q and want to show that orb is dense7 ie7 that clorb X So suppose not Recall that z is cyclic for T Step ll7 so 73Tx is dense in X7 and therefore one can nd a subcollection Q C 730 of polynomials such that QTz is a dense subset of the nonvoid open set Xclorb p7 and therefore a dense subset of Xclorb We showed in Step III that this latter set is T invariant7 so QTorb C Xclorb p7 hence by continuity of T 25 Xclorb D QT orb D QT clorb For convenience7 write 73 for 730 LINEAR DYNAMICS 59 CLAIM p E 73 i pTx 3clorbo Suppose we have proved this Claim The vector subspace 73Tz7 being dense in X7 is in nite dimensional7 so there is no doubt that CIDT is connected3 This connected set is the disjoint union of two subsets G PT clorb and H PT Xclorb Clearly G is relatively open in 73 Tx7 and thanks to the Claim7 so is H Furthermore7 neither G nor H is empty z 6 G7 and QTz C H This contradicts the connectedness of 73 Tx7 and nishes the proof It remains to prove the Claim Suppose for the sake of contradiction that pTz E 6clorb for some p E PT Consider the set D clorbO U QTz7 which is dense in X because the collection Q of polynomials has been chosen to make QT dense in the complement of clorb Now pTD pTclorb U 10TQTx7 and by the T invariance of Xclorbo Step lll7 the second term on the right lies in Xclorb But so does the rst term on the right lndeed7 pTx clorb because its assumed to be in 6clorb so by a now familiar argument7 pTclorb C Xclorb Thus pTD lies entirely in Xclorb s7 so is disjoint from the nonvoid open set clorb s7 thus contradicting the density of D This completes the proof of the Claim7 and with it7 the proof of the Bourdon Feldman Theorem I The Bourdon Feldman Theorem is a beautiful piece of work7 but dont let my introduction to it fool you into thinking it makes trivial the results of Ansari and Costakis Peris The proof of the Bourdon Feldman Theorem has many elements that went into proving the previous two results As we mentioned earlier7 the original question it answers was asked by Peris in 27 furthermore the phenomena of cyclicity and connectedness play an important role in Ansari7s proof 3Since we work with complex scalars here7 all thatls really needed is that the vector space have dimension 07 however the argument above shows that this part of the argument works even for real scalarsi 6O JOEL H SHAPIRO REFERENCES 1 So 1 Ansari Hypercyclic and cyclic vectors 1 Functional Analysis 128 1995 37473831 2 1 N1 Baker and Ch Pommerenke On the iteration of analytic functions in a halfplane II J London Math Soc 2 20 1979 25572581 J1 Banks J1 Brooks Go Cairns Go Davis and P Stacey On Devaney s de nition of chaos American Mathi Monthly 99 1992 3327334 J Res and A1 Peris Hereditarily hypercyclic operators 1 Functional Analysis 167 1999 947112 Go D1 Birkhoff Demonstration d un theoreme elementaire sur les fonctions entieres ClRi Acad Sci Paris 189 1929 47374751 P So Bourdon and N S Feldman Somewhere dense orbits are everywhere dense preprint Washington and Lee University January 20011 P So Bourdon and J1 H1 Shapiro Cyclic phenomena for composition operators Memoirs Amer Math Soc 596 American Mathematical Society Providence R1 19971 8 Kl C1 Chan and 1 H1 Shapiro The cyclic behavior of translation operators on Hilbert spaces of entire functions Indiana Univ Math J 40 1991 142171449 9 Go Costakis On a conjecture of D Herrero concerning hypercyclic operators C1 R1 Acadi Scili Paris 330 2000 17971821 10 C1 C1 Cowen Iteration and the solution of functional equations for functions analytic in the unit disc Trans Amer Math Soc 265 1981 697951 11 C1 C1 Cowen Composition operators on H2 1 Operator Th 9 19837771061 12 C1 C1 Cowen and T L Kriete lll Subnor39mality and composition operators on H2 1 Functl Anali 81 1988 298 319 13 C1 C1 Cowen and B1 D1 MacCluer Composition Operators on Spaces of Analytic Functions CRC Press Boca Raton 19951 14 A1 Denjoy Sur l it ration des fonctions analytiques ClRi Acad Sci Paris 182 1926 25572571 15 RiLl Devaney An Introduction to Chaotic Dynamical Systems second ed AddisonWesley Reading Mass 1989 16 Pl En o On the invariant subspace problem for Banach spaces Acta Math 158 1987 21373131 17 R1M1 Gethner and J1H1 Shapiro Universal vectors for operators on spaces of holomorphic functions Proc Amer Math Soc 100 1987 28172881 18 Go Godefroy and J H Shapiro Operators with dense invariant cyclic vector manifolds J1 Functl Anali 98 1991 22972691 19 C1 Go Grosse Erdmann Universal families and hypercyclic operators Bull l Amer Math Soc 36 1999 34573811 20 Do A Herrero Hypercyclic operators and chaos J1 Operator Theory 28 1992 937103 21 Do A Herrero and Z Wang Compact perturbations of hypercyclic and supercyclic operators Indiana Univ Math 1 39 1990 8198301 2 C1 Kitai Invariant closed sets for linear operators Thesis Univ of Toronto 1982 23 Go Koenigs Recherches sur les int grales de certaines quationes functionelles Annales Ecole Normale Superior 3 1 1884 Supplement 3741 Do H1 Luecking and L A Rubel Complex Analysis A Functional Analysis Approach SpringerVerlag 1984 E U SE B amp 25 Go R1 MacLane Sequences of derivatives and normal families J1 D7Analyse Math 2 1952 727871 26 A1 1 Markushevich Theory of Functions of a Complex Variable Vol 111 Prentice Hall 19671 27 A1 Peris Multihypercyclic operators are hypercyclic Math Z 236 2001 77977861 28 Chi Pommerenke On the iteration of analytic functions in a halfplane I J London Math Soc 2 19 1979 4397441 29 C Read On the invariant subspace problem for Banach spaces Israel 1 Math 63 1988 1401 30 C Robinson Dynamical Systems CRC Press 1995 31 So RoleWicz On orbits of elements Studia Math 33 1969 177221 32 VW Rudin Real and Complex Analysis Third Edition McGraWHill New York 19871 33 H1 Salas A hypercyclic operator whose adjoint is also hypercyclic Proc Amer Math Soc 112 1991 76577701

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