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Nonlinear Dynamics

by: Kylie Bartoletti DVM

Nonlinear Dynamics PHYS 7224

Kylie Bartoletti DVM

GPA 3.67

Predrag Cvitanovic

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Predrag Cvitanovic
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This 0 page Class Notes was uploaded by Kylie Bartoletti DVM on Monday November 2, 2015. The Class Notes belongs to PHYS 7224 at Georgia Institute of Technology - Main Campus taught by Predrag Cvitanovic in Fall. Since its upload, it has received 35 views. For similar materials see /class/234272/phys-7224-georgia-institute-of-technology-main-campus in Physics 2 at Georgia Institute of Technology - Main Campus.


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Date Created: 11/02/15
A brief history of chaos Laws of attribution 1 Arnol d s Law everything that is discovered is named after someone else including Arnol d s law 2 Berry s Law sometimes the sequence of antecedents seems endless S0 nothing is discovered for the rst time 3 Whiteheads s Law Eherything of importance has been said before by someone who did not discover it MV Berry A 1 Chaos is born R Mainieri Trying to predict the motion ofthe Moon has preoccupied astronomers since antiquity Accurate understanding of its motion was important for determining the longitude of ships while traversing open seas Kepler s Rudolphine tables had been a great improvement over previ ous tables and Kepler was justly proud of his achievements He wrote in the introduction to the announcement of Keplerls third law Harmonice Mundz39 Linz 1619 in a style that would not y with the contemporary Physical Review Letters editors What 1 prophesied twoandtwenty years ago as soon as 1 dis covered the ve solids among the heavenly orbitsiwhat I rmly believed long before 1 had seen Ptolemy s Harmonickwhat 1 had promised my friends in the title of this book which I named be fore l was sure of my discoveryiwhat sixteen years ago 1 urged as the thing to be soughtithat for which ljoined Tycho Brah for which I settled in Prague for which 1 have devoted the best part of my life to astronomical contemplations at length l have brought to light and recognized its truth beyond my most san guine expectations It is not eighteen months since 1 got the rst glimpse of light three months since the dawn very few days since the unveiled sun most admirable to gaze upon burst upon me Nothing holds me 1 will indulge my sacred fury 1 will triumph over mankind by the honest confession that l have stolen the golden vases 0f the Egyptians to build up a tabernacle for my God far away from the con nes of Egypt If you forgive me 1 rejoice if you are angry I can bear it the die is cast the book is A1 Chaos is born 129 A2 Chaos grows up 132 A3 Chaos with us 134 Further reading 136 A4 Periodic orbit theory 137 A5 Death of the Old Quantum Theory 137 Further reading 139 130 A brief history of chaos written to be read either now or in posterity I care not which it may well wait a century for a reader as God has waited six thousand years for an observer Then came Newton Classical mechanics has not stood still since Newton The formalism that we use today was developed by Euler and Lagrange By the end of the 1800 s the three problems that would lead to the notion of chaotic dynamics were already known the threebody problem the ergodic hypothesis and nonlinear oscillators A11 Threebody problem Bernoulli used Newton7s work on mechanics to derive the elliptic orbits of Kepler and set an example of how equations of motion could be solved by integrating But the motion of the Moon is not well approximated by an ellipse with the Earth at a focus at least the effects of the Sun have to be taken into account if one wants to reproduce the data the classical Greeks already possessed To do that one has to consider the motion of three bodies the Moon the Earth and the Sun When the planets are replaced by point particles of arbitrary masses the problem to be solved is known as the threebody problem The threebody problem was also a model to another concern in astronomy In the Newtonian model of the solar system it is possible for one of the planets to go from an elliptic orbit around the Sun to an orbit that escaped its dominion or that plunged right into it Knowing if any of the planets would do so became the problem of the stability of the solar system A planet would not meet this terrible end if solar system consisted of two celestial bodies but whether such fate could befall in the threebody case remained unclear After many failed attempts to solve the threebody problem natu ral philosophers started to suspect that it was impossible to integrate The usual technique for integrating problems was to nd the conserved quantities quantities that do not change with time and allow one to relate the momenta and positions different times The rst sign on the impossibility of integrating the threebody problem came from a result of Burns that showed that there were no conserved quantities that were polynomial in the momenta and positions Burns7 result did not preclude the possibility of more complicated conserved quantities This problem was settled by Poincare and Sundman in two very different ways In an attempt to promote the journal Acta Mathematica Mittag Lef er got the permission of the King Oscar H of Sweden and Norway to establish a mathematical competition Several questions were posed although the king would have preferred only one and the prize of 2500 kroner would go to the best submission One of the questions was formulated by Weierstrass Given a system of arbitrary mass points that attract each other according to Newton s laws under the assumption that no two points ever collide try to nd a representation of the coordi nates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniform y This problem whose solution would considerably extend our understanding of the solar system Poincar ls submission won the prize He showed that conserved quanti ties that were analytic in the momenta and positions could not exist To show that he introduced methods that were very geometrical in spirit the importance of state space flow the role of periodic orbits and their cross sections the homoclinic points The interesting thing about Poincar ls work was that it did not solve the problem posed He did not nd a function that would give the coor dinates as a function of time for all times He did not show that it was impossible either but rather that it could not be done with the Bernoulli technique of nding a conserved quantity and trying to integrate lnte gration would seem unlikely from Poincar ls prizewinning memoir but it was accomplished by the Finnishborn Swedish mathematician Sund man Sun man showed that to integrate the threebody problem one had to confront the twobody collisions He did that by making them go away through a trick known as regularization of the collision manifold The trick is not to expand the coordinates as a function of time t but rather as a function of To solve the problem for all times he used a conformal map into a strip This allowed Sundman to obtain a series expansion for the coordinates valid for all times solving the problem that was proposed by Weirstrass in the King Oscar llls competition e Sundmanls series are not used today to compute the trajectories of any threebody system That is more simply accomplished by numer ical methods or through series that although divergent produce better numerical results The conformal map and the collision regularization mean that the series are effectively in the variable 1 7 e 3 Quite rapidly this gets exponentially close to one the radius of convergence of the series Many terms more terms than any one has ever wanted to compute are needed to achieve numerical convergence Though Sund manls work deserves better credit than it gets it did not live up to Weirstrassls expectations and the series solution did not considerably extend our understanding of the solar system77 The work that followed from Poincare di A12 Ergodic hypothesis The second problem that played a key role in development of chaotic dynamics was the ergodic hypothesis of Boltzmann Maxwell and Boltz mann had combined the mechanics of Newton with notions ofprobability in order to create statistical mechanics deriving thermodynamics from the equations of mechanics To evaluate the heat capacity of even a simple system Boltzmann had to make a great simplifying assumption of ergodicity that the dynamical system would visit every part of the phase space allowed by conservation laws equally often This hypothesis was extended to other averages used in statistical mechanics and was A1 Chaos is born 131 132 A brief history of chaos called the ergodic hypothesis It was reformulated by Poincare to say that a trajectory comes as close as desired to any phase space pointi Proving the ergodic hypothesis turned out to be very dif cult By the end of twentieth century it has only been shown true for a few systems and wrong for quite a few others Early on as a mathematical neces sity the proof of the hypothesis was broken down into two parts First one would show that the mechanical system was ergodic it would go near any point and then one would show that it would go near each point equally often and regularly so that the computed averages made mathematical sensei Koopman took the first step in proving the ergodic hypothesis when he noticed that it was possible to reformulate it using the recently developed methods of Hilbert spaces This was an impor tant step that showed that it was possible to take a finitedimensional nonlinear problem and reformulate it as a infinitedimensional linear problemi This does not make the problem easier but it does allow one to use a different set of mathematical tools on the problemi Shortly after Koopman started lecturing on his method von Neumann proved a version of the ergodic hypothesis giving it the status of a theoremi He proved that if the mechanical system was ergodic then the computed averages would make sense Soon afterwards Birkhoff published a much stronger version of the theoremi A13 Nonlinear oscillators The third problem that was very in uential in the development of the theory of chaotic dynamical systems was the work on the nonlinear os cillatorsi The problem is to construct mechanical models that would aid our understanding of physical systems Lord Rayleigh came to the problem through his interest in understanding how musical instruments generate sound In the first approximation one can construct a model of a musical instrument as a linear oscillatori But real instruments do not produce a simple tone forever as the linear oscillator does so Lord Rayleigh modified this simple model by adding friction and more realis tic models for the spring By a clever use of negative friction he created two basic models for the musical instruments These models have more than a pure tone and decay with time when not strokedi In his book The Theory of Sound Lord Rayleigh introduced a series of methods that would prove quite general such as the notion of a limit cycle a periodic motion a system goes to regardless of the initial conditions A2 Chaos grows up Rf Mainieri The theorems of von Neumann and Birkhoff on the ergodic hypothe sis were published in 1912 and 1913 This line of enquiry developed in two directions One direction took an abstract approach and considered dynamical systems as transformations of measurable spaces into them selves Could we classify these transformations in a meaningful way This lead Kolmogorov to the introduction of the concept of entropy for dynamical systems With entropy as a dynamical invariant it became possible to classify a set of abstract dynamical systems known as the Bernoulli systems The other line that developed from the ergodic hypothesis was in trying to nd mechanical systems that are ergodici An ergodic system could not have stable orbits as these would break ergodicityi So in 1898 Hadamard published a paper with a playful title of in billiards M where he showed that the motion of balls on surfaces of constant negative curvature is everywhere unstablei This dynamical system was to prove very useful and it was taken up by Birkhoffi Morse in 1923 showed that it was possible to enumerate the orbits of a ball on a surface of constant negative curvaturei He did this by introducing a symbolic code to each orbit and showed that the number of possible codes grew exponentially with the length of the coder With contribu tions by Artin Hedlund and Hi Hopf it was eventually proven that the motion of a ball on a surface of constant negative curvature was ergodici The importance of this result escaped most physicists one exception being Krylov who understood that a physical billiard was a dynamical system on a surface of negative curvature but with the curvature con centrated along the lines of collision Sinai who was the rst to show that a physical billiard can be ergodic knew Krylovls work well The work of Lord Rayleigh also received vigorous development It prompted many experiments and some theoretical development by van der Pol Duf ng and Hayashii They found other systems in which the nonlinear oscillator played a role and classi ed the possible motions of these systems This concreteness of experiments and the possibility of analysis was too much of temptation for Mary Lucy Cartwright and JiEi Littlewood who set out to prove that many of the structures conjectured by the experimentalists and theoretical physicists did indeed follow from the equations of motion Birkhoff had found a remarkable curve77 in a two dimensional map it appeared to be nondifferentiable and it would be nice to see if a smooth flow could generate such a curve The work of Cartwright and Littlewood lead to the work of Levinson which in turn provided the basis for the horseshoe construction of Si Smalei In Russia Lyapunov paralleled the methods of Poincare and initiated the strong Russian dynamical systems schooli Andronov carried on with the study of nonlinear oscillators and in 1937 introduced together with Pontryagin the notion of coarse systems They were formalizing the understanding garnered from the study of nonlinear oscillators the un derstanding that many of the details on how these oscillators work do not affect the overall picture of the state space there will still be limit cycles if one changes the dissipation or spring force function by a little bit And changing the system a little bit has the great advantage of eliminating exceptional cases in the mathematical analysis Coarse sys tems were the concept that caught Smalels attention and enticed him to study dynamical systems A2 Chaos grows up 133 134 A brief history of chaos A3 Chaos with us RI Mainieri In the fall of 1961 Steven Smale was invited to Kiev where he met Arnol7d Anosov Sinai and Novikovi He lectured there and spent a lot of time with Anosovi He suggested a series of conjectures most of which Anosov proved within a year It was Anosov who showed that there are dynamical systems for which all points as opposed to a nonwandering set admit the hyperbolic structure and it was in honor of this result that Smale named these systems AxiomAl In Kiev Smale found a re ceptive audience that had been thinking about these problems Smale s result catalyzed their thoughts and initiated a chain of developments that persisted into the 1970 s Smale collected his results and their development in the 1967 review article on dynamical systems entitled Differentiable dynamical sys tems77i There are many great ideas in this paper the global foliation of invariant sets of the map into disjoint stable and unstable parts the existence of a horseshoe and enumeration and ordering of all its orbits the use of zeta functions to study dynamical systems The emphasis of the paper is on the global properties of the dynamical system on how to understand the topology of the orbits Smale s account takes you from a local differential equation in the form of vector elds to the global topological description in terms of horseshoesi The path traversed from ergodicity to entropy is a little more confus ing The general character of entropy was understood by Weiner who seemed to have spoken to Shannon In 1948 Shannon published his re sults on information theory where he discusses the entropy of the shift transformationi Kolmogorov went far beyond and suggested a de nition of the metric entropy of an area preserving transformation in order to classify Bernoulli shifts The suggestion was taken by his student Sinai and the results published in 1959 In 1960 Rohlin connected these results to measuretheoretical notions of entropy The next step was published in 1965 by Adler and Palis and also Adler Konheim McAndrew these papers showed that one could de ne the notion of topological entropy and use it as an invariant to classify continuous mapsi In 1967 Anosov and Sinai applied the notion of entropy to the study of dynamical sys tems It was in the context of studying the entropy associated to a dynamical system that Sinai introduced Markov partitions in 1968 Markov partitions allow one to relate dynamical systems and statis tical mechanics this has been a very fruitful relationship It adds mea sure notions to the topological framework laid down in Smalels paperi Markov partitions divide the state space of the dynamical system into nice little boxes that map into each other Each box is labeled by a code and the dynamics on the state space maps the codes around inducing a symbolic dynamicsi From the number of boxes needed to cover all the space Sinai was able to de ne the notion of entropy of a dynamical systemi In 1970 Bowen came up independently with the same ideas although there was presumably some ow of information back and forth before these papers got published Bowen also introduced the important concept of shadowing of chaotic orbitsi We do not know whether at this point the relations with statistical mechanics were clear to every one They became explicit in the work of Ruellei Ruelle understood that the topology of the orbits could be speci ed by a symbolic code and that one could associate an energy to each orbit The energies could be formally combined in a partition function to generate the invariant measure of the systemi After Smale Sinai Bowen and Ruelle had laid the foundations of the statistical mechanics approach to chaotic systems research turned to studying particular cases The simplest case to consider is one dimensional maps The topology ofthe orbits for parabolalike maps was worked out in 1973 by Metropolis Stein and Stein The more general onedimensional case was worked out in 1976 by Milnor and Thurston in a widely circulated preprint whose extended version eventually got published in 1988 A lecture of Smale and the results of Metropolis Stein and Stein in spired Feigenbaum to study simple mapsi This lead him to the discovery of the universality in quadratic maps and the application of ideas from eldtheory to dynamical systems Feigenbaum s work was the culmina tion in the study of onedimensional systems a complete analysis of a nontrivial transition to chaos Feigenbaum introduced many new ideas into the eld the use of the renormalization group which lead him to introduce functional equations in the study of dynamical systems the scaling function which completed the link between dynamical systems and statistical mechanics and the use of presentation functions as the dynamics of scaling functions The work in more than one dimension progressed very slowly and is still far from completed The rst result in trying to understand the topology of the orbits in two dimensions the equivalent of Metropo lis Stein and Stein or Milnor and Thurstonls work was obtained by Thurstoni Around 1975 Thurston was giving lectures On the geometry and dynamics of diffeomorphisms of surfaces i Thurston7s techniques exposed in that lecture have not been applied in physics but much of the classi cation that Thurston developed can be obtained from the no tion of a pruning front developed independently by Cvitanovici Once one develops an understanding for the topology of the orbits of a dynamical system one needs to be able to compute its properties Ruelle had already generalized the zeta function introduced by Artin and Mazur so that it could be used to compute the average value of observables The dif culty with Ruellels zeta function is that it does not converge very well Starting out from Smale s observation that a chaotic dynamical system is dense with a set of periodic orbits Cvitanovic used these orbits as a skeleton on which to evaluate the averages of observables and organized such calculations in terms of rapidly converging cycle expansionsi This convergence is attained by using the shorter orbits used as a basis for shadowing the longer orbitsi This account is far from complete but we hope that it will help get A3 Chaos with us 135 136 Further reading a sense of perspective on the eld It is not a fad and it will not die anytime soon Further reading Notion of global foliations For each paper cited in dy namical systems literature there are many results that went into its development As an example take the no tion of global foliations that we attribute to Smale As fa we can trace the idea it goes back to Rene Thom local foliations were already used by Hadamard Smale attended a seminar of Thom in 1958 or 1959 In that seminar Thom was explaining his notion of transversal ity One of Thom s disciples introduced Smale to Brazil iar mathematician Peixoto Peixoto who had learned the results of the AndronovPontryagin school from Lef schetz was the closest Smale had ever come until then to the AndronovPontryagin school It was from Peixoto that Smale learned about structural stability a notion that got him enthusiastic about dynamical systems as it blended well with his topological background It was from discussions with Peixoto that Smale got the prob lems in dynamical systems that lead him to his 1960 pa per on Morse inequalities The next year Smale published his result on the hyperbolic structure of the nonwander ing set Smale was not the rst to consider a hyperbolic point Poincare had already done that but Smale was the rst to introduce a global hyperbolic structure By 1960 Smale was already lecturing on the horseshoe as a structurally stable dynamical system with an in nity of periodic points and promoting his global viewpoint R Mainieri Levels of ergodicity 1n the mid 1970 s A Kar tok and YaB Pesin tried to use geometry to establish positive Lyapunov exponents A Katok and JM Strel cyn carried out the program and developed a theory of general dynamical systems with singularities They stud ied uniformly hyperbolic systems as strong as Anosov s but with sets of singularities Under iterations a dense set of points hits the singularities Even more important are the points that never hit the singularity set In order to establish some control over how they approach the set one looks at trajectories that apporach the set by some given 5 or faster YaG Sinai L Bunimovich arid Chernov studied the geometry of billiards in a very detailed way A Katok and YaB Pesin s idea was much more robust Look at the discontinuity set geometry of it matters not at all take an 5 neighborhood around it Given that the Lebesgue measure is 5 and the stability grows not faster than distancequot A Katok arid JM Strelcyn prove that the Lyapunov exponent is nonzero In mid 1980 s YaB Pesin studied the dissipative case Now the problem has no invariant Lebesgue measure As suming uniform hyperbolicity with singularities arid ty ing together Lebesgue measure and discontinutities and given that the stability grows not faster than distancequot YaB Pesin proved that the Lyapunov exponent is non zero and that SRB measure exists He also proved that the Lorenz Lozi arid Byelikh attractors satisfy these con dition 1n the the systems were uniformly hyperbolic all trou ble was in differentials For the H non attractor already the differentials are nonhyperbolic The points do not separate uniformly but the analogue of the singularity set can be obtained by excizing the regions that do not separate Hence there are 3 levels of ergodic systems 1 Anosov ow 2 Anosov ow singularity set 0 the Hamiltonian systems general case A Kar d tok an JM Strelcyn billiards YaG Sinai and L Bunimovich o the dissipative case YaB Pesin 3 H non o The rst proof was given by M Benedicks and L Carleson o A more readable roof is M Benedicks and LS Young 12 given in based on YaB Pesin s comments A4 Periodic orbit theory Pure mathematics is a branch of applied mathematics Joe Keller after being asked to de ne applied mathematics The history of the periodic orbit theory is rich and curious and the recent advances are to equal degree inspired by a century of separate development of three disparate subjects 1 classical chaotic dynamics initiated by Poincare and put on its modern footing by Smale Ruelle and many others 2 quantum theory initiated by Bohr with the mod ern chaotic formulation by Gutzwiller and 3 analytic number theory initiated by Riemann and formulated as a spectral problem by Selbergr Following totally different lines of reasoning and driven by very differ ent motivations the three separate roads all arrive at formally nearly identical trace formulas zeta functions and spectral determinants That these topics should be related is far from obviousr Connection between dynamics and number theory arises from Selbergls observation that description of geodesic motion and wave mechanics on spaces of constant negative curvature is essentially a numbertheoretic problemr A posteriori one can say that zeta functions arise in both classical and quantum mechanics because in both the dynamical evolution can be described by the action of linear evolution or transfer operators on in nitedimensional vector spaces The spectra of these operators are given by the zeros of appropriate determinants One way to evaluate determinants is to expand them in terms of traces log det tr log and in this way the spectrum of an evolution operator becames related to its traces irer periodic orbits A perhaps deeper way of restating this is to observe that the trace formulas perform the same service in all of the above problems they relate the spectrum of lengths local dynamics to the spectrum of eigenvalues global averages and for nonlinear geome tries they play a role analogous to that the Fourier transform plays for the circle A5 Death of the Old Quantum Theory In 1913 Otto Stern and Max Theodor Felix von Laue went up for a walk up the Uetliberg On the top they sat down and talked about physics In particular they talked about the new atom model of Bohr There and then they made the Uetli Schwur If that crazy model of Bohr turned out to be right then they would leave physics It did and they didn t A Pais Inward Bound of Matter and Forces in the Physical World In an afternoon of May 1991 Dieter Wintgen is sitting in his of ce at the Niels Bohr Institute beaming with the unparalleled glee of a boy who has just committed a major mischief The starting words of the manuscript he has just penned are A4 Periodic orbit theory 137 138 Further reading The failure of the Copenhagen School to obtain a reasonable 34 years old at the time Dieter was a scruffy kind of guy always in san dals and holed out jeans a left winger and a mountain climber working around the clock with his students Gregor and Klaus to complete the work that Bohr himself would have loved to see done back in 1916 a planetary calculation of the helium spectrum Never mind that the Copenhagen School refers not to the old quan tum theory but to something else The old quantum theory was no theory at all it was a set of rules bringing some order to a set of phenom ena which de ed logic of classical theory The electrons were supposed to describe planetary orbits around the nucleus their wave aspects were yet to be discovered The foundations seemed obscure but Bohrls an swer for the onceionized helium to hydrogen ratio was correct to ve signi cant gures and hard to ignore The old quantum theory marched on until by 1924 it reached an impasse the helium spectrum and the Zeeman effect were its death knell Since the late 1890 s it had been known that the helium spectrum consists of the orthohelium and parahelium lines In 1915 Bohr sug gested that the two kinds of helium lines might be associated with two distinct shapes of orbits a suggestion that turned out to be wrong In 1916 he got Kramers to work on the problem and wrote to Rutherford l have used all my spare time in the last months to make a serious attempt to solve the problem of ordinary helium spectrum 1 think really that at last 1 have a clue to the problem To other colleagues he wrote that the theory was worked out in the fall of 1916 and of having obtained a partial agreement with the measurements Nevertheless the BohrSommerfeld theory while by and large successful for hydrogen was a disaster for neutral helium Heroic efforts of the young generation including Kramers and Heisenberg were of no avail For a while Heisenberg thought that he had the ionization potential for helium which he had obtained by a simple perturbative scheme He wrote enthusiastic letters to Sommerfeld and was drawn into a col laboration with Max Born to compute the spectrum of helium using Bornls systematic perturbative scheme In rst approximation they re produced the earlier calculations The next level of corrections turned out to be larger than the computed effect The concluding paragraph of Max Bornls classic Vorlesungen iiber Atommechanik from 1925 sums it up in a somber tone the systematic application of the principles of the quantum theory gives results in agreement with experiment only in those cases where the motion of a single electron is considered it fails even in the treatment of the motion of the two electrons in the helium atom This is not surprising for the principles used are not really consistent complete systematic transformation of the classical mechanics into a discontinuous mechanics is the goal to wards which the quantum theory strives That year Heisenberg suffered a bout of hay fever and the old quan tum theory was dead In 1926 he gave the rst quantitative explanation of the helium spectrumi He used wave mechanics electron spin and the Pauli exclusion principle none of which belonged to the old quantum theory and planetary orbits of electrons were cast away for nearly half a century Why did Pauli and Heisenberg fail with the helium atom It was not the fault of the old quantum mechanics but rather it re ected their lack of understanding of the subtleties of classical mechanics Today we know what they missed in 191324 the role of conjugate points topological indices along classical trajectories was not accounted for and they had no idea of the importance of periodic orbits in nonintegrable systemsi Since then the calculation for helium using the methods of the old quantum mechanics has been xed Leopold and Percival added the topological indices in 1980 and in 1991 Wintgen and collaborators or bitsi Dieter had good reasons to gloat while the rest of us were prepar ing to sharpen our pencils and supercomputers in order to approach the dreaded 3body problem they just went ahead and did it What it tookiand much elseiis described in this book One is also free to ponder what quantum theory would look like today if all this was worked out in 1917 Further reading 139 Further reading Sources This tale aside from a few personal recollec Jammer s account The helium spectrum is taken up tions is in large part lifted from Abraham Pais7 accounts in Chapter 7 In August 1994 Dieter Wintgen died in a of the demise of the old quantum theory 78 as well as climbing accident in the Swiss Alps Why does it work Bloch Space is the eld of linear operators Nonsense space is blue and birds y through it Felix Bloch Heisenberg and the early days of quantum mechanics Heisenberg R Artuso HH Rugh and P Cvitanovic39 As we shall see the trace formulas and spectral determinants work well sometimes very well The question is Why And it still is The heuristic manipulations of Chapters 16 and 6 were naive and reckless as we are facing in niteidimensional vector spaces and singular intei gral kernels We now outline the key ingredients of proofs that put the trace and determinant formulas on solid footing This requires taking a closer look at the evolution operators from a mathematical point ofview since up to now we have talked about eigenvalues without any reference to what kind ofa function space the corresponding eigenfunctions belong to We shall restrict our considerations to the spectral properties of the PerroniErobenius operator for maps as proofs for more general evolu7 tion operators follow along the same lines What we refer to as a the set of eigenvalues acquires meaning only within a precisely speci ed functional setting this sets the stage for a discussion of the analyticity properties of spectral determinants ln Example 211 we compute ex plicitly the eigenspectrum for the three analytically tractable piecewise linear examples In Section 213 we review the basic facts of the clas7 sical Fredholm theory of integral equations The program is sketched in Section 214 motivated by an explicit study of eigenspectrum of the Bernoulli shift map and in Section 215 generalized to piecewise real analytic hyperbolic maps acting on appropriate densities We show on a very simple example that the spectrum is quite sensitive to the regui larity properties of the functions considere For expanding and hyperbolic niteisubshift maps analyticity leads to a very strong result not only do the determinants have better analytic ity properties than the trace formulas but the spectral determinants are singled out as entire functions in the complex 3 plane The goal of this chapter is not to provide an exhaustive review of the rigorous theory of the PerroniErobenius operators and their spectral determinants but rather to give you a feeling for how our heuristic considerations can be put on a rm basis The mathematics underpin ning the theory is both hard and profound If you are primarily interested in applications of the periodic orbit 211 Linear maps exact spectra 322 212 Evolution operator in a matrix rep resentation 326 213 Classical Fredholm theory 328 14 Analyticin of spectral determine ants 330 215 Hyperbolicmaps 335 216 The physics of eigenvalues and eigenfunctions 337 217 Troubles ahead 339 Summary 340 Further reading 341 Exercises 342 References 342 W Remark 21 4 322 CHAPTER 21 WHY DOES IT WORK theory you should skip this chapter on the rst reading W fast track Chapter 12 p 167 211 Linear maps exact spectra unstable xed point Ref 6 shows that this can be carried over to drdimensions Not only that but in Example 215 We compute the ex act spectrum for the simplest example of a dynamical system with an in nity of unstable periodic orbits the Bernoulli shift Example 211 The simplest eigenspectrum r a single xed point In orderto get some feeling for the determinants de ned so formally in Sec tion172 let us Work out a trivial example a repeller with only one expand ing linear branch frm lAlgt1 Hie n 1410is 1 my mtg7 M W WNW 211 Zly39c A 1Acgc 101 212 equation k labels the kth eigenfunction rk in the same spirit in which henumberofnr lkl m Ltln L I c A quantumrmechanical amplitude with more nodes has more variabilr ity hence a higher kinetic energy Analogously for a Perroanrobenius operator a higher k eigenvalue llAlAk is getting exponentially smaller because densities that vary more rapidly decay more rapidly under the expanding action of the map Example 212 The trace formula for a single xed point The eigenvalues A quot1 fall off exponentially with k so the trace ofll is a convergent sum 1 k 1 1 trll i A 7 4 w k M 7A4 win711 in agreement with 167 A similar result follows for powers of L yielding the singler xed point version ofthe trace formula for maps 1610 255k 7 2m 5671 kzolizeskiglliVl 5 WWW 2139s semen iengme GuasBmknrgvaswnllBZAung mm 211 LINEAR MAPS EXACT SPECTRA 323 The left hand side of 213 is a meromorphic function with the lead ing zero at z So what Example 213 Meromorphic functions and exponential convergence As an illustration of how exponential convergence of a truncated series is related to analytic properties of functions consider as the simplest possible example of a meromorphic function the ratio Z i a z 7 b with 11 real and positive and a lt 1 Within the spectral radius lzl lt b the function 11 can be represented by the power series hz f akzk k0 hz where 70 117 and the higher order coef cients are given by 7739 a 7 bbj1 Consider now the truncation of order N of the power series a zaib lizN bN hNzZakzk Let 2N be the solution of the truncated series hN N 0 To estimate the distance between a and 2N it is suf cient to calculate I39ma It is of order abN1 so nite order estimates converge exponentially to the asymptotic value This example shows that 1 an estimate of the leading pole the leading eigenvalue ofL from a nite truncation of a trace formula cone verges exponentially and 2 the nonileading eigenvalues of lie out side of the radius of convergence of the trace formula and cannot be computed by means of such cycle expansion However as we shall now see the whole spectrum is reachable at no extra effort by compute ing it from a determinant rather than a trace Example 214 The spectral determinant for a single xed point The spectral determinant 173 follows from the trace formulas of Exam ple 212 det17z H lt1 7 27tquotQn t 214 k0 O where the cummulants QT are given explicitly by the Euler formula 1 A 1 A quot W W If you cannot gure out how to derive this formula the solutions on p 7 offer several proofs g 21 3 page 342 Q 215 The main lesson to glean from this simple example is that the cum mulants Qn decay asymptotically fasterthan exponentially as A Mn lVQ For example if we approximate series such as 214 by the rst 10 terms the error in the estimate of the leading zero is m 1A50 ChaasBook mgversmnll a ZAug212007 convegi lEaugZOOE 324 CHAPTER 21 WHY DOES IT WORK So far all is well for a rather boring example a dynamical system with a single repelling xed point What about chaos Systems where the number of unstable cycles increases exponentially with their length We now turn to the simplest example of a dynamical system with an in nity of unstable periodic orbits Example 215 Bernoulli shift Consider next the Bernoulli shift map xgt gt2x mod 1 xEO1 216 The associated PerroniFrobenius operator 149 assambles py from its two preimages MW 9 9 g 217 For this simple example the eigenfunctions can be written down explicitly they coincide up to constant prefactors with the Bernoullipolynomials Bnx These polynomials are generated by the Taylor expansion of the generating function 7 text gt0 tk Qzt r 71 Z BANE 3095 1 3195 I k0 NMH The PerroniFrobenius operator 217 acts on the generating function 9 as 1 text2 tetZgactZ t ext2 ltgtltgt t2k 90m t 5m Z 3W k 2 hence each Bkx is an eigenfunction of with eigenvalue 127 The full operator has two components corresponding to the two branches For the n times iterated operator we have a full binary shift and for each of the 2quot branches the above calculations carry over yielding the same trace 2quot 7 1 1 for every cycle on length 71 Without further ado we substitute everything back and obtain the determinant detliz explt72n271gt H1721k 218 k0 Verifyl39llgthattheBernoulliJ quot 39 I 39 lLli 39 11212quot The Bernoulli map spectrum looks reminiscent of the single xed point spectrum 212 with the difference that the leading eigenvalue here is 1 rather than llAl The difference is signi cant the single xedipoint map is a repeller with escape rate 16 given by the lead ing eigenvalue 7 ln lA while there is no escape in the case of the Bernoulli map As already noted in discussion of the relation 1723 for bound systems the local expansion rate here ln lAl ln 2 is bale anced by the entropy here ln 2 the log of the number of preimages F5 yielding zero escape rate So far we have demonstrated that our periodic orbit formulas are correct for two piecewise linear maps in l dimension one with a single xed point and one with a full binary shift chaotic dynamics For a convag r lEaugZOOE ChaosBook mgversmnll a ZAug212007 211 LINEAR MAPS EXACT SPECTRA 325 single xed point eigenfunctions are monomials in I For the chaotic example they are orthogonal polynomials on the unit interval What about higher dimensions We check our formulas on a 2d hyperbolic map next Example 216 The simplest of 2d maps 7 a single hyperbolic xed We start by considering a very simple linear hyperbolic map with a single hyperbolic xed point KI f1I17I27f2I17I2 AsIuAuM 0 lt lAsl lt 17 Mail gt 1 The PerroniFrobenius operator 1410 acts on the 27d density functions as p12 IiAs 2Au 219 7 1 As An What are good eigenfunctions Cribbing the 17d eigenfunctions for the sta7 ble contracting x1 direction from Example 211 is not a good idea as under the iteration of the high terms in a Taylor expansion of px1 2 in the x1 variable would get multiplied by exponentially exploding eigenvalues 1 This makes sense as in the contracting directions hyperbolic dynam7 ics crunches up initial densities instead of smoothing them So we guess instead that the eigenfunctions are of form Whahm anyw k1 k2 012 2110 a mixture of the Laurent series in the contraction x1 direction and the Tayi lor series in the expanding direction the x2 variable The action of Perroni Erobenius operator on this set of basis functions 7 AE1 k1k2172 WWWMMWMML 0AslAsl is smoothing with the higher In k2 eigenvectors decaying exponentially faster by AE1 Af21 factor in the eigenvalue One veri es by an explicit cal culation undoing the geometric series expansions to lead to 179 that the trace of indeed equals 1ldet1 7Ml1 17Au17Aslfromwhich it follows that all our trace and spectral determinant formulas apply The are gument applies to any hyperbolic map linearized around the xed point of form fx1 xd A1I1A2I2 Adxd So far we have checked the trace and spectral determinant formui las derived heuristically in Chapters 16 and 17 but only for the case of 17 and 27d linear maps But for in niteidimensional vector spaces this game is fraught with dangers and we have already been mislead by piecewise linear examples into spectral confusions contrast the spec tra of Example 141 and Example 151 with the spectrum computed in Example 161 We show next that the above results do carry over to a sizable class of piecewise analytic expanding maps ChaasBook mgversionll a ZAug212007 convegi lEaugZOOE 326 CHAPTER 21 WHY DOES IT WORK 212 Evolution operator in a matrix representation The standard and for numerical purposes sometimes very effective way to look at operators is through their matrix representations Evoi lution operators are moving density functions de ned over some state space and as in general we can implement this only numerically the temptation is to discretize the state space as in Section 143 The prob lem with such state space discretization approaches that they some times yield plainly wrong spectra compare Example 151 with the re sult of Example 161 so we have to think through carefully what is it that we really measure An expanding map fz takes an initial smooth density 15711 de ned on a subinterval stretches it out and overlays it over a larger in terval resulting in a new smoother density q5n1x Repetition of this process smoothes the initial density so it is natural to represent densi7 ties 15 by their Taylor series Expanding 00 k 00 2 My k 152 0 mm 11lt0gt 0 0 15310 dz 67 fr nz y0 z 1040 and substitute the two Taylor series into 146 15211 1571 y Mdz 6y 7 f1 n1 The matrix elements follow by evaluating the integral 82 k L Wdz yz 2111 y0 we obtain a matrix representation of the evolution operator 1k 4 dgcrygcF ZELW my 012 k which maps the 2k component of the density of trajectories 15 into the yk component of the density n1y one time step later with y f I We already have some practice with evaluating derivatives 62 Z Bail56y from Section 142 This yields a representation of the evolution operator centered on the xed point evaluated recursively in terms of derivatives of the map f Lm dz6ltlgtltz 7 m wf 7 1 lt4 1 gtzk i if H 111 MW km convag r lEaugZOOE ChaosBook mgversionll a ZAug212007 2112 212 EVOLUTION OPERATOR IN A MATRIX REPRESENTATION327 The matrix elements vanish for Z lt k so L is a lower triangular matrix The diagonal and the successive off7diagonal matrix elements are easily evaluated iteratively by computer algebra 1H 2f EMMA Eor chaotic systems the map is expanding Al gt 1 Hence the diagonal terms drop off exponentially as 1lAlkH the terms below the diagonal fall off even faster and truncating L to a nite matrix introduces only exponentially small errors The trace formula 213 takes now a matrix form t zL t L r172 r1izL In order to illustrate how this works we work out a few examples In Example 217 we show that these results carry over to any ana7 lytic single7branch 17d repeller Eurther examples motivate the steps that lead to a proof that spectral determinants for general analytic 17 dimensional expanding maps and 7 in Section 215 for 27dimensional hyperbolic mappings 7 are also entire functions 1 Lick W7 Lk1k 2113 Example 217 Perron7Frobenius operator in a matrix representation As in Example 211 we start with a map with a single xed point but this time with a nonlinear piecewise analytic map f with a nonlinear inverse F f4 sign of the derivative 7 0F F 1F l and the Perron7Frobenius operator acting on densities analytic in an open domain enclosing the xed point x w QM dt 61 fI gtI t7 F W Fy7 Assume that F is a contraction ofthe unit disk in the complex plane ie lFzl lt 9 lt 1 and IF zI lt C lt 00 for lzl lt1 2114 and expand d in a polynomial basis with the Cauchy integral formula 2sz diw ltwgt dw ltwgt 7 27ri w7z7 m7 27ri wquot1 Combining this with 2122 we see that in this basis Perron7Frobenius oper7 ator is represented by the matrix W dw a F ltwgtltFltwgtgt won gm Lmnm Ln f W 2115 Taking the trace and summing we get tr L Z L fl n20 This integral has but one simple pole at the unique xed point 11 Fw fw Hence din aF w 27ri w7Fw aF w 1 quot f 1 7 PW WW 7 11 ChaasBook mgversmnll a ZAug212007 convexg7 lEaugZOOE Fig 211 A nonlinear one7branch repeller with a single xed point 21 g 21 6 page 342 328 CHAPTER 21 WHY DOES IT WORK This superrexponential decay of cummulants Qk ensures that for a repeller consisting of a single repelling point the spectral determinant 214 is entire in the complex 2 plane In retrospect the matrix representation method for solving the den sity evolution problems is eminently sensible 7 after all that is the Way one solves a close relative to classical density evolution equations the Schrodinger equation When available matrix representations for L enable us to compute many more orders of cumulant expansions of spectral determinants and many more eigenvalues of evolution operr ators than the cycle expensions approac Now if the spectral determinant is entire formulas such as 1725 imply that J 39 39 39 e practical import of this observation is that it guarantees that nite or der estimates of zeroes of dynamical zeta functions and spectral detr erminants converge exponentially or r in cases such as 214 7 super exponentially to the exact values and so the cycle expansions to be dis cussed in Chapter 18 represent a true perturbative approach to chaotic dynamics Before turning to speci cs We summarize a few facts about classir cal theory of integral equations something you might prefer to skip on rst reading The purpose of this exercise is to understand that the Fredholm theory a theory that Works so Well for the Hilbert spaces of quantum mechanics does not necessarily Work for deterministic dy namics r the ergodic theory is much harder Wtasttrack Section 214 330 213 Classical Fredholm theory function i He who would valiant be 39Gainst all disaster John Bunyan Pilgrim 395 Progress e The Perroanrobenius operator we dyaltz r fy y has the same appearance as a classical Fredholm integral operator Mir M dy Krywyr 2116 and one is tempted to resort too classical Fredholm theory in order to establish analyticity properties of spectral determinants This path to enlightenment is blocked by the singular nature of the kernel which is a coma lhu tm GiacsBmknrgvaswnllBZAung mm 213 CLASSICAL FREDHOLM THEORY 329 distribution whereas the standard theory of integral equations usually concerns itself with regular kernels Cz7 y E L2 Here we brie y recall some steps of Fredholm theory before working out the example of Example 215 The general form of Fredholm integral equations of the second kind is was M dmwm an am where 51 is a given function in L2 and the kernel Cz7 y E L2 M2 HilbertASchmidt condition The natural object to study is then the lin ear integral operator 2116 acting on the Hilbert space L2M the fundamental property that follows from the L2 Q nature of the kernel is that such an operator is compact that is close to a nite rank operi ator see Appendix 7 A compact operator has the property that for every 6 gt 0 only a nite number of linearly independent eigenvectors exist corresponding to eigenvalues whose absolute value exceeds 6 so we immediately realize Fig 214 that much work is needed to bring PerroniFrobenius operators into this picture We rewrite 2117 in the form Tso T17K 2118 The Fredholm alternative is now applied to this situation as follows the equation T90 5 has a unique solution for every 5 E L2 or there exists a nonzero solution of Tgoo 0 with an eigenvector of K corresponding to the eigenvalue 1 The theory remains the same if instead of T we consider the operator TA ll 7 AK with A y 0 As K is a compact operator there is at most a denumerable set of for which the second part of the Fredholm alternative holds apart from this set the inverse operator ll 7 AT 1 exists and is bounded in the operator sense When A is suf ciently small we may look for a perturbative expression for such an inverse as a geometric series liAKfl 1AKA2K2 11AVV7 2119 where K is a compact integral operator with kernel K zy 1 dzl dzn1Kz21Kzn1y7 Mn and W is also compact as it is given by the convergent sum ofcompact operators The problem with 2119 is that the series has a nite radius of convergence while apart from a denumerable set of As the inverse operator is well de ned A fundamental result in the theory of integral equations consists in rewriting the resolving kernel W as a ratio of two analytic functions of 73179 A D ChaasBook mgversionll a ZAug212007 convegi lEaugZOOE WWW 330 CHAPTER 21 WHY DOES IT WORK If we introduce the notation my 7 CIlvyl CIlvyn Czmyl Czmyn we may write the explicit expressions 7 7 n 21mg D 7 lnE1 l n Mnd21danltZlu 2ngt 00 Am exp 7 E trlCmgt 2120 yimyn quot11 Dz y39A K I i n d2 dz K I 21 Z 7 y 71 ml 1 n y 21 2n The quantity D is known as the Fredholm determinant see 1724 and Appendix 7 it is an entire analytic function of A and D 0 if and only iflA is an eigenvalue of K Worth emphasizing again the Fredholm theory is based on the compactness of the integral operator ie on the functional properties summability of its kernel As the PerroniFrobenius operator is not compact there is a bit of wishful thinking involved here 214 Analyticity of spectral determinants They savored the strange warm glow of being much more ignoi rant than ordinary people who were only ignorant of ordinary things Terry Pratchett Spaces of functions integrable L1 or squareiintegrable L2 on inter val 07 1 are mapped into themselves by the PerroniFrobenius operator and in both cases the constant function 150 E 1 is an eigenfunction with eigenvalue 1 lfwe focus our attention on L1 we also have a family of L1 eigenfunctions 1 My gexpemww 2121 0 with complex eigenvalue 2 9 parametrized by complex 6 with Re 6 gt 0 By varying 6 one realizes that such eigenvalues ll out the entire unit disk Such essential spectrum the case k 0 of Fig 214 hides all ne details of the spectrum What s going on Spaces L1 and L2 contain arbitrarily ugly func7 tions allowing any singularity as long as it is square integrable A and there is no way that expanding dynamics can smooth a kinky function withanonix 39 39 i let s sayaxquot 39 step and 11 1 convag r lEaugZOOE ChaosBook mgversionll a ZAug212007 214 ANALYTICITY OE SPECTRAL DETERMINANTS 331 that is why the eigenspectrum is dense rather than discrete Mathematii cians love to wallow in this kind of muck but there is no way to prepare a nowhere differentiable L1 initial density in a laboratory The only thing we can prepare and measure are piecewise smooth realianalytic density functions For a bounded linear operator A on a Banach space 9 the spectral radius is the smallest positive number pspec such that the spectrum is inside the disk of radius pspec while the essential spectral radius is the smallest positive number p855 such that outside the disk of radius pass the spectrum consists only ofisolated eigenvalues of nite multiplicity see Fig 214 We may shrink the essential spectrum by letting the PerroneErobenius operator act on a space of smoother functions exactly as in the one branch repeller case of Section 211 We thus consider a smaller space CH the space of k times differentiable functions whose k th deriva7 tives are Holder continuous with an exponent 0 lt a g 1 the expan7 sion property guarantees that such a space is mapped into itself by the PerroniErobenius operator In the strip 0 lt Re 6 lt k a most 159 will cease to be eigenfunctions in the space CCIHD the function 15 survives only for integer valued 6 n In this way we arrive at a nite set of isolated eigenvalues 1 2 1 2 16 and an essential spectral radius pm Tait We follow a simpler path and restrict the function space even further namely to a space of analytic functions ie functions for which the Taylor expansion is convergent at each point of the interval 0 1 With this choice things turn out easy and elegant To be more speci c let 15 be a holomorphic and bounded function on the disk D B0 R ofradius R gt 0 centered at the origin Our PerroniErobenius operator preserves the space of such functions provided 1 R2 lt R so all we need is to choose R gt 1 If F5 3 6 01 denotes the s inverse branch of the Bernoulli shift 216 the corresponding action of the PerroniErobenius operator is given by 5hy a F h o F using the Cauchy integral formula along the 8D boundary contour dw hwFs y 5hy a 73Dw7Fsy 2122 27Ti For reasons that will be made clear later we have introduced a sign a i1 of the given real branch lF yl a F y For both branches of the Bernoulli shift 3 1 but in general one is not allowed to take absolute values as this could destroy analyticity In the above formula one may also replace the domain D by any domain containing 0 1 such that the inverse branches maps the closure of D into the interior of D Why simply because the kernel remains nonisingular under this con dition ie w 7 Fy y 0 whenever w 6 8D and y 6 Cl D The problem is now reduced to the standard theory for Eredholm determii nants Section 213 The integral kernel is no longer singular traces and determinants are wellide ned and we can evaluate the trace of LF by ChaasBook mgversionll a ZAug212007 convegi lEaugZOOE E 21 5 page 342 g 216 page 342 Eff 216 page 342 332 CHAPTER 21 WHY DOES IT WORK means of the Cauchy contour integral formula dw O39F t L 7 7 r F 27ri w 7 F Elementary complex analysis shows that since F maps the closure ofD into its own interior F has a unique realevalued xed point 1 with a multiplier strictly smaller than one in absolute value Residue calculus therefore yields aFz FW 7 l 7 WW 1V justifying our previous ad hoc calculations of traces using Dirac delta functions Example 218 PerroneFrobenius operator in a matrix representation As in Example 211 we start with a map with a single xed point but this time with a nonlinear piecewise analytic map f with a nonlinear inverse F f 1sign ofthe derivative 7 0F F lF l we dz 6ltz 7 M W a 1 ltFltzgtgt Assume that F is a contraction of the unit disk ie lt 9 lt1 and lF zl lt C lt 00 for lzl lt 1 2123 and expand d in a polynomial basis by means of the Cauchy formula LU gtw dw gtw 27ri w 7 z 27ri wquot1 W 2 m n20 4m Combining this with 2122 we see that in this basis is represented by the matrix diw a F ltwgtltFltwgtgt 27ri w 1 gtw Zmemm L f 2124 Taking the trace and summing we get tr ZLm Lw L00 n20 27ri w7Fw This integral has but one simple pole at the unique xed point uquot Fw fw Hence a F w 1 t L 7 1 Flaw WW 7 1x We worked out a very speci c example yet our conclusions can be generalized provided a number of restrictive requirements are met by the dynamical system under investigation convag r lEaugZOOE ChaosBook mgversmnll a zAug212007 214 ANALYTICITY OE SPECTRAL DETERMINANTS 333 1 the evolution operator is multiplicative along the ow 2 the symbolic dynamics is a nite subshift 3 all cycle eigenvalues are hyperbolic exponentially bound ed in magnitude away from 1 4 the map or the ow is real analytic ie it has a piece wise analytic continuation to a complex extension of the state space These assumptions are romantic expectations not satis ed by the dye namical systems that we actually desire to understand Still they are not devoid of physical interest for example nice repellers like our 37 disk game of pinball do satisfy the above requirements Properties 1 and 2 enable us to represent the evolution operator as a nite matrix in an appropriate basis properties 3 and 4 enable us to bound the size of the matrix elements and control the eigenvalues To see what can go wrong consider the following examples Property 1 is violated for ows in 3 or more dimensions by the fol lowing weighted evolution operator y7r lA Il 5y 7 f r 7 where A is an eigenvalue of the fundamental matrix transverse to the ow Semiclassical quantum mechanics suggest operators of this form with B 12 see Chapter 7 The problem with such operators arises from the fact that when considering the fundamental matrices Jab 1an for two successive trajectory segments a and b the corre7 sponding eigenvalues are in general not multiplicative Aab y AaAb unless a b are iterates of the same prime cycle 0 so 1an JTVV Consequently this evolution operator is not multiplicative along the trajectory The theorems require that the evolution be represented as a matrix in an appropriate polynomial basis and thus cannot be ap plied to nonimultiplicative kernels ie kernels that do not satisfy the semiigroup property UU U The cure for this problem in this particular case is given in Appendix 7 Property 2 is violated by the 17d tent map see Fig 214 a 1610z171172zl7 l2ltaltl All cycle eigenvalues are hyperbolic but in general the critical point 10 12 is not a preiperiodic point so there is no nite Markov pare tition and the symbolic dynamics does not have a nite grammar see Section 115 for de nitions In practice this means that while the lead ing eigenvalue of might be computable the rest of the spectrum is very hard to control as the parameter a is varied the nonileading 2e ros of the spectral determinant move wildly about Property 3 is violated by the map see Fig 214 b 7 z212 7IEIO07 f172721 zell 1 ChaasBook mgvexsmnll a ZAug212007 corn2gi lEaugZOOE 334 CHAPTER 21 WHY DOES IT WORK fx 0 5 a b Fig 212 a A hyperbolic tent map without a nite Markov partition b A Markov map with a marginal xed point Here the interval 01 has a Markov partition into two subintervals Io and 11 and f is monotone on each However the xed point at z 0 has marginal stability A0 l and violates condition 3 This type of map is called intermittent and necessitates much extra work The problem is that the dynamics in the neighborhood of a marginal xed point is very slow with correlations decaying as power laws rather than exponentially We will discuss such ows in Chapter 22 Property 4 is required as the heuristic approach of Chapter 16 faces two major hurdles l The trace 168 is not well de ned because the integral kernel is singular 2 The existence and properties of eigenvalues are by no means clear Actually property 4 is quite restrictive but we need it in the present approach so that the Banach space of analytic functions in a disk is preserved by the PerroniFrobenius operator In attempting to generalize the results we encounter several prob lems First in higher dimensions life is not as simple Multiidimensional residue calculus is at our disposal but in general requires that we nd polyidomains direct product of domains in each coordinate and this need not be the case Second and perhaps somewhat surprisingly the counting of periodic orbits presents a dif cult problem For example instead of the Bernoulli shift consider the doubling map of the circle I gt gt 21 mod 1 z E RZ Compared to the shift on the interval 01 the only difference is that the endpoints 0 and l are now glued together Be cause these endpoints are xed points of the map the number of cycles of length n decreases by l The determinant becomes det172 exp 7 7 71gt 17 2 2125 711 The value 2 1 still comes from the constant eigenfunction but the Bernoulli polynomials no longer contribute to the spectrum as they are convag r lEaugZOOE ChaosBook mgversionll a ZAug212007 215 HYPERBOLIC MAPS 335 not periodic Proofs of these facts however are dif cult if one sticks to the space of analytic functions Third our Cauchy formulas a priori work only when considering purely expanding maps When stable and unstable directions coiexist we have to resort to stranger function spaces as shown in the next sec tion 215 Hyperbolic maps 1 can give you a de nion of a Banach space but I do not know what that means Federico Bonnetto Banach space HH Rugh Proceeding to hyperbolic systems one faces the following paradox If f is an areaipreserving hyperbolic and realianalytic map of for exam ple a 27dimensional torus then the PerroniFrobenius operator is uni tary on the space of L2 functions and its spectrum is con ned to the unit circle On the other hand when we compute determinants we nd eigenvalues scattered around inside the unit disk Thinking back to the Bernoulli shift Example 215 one would like to imagine these eigenvalues as popping up from the L2 spectrum by shrinking the func7 tion space Shrinking the space however can only make the spectrum smaller so this is obviously not what happens lnstead one needs to introduce a mixed function space where in the unstable direction one resorts to analytic functions as before but in the stable direction one instead considers a dual space of distributions on analytic functions Such a space is neither included in nor includes L2 and we have thus resolved the paradox However it still remains to be seen how traces and determinants are calculated The linear hyperbolic xed point Example 216 is somewhat mislead ing as we have made explicit use of a map that acts independently along the stable and unstable directions For a more general hyperbolic map there is no way to implement such direct product structure and the whole argument falls apart Her comes an idea use the analyticity of the map to rewrite the PerroniFrobenius operator acting as follows where 0 denotes the sign of the derivative in the unstable direction h21722 ff 217161 WWW dwl d1 2126 w17w2f2w17w2 7 22 27ri 27ri Here the function 15 should belong to a space of functions analytic re spectively outside a disk and inside a disk in the rst and the second co ordinates with the additional property that the function decays to zero as the rst coordinate tends to in nity The contour integrals are along the boundaries of these disks It is an exercise in multiidimensional residue calculus to verify that for the above linear example this expres7 sion reduces to 219 Such operators form the building blocks in the ChaasBook mgversmnll a ZAug212007 convegi lEaugZOOE Ia Remark 214 336 CHAPTER 21 WHY DOES IT WORK Fig 213 For an analytic hyperbolic map specifying the contracting coordinate wh at the initial rectangle and the expanding coordinate m at the image rectangle de nes a unique trajectory between the two rectangles In particular wu and 2 not shown are uniquely speci ed calculation of traces and determinants One can prove the following Theorem The spectral detenninantfor 27d hyperbolic analytic maps is entire The proof apart from the Markov property that is the same as for the purely expanding case relies heavily on the analyticity of the map in the explicit construction of the function space The idea is to view the hyperbolicity as a cross product of a contracting map in forward time and another contracting map in backward time In this case the Markov property introduced above has to be elaborated a bit Instead of dividing the state space into intervals one divides it into rectane gles The rectangles should be viewed as a direct product of intervals say horizontal and vertical such that the forward map is contract ing in for example the horizontal direction while the inverse map is contracting in the vertical direction For Axiom A systems see Ref mark 214 one may choose coordinate axes close to the stableunstable manifolds of the map With the state space divided into N rectangles M1M2MN Mi If X If one needs a complex extension D X DE with which the hyperbolicity condition which simultanee ously guarantees the Markov property can be formulated as follows Analytic hyperbolic property Either lntVlj 0 or for each pair wk 6 C1Df 21 E C1D there exist unique analytic funce tions of wmzv wv wv wmzv E lntD 2h 2hwhzv E lntD such that fwh7 wv 27721 Furthermore if wk E If and 21 E If then wv E If and 2h 6 If see Fig 21 In plain English this means for the iterated map that one replaces the coordinates 27721 at time n by the contracting pair 27 wv where wv is the contracting coordinate at time n l for the partial inverse map In two dimensions the operator in 2126 acts on functions analytic outside D in the horizontal direction and tending to zero at in nity and inside D in the vertical direction The contour integrals are pre cisely along the boundaries of these domains convag r lEaugZOOE ChaosBook mgversionll a ZAug212007 216 THE PHYSICS OF EIGENVALUES AND EIGENFUNCTIONS337 A map f satisfying the above condition is called analytic hyperbolic and the theorem states that the associated spectral determinant is en tire and that the trace formula 168 is correct Examples of analytic hyperbolic maps are provided by small analytic perturbations of the cat map the Sedisk repeller and the 27d baker s map 216 The physics of eigenvalues and eigenfunctions We appreciate by now that any honest attempt to look at the spectral properties of the PerroneFrobenius operator involves hard mathe ematics but the effort is rewarded by the fact that we are nally able to control the analyticity properties of dynamical zeta functions and spectral determinants and thus substantiate the claim that these ob jects provide a powerful and wellefounded perturbation theory Often see Chapter 15 physically important part of the spectrum is just the leading eigenvalue which gives us the escape rate from a re peller or for a general evolution operator formulas for expectation values of observables and their higher moments Also the eigenfunce tion associated to the leading eigenvalue has a physical interpretation see Chapter 14 it is the density ofthe natural measures with singular measures ruled out by the proper choice of the function space This conclusion is in accord with the generalized PerroneFrobenius theorem for evolution operators In the nite dimensional setting such a theo rem is formulated as follows 0 PerroneFrobenius theorem Let Lij be a nonnegative matrix such that some n exists for which L ij gt 0 Vi7j then 1 The maximal modulus eigenvalue is nonedegenerate real and positive 2 The corresponding eigenvector de ned up to a constant has nonnegative coordinates We may ask what physical information is contained in eigenvalues be yond the leading one suppose that we have a probability conserving system so that the dominant eigenvalue is 1 for which the essential spectral radius satis es 0 lt p855 lt 6 lt 1 on some Banach space E Denote by P the projection corresponding to the part of the spectrum inside a disk of radius 9 We denote by A1 A2 7 AM the eigenvalues outside of this disk ordered by the size of their absolute value with A1 1 Then we have the following decomposition M so ZAMwa P go 2127 i1 ChaasBook mgversionll a ZAug212007 convegi lEaugZOOE W Remark 214 g 217 page 342 338 CHAPTER 21 WHY DOES IT WORK when Li are nite matrices in Jordan canomical form L0 0is a 1X1 matrix as A0 is simple due to the PerroniErobenius theorem whereas 1 is a row vector whose elements form a basis on the eigenspace cor responding to M and 1 is a column vector of elements of 3 the dual space of linear functionals over 8 spanning the eigenspace of cor responding to M For iterates of the PerroniErobenius operator 2127 becomes M W ZANzL so Pm 2128 i1 If we now consider for example correlation between initial g0 evolved n steps and nal 5 cm M mm W y M dw lt5 0 fngtltwgtsoltwgt 2129 it follows that L mm Alw w ZWWM 0w 2130 i2 where MW dysltygtwiL2W M The eigenvalues beyond the leading one provide two pieces of in formation they rule the convergence of expressions containing high owers of the evolution operator to leading order the A1 contribution Moreover if w15g0 0 then 2129 de nes a correlation function as each term in 2130 vanishes exponentially in the n A 00 imit the eigenvalues A2 7AM determine the exponential decay of correla7 tions for our dynamical system The prefactors w depend on the choice of functions whereas the exponential decay rates given by logarithms of AZ do not the correlation spectrum is thus a universal property of the dynamics once we x the overall functional space on which the PerroniErobenius operator acts Example 219 Bernoulli shift eigenfunctions Let us revisit the Bernoulli shift example 216 on the space of analytic func7 tions on a disk apart from the origin we have only simple eigenvalues Ak 27k k 0 1 The eigenvalue 0 1 corresponds to probability conseri vation the corresponding eigenfunction Bow 1 indicates that the natural measure has a constant density over the unit interval If we now take any analytic function 77x with zero average with respect to the Lebesgue mea7 sure it follows that M10 77 0 and from 2130 the asymptotic decay of the correlation function is unless also w1n 77 0 Oan L N exp7nl0g2 2131 Thus 7 log 1 gives the exponential decay rate of correlations with a ref actor that depends on the choice of the function Actually the Bernoulli convag r lEaugZOOE ChaosBook mgversmnll a ZAug212007 217 TROUBLES AHEAD 339 shift case may be treated exactly as for analytic functions we can employ the EuleriMacLaurin summation formu a 1 mil 7 mil W O dwnw 2 who 2132 As e 39 jug functions with O we have from 2129 and the fact that Bernoulli polynomials are eigenvectors of the PerroniFrobenius operator that as 7m m 7 m 1 0nnltngtZ 2 7 771 7 0 dznltzgtBmltzgtr The decomposition 2132 is also useful in realizing that the linear function als 1 are singular objects ifwe write it as W Z Bantam we see that these functionals are of the form 1 1115 dw mason 0 where 0271 23971 7 7 23971 i a w 1 6 M wheni 2 1 and 11001 1 This representation is only meaningful when the function E is analytic in neighborhoods ofw w 7 Mm 2133 217 Troubles ahead The above discussion con rms that for a series of examples of increas ing generality formal manipulations with traces and determinants are justi ed the PerroniFrobenius operator has isolated eigenvalues the trace formulas are explicitly veri ed and the spectral determinant is an entire function whose zeroes yield the eigenvalues Real life is harder as we may appreciate through the following considerations 0 Our discussion tacitly assumed something that is physically en tirely reasonable our evolution operator is acting on the space of analytic functions ie we are allowed to represent the initial density Mr by its Taylor expansions in the neighborhoods of pee riodic points This is however far from being the only possi7 ble choice mathematicians often work with the function space UH ie the space of k times differentiable functions whose k th derivatives are Holder continuous with an exponent 0 lt a g 1 then every yquot with 9 gt k is an eigenfunction of the Perrone Frobenius operator and we have Lyquot 776C 1 7 A An y 7 ChaasBook mgversmnll a ZAug212007 convegi lEaugZOOE essential spec um spmalmdius iso ed eige value Fig 214 Spectrum of the Perronr Frobenius operator acting on the space of C Holderrcontinuous functions only k isolated eigenvalues remain be tween the spectral radius and the essenr tial spectral radius which bounds the es sentialquot continuous spectrum 2 1 1 Dane 342 g 21 2 page 342 340 CHAPTER 21 WHY DOES IT WORK This spectrum differs markedly from the analytic case only a small number of isolated eigenvalues remain enclosed between the spectral radius and a smaller disk ofradius llAlk1 see Fig 21 In literature the radius of this disk is called the essential spectral rae dius In Section 214 we discussed this point further with the aid of a less trivial ledimensional example The physical point ofview is complementary to the standard setting of ergodic theory where many chaotic properties of a dynamical system are encoded by the presence of a continuous spectrum used to prove asymptotic decay of correlations in the space of L2 squareeintegrable funce tions A deceptively innocent assumption is hidden beneath many fea tures discussed so far that 211 maps a given function space into itself This is strictly related to the expanding property of the map if fx is smooth in a domain D then fzA is smooth on a larger domain provided lAl gt 1 This is not obviously the case for hyperbolic systems in higher dimensions and as we saw in Sec tion 215 extensions of the results obtained for expanding maps are highly nontrivial o It is not at all clear that the above analysis ofa simple oneebranch one xed point repeller can be extended to dynamical systems with a Cantor set ofperiodic points we showed this in Section 214 Summary Examples of analytic eigenfunctions for led maps are seductive and make the problem of evaluating ergodic averages appears easy just integrate over the desired observable weightes by the natural measure right No generic natural measure sits on a fractal set and is singular everywhere The point of this book is that you never need to construct the natural measure cycle expansions will do thatjob A theory of evaluation of dynamical averages by means of trace for mulas and spectral determinants requires a deep understanding of their analyticity and convergence We work here through a series of examples l exact spectrum but for a single xed point of a linear map 2 exact spectrum for a locally analytic map matix representation 3 rigorous proof of existence of dicrete spectrum for 27d hyperbolic maps In the case of especially wellebehaved Axiom A systems where both the symbolic dynamics and hyperbolicity are under control it is possible to treat traces and determinants in a rigorous fashion and strong results about the analyticity properties of dynamical zeta funce tions and spectral determinants outlined above follow convag r lEaugZOOE ChaosBook mgversmnll a ZAug212007 F Further reading 341 Most systems of interest are not of the axiom A category they are neither purely hyperbolic nor as we have seen in Chapters 10 and 11 do they have nite grammar The importance of symbolic dynamics is generally grossly unappreciated the crucial ingredient for nice anai lyticity properties of zeta functions is the existence of a nite grammar coupled with uniform hyperbolicity The dynamical systems which are really interesting 7 for example smooth bounded Hamiltonian poi tentials 7 are presumably never fully chaotic and the central question remains How do we attack this problem in a systematic and control lable fashion Further reading Surveys of rigorous theory We recommend the references listed in Section 7 for an introduction to the mathemati ical literature on this subject For a physicist Driebe s monograph 33 might be the most accessible introduction into mathematics discussed brie ey in this chapter There are a number of reviews of the mathematical approach to dynamical zeta functions and spectral determinants with pointers to the original references such as Refs 1 Z An alternative approach to spectral properties of the Perroni Frobenius operator is given in Re Ergodic theory as presented by Sinai 14 and oth ers tempts one to describe the densities on which the evolution operator acts in terms of either integrable or squareiintegrable functions For our purposes as we have already seen this space is not suitable An in troduction to ergodic theory is given by Sinai Kornfeld and Fomin 15 more advanced oldifashioned presentai tions are Walters 12 and Denker Grillenberger and Sig mund 16 and a more formal one is given by Peter son 17 Fredholm theory Our brief summary of Fredholm theory is based on the exposition of Ref 4 A technical introduction of the theory from an operator point of view is given in Ref 5 The theory is presented in a more gen eral form in Ref 6 Bernoulli shift For a more detailed discussion cone sult chaper 3 of Ref 33 The extension of Fredholm the ory to the case or Bernoulli shift on CW in which the PerroniFrobenius operator is not compact 7 technically it is only quasiicompact That is the essential spectral radius is strictly smaller than the spectral radius has been given by Ruelle 7 a concise and readable statement of the re sults is contained in Ref 8 Hyperbolic dynamics When dealing with hyperbolic ChaasBook mgversmnll a ZAug212007 systems one might try to reduce to the expanding case by projecting the dynamics along the unstable directions As mentioned in the text this can be quite involved technii cally as such unstable foliations are not characterized by strong smoothness properties For such an approach see f 3 70 1 Spectral determinants for smooth ows The theoi rem on page 335 also applies to hyperbolic analytic maps in d dimensions and smooth hyperbolic analytic ows in d 1 dimensions provided that the ow can be re duced to a piecewise analytic map by a suspension on a Poincare section complemented by an analytic ceiling function 35 that accounts for a variation in the section return times For example if we take as the ceiling func7 tion gx 551quot where Tx is the next Poincare section time for a trajectory staring at x we reproduce the ow spectral determinant 1713 Proofs are beyond the scope of this chapter Explicit diagonalization For 17d repellers a diagonal ization of an explicit truncated LWW matrix evaluated in a judiciously chosen basis may yield many more eigenvali ues than a cycle expansion see Refs 1011 The reasons why one persists in using periodic orbit theory are par tially aesthetic and partially pragmatic The explicit calcu7 lation of LWW demands an explicit choice of a basis and is thus noniinvariant in contrast to cycle expansions which utilize only the invariant information of the ow In addi7 tion we usually do not know how to construct Lmn for a realistic highidimanensional ow such as the hyperbolic Sidisk game of pinball ow of Section 13 whereas perii odic orbit theory is true in higher dimensions and straight forward to apply PerroniFrobenius theorem A proof of the Perroni Frobenius theorem may be found in Ref 12 For positive convegi lEaugZOOE 342 transfer operators this theorem has been generalized by Ruelle 13 Axiom A systems The proofs in Section 215 follow the Thesis work ofHH Rugh 91819 For a math ematical introduction to the subject consult the excellent review by V Baladi 1 It would take us too far a eld to give and explain the de nition of Axiom A systems see Refs 22 23 Axiom A implies however the existence of a Markov partition of the state space from which the properties 2 and 3 assumed on page 324 follow Exponential mixing speed of the Bernoulli shift We see from 2131 that for the Bernoulli shift the exponential decay rate of correlations coincides with the Lyapunov ex ponent while such an identity holds for a number of syse Exercises tems it is by no means a general result and there exist explicit counterexamples Left eigenfunctions We shall never use an explicit form of left eigenfunctions corresponding to highly sine gular kernels like 2133 Many details have been elabe orated in a number of papers such as Ref 20 with a daring physical interpretation Ulam s idea The approximation of PerroneFrobenius operator de ned by 1414 has been shown to reproduce the spectrum for expanding maps once ner and ner Markov partitions are used 21 The subtle point of choose ing a state space partitioning for a generic case is dis cussed in Ref Exercises 211 What space does L act on Show that 212 is a complete basis on the space of analytic functions on a disk and thus that we found the complete set of eigenvalues What space does act on What can be said about the spectrum of 211 on L10 1 Compare the result with Fig 214 u2l 4n 722 17 u217 u2217 u is a polynomial in u and the coef cients fall off asymptotically as 0 z uquot3 2 Verify if you have a proof to all orders email it to the authors See also Solution 213 21 s Elder fomllda39 Dame The Email forml a 215 215 Bernoulli shift on L spaces Check that the family k t 7327 231121 belongs to L1O What can be said about H1m 1EW107ugtlt1 5 39e39ctral radius on L2O 1 Auseful he Mk4 reference is 4 x k u Z t a W 2116 Cauchy int lrai Rework all complex analysis k0 steps used in the Bernoulli shift example on analytic functions on a disk 214 27d product expansion We conjecture that the expansion corresponding to 2134 is in this case 217 Escape rate Consider the escape rate from a X X strange repeller nd a choice of trial functions 5 Hlt1 mk Elk gt733 and 40 such that 2129 gives the fraction on partie he he 1 7021 7 2 39 39 391 i 2 cles surviving after 71 iterations if their initial den 1 2n 2 sity distribution is pox Discuss the behavior of 1 17 702 t 17 7021 7 u22t such an expression in the long time limit References l V Baladi A brief introduction to dynamical zeta functions in DMV7 Seminar 27 Classical Nonintegrability Quantum Chaos A Knauf and YaG Sinai eds Birkhauser 1997 refsConverg 7 ZgjanZUUI ChaosBook orgversmnll a 2 Aug 212007 Swings


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