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Ergodic Theory in the Perspective of Functional Analysis

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Ergodic Theory in the Perspective of Functional Analysis 13 Lectures by Roland Derndinger, Rainer Nagel, GÄunther Palm (uncompleted version) In July 1984 this manuscript has been accepted for publication in the series \Lecture Notes in Mathematics" by Springer-Verlag. Due to a combination of unfortunate circumstances none of the authors was able to perform the necessary ¯nal revision of the manuscript. TÄubingen,July 1987 1 2 I. What is Ergodic Theory? The notion \ergodic" is an arti¯cial creation, and the newcomer to \ergodic theory" will have no intuitive understanding of its content: \elementary ergodic theory" neither is part of high school- or college- mathematics (as does \algebra") nor does its name explain its subject (as does \number theory"). Therefore it might be useful ¯rst to explain the name and the subject of \ergodic theory". Let us begin with the quotation of the ¯rst sentence of P. Walters' introductory lectures (1975, p. 1): \Generally speaking, ergodic theory is the study of transformations and flows from the point of view of recurrence properties, mixing properties, and other global, dynamical, properties connected with asymptotic behavior." Certainly, this de¯nition is very systematic and complete (compare the beginning of our Lectures III. and IV.). Still we will try to add a few more answers to the question: \What is Ergodic Theory ?" Naive answer: A container is divided into two parts with one part empty and the other ¯lled with gas. Ergodic theory predicts what happens in the long run after we remove the dividing wall. First etymological answer: ergodhc=di±cult. Historical answer: 1880 - Boltzmann, Maxwell - ergodic hypothesis 1900 - Poincar¶e - recurrence theorem 1931 - von Neumann - mean ergodic theorem 1931 - Birkho® - individual ergodic theorem 1958 - Kolmogorov - entropy as an invariant 1963 - Sinai - billiard flow is ergodic 1970 - Ornstein - entropy classi¯es Bernoulli shifts 1975 - Akcoglu - individual Lp-ergodic theorem Naive answer of a physicist: Ergodic theory proves that time mean equals space mean. I.E. Farquhar's [1964] answer: \Ergodic theory originated as an o®shot of the work of Boltzmann and of Maxwell in the kinetic theory of gases. The impetus provided by the physical problem led later to the development by pure mathemati- cians of ergodic theory as a branch of measure theory, and, as is to be expected, the scope of this mathematical theory extends now far beyond the initial ¯eld of interest. However, the chief physical problems to which ergodic theory has rel- evance, namely, the justi¯cation of the methods of statistical mechanics and the relation between reversibility and irreversibility have been by no means satisfacto- rily solved, and the question arises of how far the mathematical theory contributes to the elucidation of these physical problems." 3 Physicist's answer: Reality Physical model Mathematical consequences A gas with n particles The \state" of the gas is a at time t 0 is given. point x in the \state space" X R6n. Time changes Time change is described by Theorem of Liouville: the Hamiltonian di®erential ' preserves the (normal- equations. Their solutions ized) Lebesgue measure ¹ Ñ yield a mapping ' : X on X. X, such that the state x0 at time t 0 becomes the state x1 'px0q at time t 1. De¯nition: An observable is a function f : X Ñ R, where fpxq can be regarded the long run behavior as the outcome of a mea- is observed. surement, when the gas is in the state P X. Problem: Find lim fp'npxqq! 1st objection: Modi¯ed problem: Find Time change is much the time mean Mtfpxq : faster than our obser- ° 1 n1 p ip qq vations. lim n i0 f ' x ! 2nd objection: In Additional hypothesis \Theorem" 1: If the er- practice, it is impos- (ergodic hypothesis): Each godic hypothesis is satis- sible to determine the particular motion will pass ³¯ed, we have Mtfpxq state x. through every state consis- f d¹ space mean, which tent with its energy (see is independent of the state P.u.T. Ehrenfest 1911). x. \Theorem" 2: The er- godic hypothesis is \never" satis¯ed. Ergodic theory looks for better ergodic hypothesis and better \ergodic theorems". Commonly accepted etymological answer: êrgon = energy –ὁδός = {path (P. and T. Ehrenfest 1911, p. 30) 4 \Correct" etymological answer: êrgon = energy –ῶδης = {like (Boltzmann 1884/85, see also III.) K. Jacobs' [1965] answer: \... als EinfÄuhrungfÄursolche Leser gedacht, die gern einmal erfahren mÄochten, womit sich diese Theorie mit dem seltsamen, aus den griechischen WÄortern er- gon (Arbeit) und odoc (Weg) zusammengesetzten Namen eigentlich beschÄaftigt. Die Probleme der Ergodentheorie kreisen um einen Begri®, der einerseits so viele reizvolle SpezialfÄalleumfa¼t, da¼ sowohl der Polyhistor als auch der stille Genie¼er auf ihre Kosten kommen, andererseits so einfach ist, da¼ sich die zentralen Ergeb- nisse und Probleme der Ergodentheorie leicht darstellen lassen; diese einfach zu formulierenden Fragestellungen erfordern jedoch bei naherer Untersuchung oft de- rartige Anstrengungen, da¼ harte Arbeiter hier ihr rechtes VergnÄugen¯nden wer- den." J. Dieudonne's [1977] answer: \Le point de d¶epartde la th¶eorieergodique provient du d¶eveloppement de la m¶ecaniquestatistique et de la theorie cin¶etiquedes gaz, o`ul'exp¶eriencesugg`ere und tendence `al'\uniformite": si l'on consid`ere`aun instant donn¶eun m¶elange h¶et¶erog`enede plusieurs gaz, l'¶evolution du m¶elangeau cours du temps tend `ale rendre homog¶ene." W. Parry's [1981] answer: \Ergodic Theory is di±cult to characterize, as it stands at the junction of so many areas, drawing on the techniques and examples of probability theory, vector ¯elds on manifolds, group actions on homogeneous spaces, number theory, statistical mechanics, etc..."" (e.g. functional analysis; added by the authors). Elementary mathematical answer: Let X be a set, ' : X Ñ X a mapping. The induced operator T' maps functions f : X Ñ R into T'f : f '. Ergodic theory investigates the asymptotic behavior n n P of ' and T' for n N. Our answer: More structure is needed on the set X, usually at least a topological or a measure theoretical structure. In both cases we can study the asymptotic behavior of the n powers T of the linear operator T T', de¯ned either on the Banach space CpXq of all continuous functions on X or on the Banach space L1pX; §; ¹q of all ¹-integrable functions on X. 5 II. Dynamical Systems Many of the answers presented in Lecture I indicate that ergodic theory deals with pairs pX; 'q where X is a set whose points represent the \states" of a physical system while ' is a mapping from X into X describing the change of states after one time unit. The ¯rst step towards a mathematical theory consists in ¯nding out which abstract properties of the physical state spaces will be essential. It is evident that an \ergodic theory" based only on set-theoretical assumptions is of little interest. Therefore we present three di®erent mathematical structures which can be imposed on the state space X and the mapping ' in order to yield \dynamical systems" that are interesting from the mathematical point of view. The parallel development of the corresponding three \ergodic theories" and the investigation of their mutual interaction will be one of the characteristics of the following lectures. II.1 De¯nition: (i) pX; §; ¹; 'q is a measure-theoretical dynamical system (briefly: MDS) if pX; §; ¹q is a probability space and ' : X Ñ X is a bi-measure-preserving transforma- tion. (ii) pX; 'q is a topological dynamical system (TDS) if X is a compact space and ' : X Ñ X is homeomorphism. (iii) pE; T q is a functional-analytic dynamical system (FDS) if E is a Banach space and T : E Ñ E is a bounded linear operator. Remarks: 1. The term \bi-measure-preserving" for the transformation ' : X Ñ X in (i) is to be understood in the following sense: There exists a subset X0 of X with ¹pX0q 1 such that the restriction '0 : X0 Ñ X0 of ' is bijective, and both '0 and its inverse are measurable and measure-preserving for the induced σ-algebra §0 : tA X X0 : A P §u. 2. If ' is bi-measure-preserving with respect to ¹, we call ¹ a '-invariant measure. 3. As we shall see in (II.4) every MDS and TDS leads to an FDS in a canonical way. Thus a theory of FDSs can be regarded as a joint generalization of the topological theory of TDSs and the probabilistic theory of MDSs. In most of the following chapters we will either start from or aim for a formulation of the main theorem(s) in the language of FDSs. 4. DDSs (\di®erentiable dynamical systems") will not be investigated in these lec- tures (see Bowen [1975], Smale [1967], [1980]). Before proving any results we present in this lecture the fundamental (types of) examples of dynamical systems which will frequently reappear in the ensuing text. The reader is invited to apply systematically every de¯nition and result to at least some of these examples. II.2. Rotations: (i) Let ¡ tz P C : |z| 1u be the unit circle, § its Borel algebra, and m the normalized Lebesgue measure on ¡. Choose an a P ¡ and de¯ne 'apzq : a z for all z P ¡: Clearly, p¡; 'aq is a TDS, and p¡; B; m; 'aq an MDS. 6 (ii) A more abstract version of the above example is the following: Take a compact group G with Borel algebra B and normalized Haar measure m. Choose h P G and de¯ne the (left)rotation 'hpgq : h g for all g P G: Again, pG; 'hq is a TDS, and pG; B; m; 'hq an MDS.
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