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Graduate Texts in Mathematics #79
An Introduction to Ergodic Theory
This text provides an introduction to ergodic theory suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 0. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text is concerned with measure-preserving transformations of probability spaces; recurrence properties, mixing properties, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy theory are discussed. Some examples are described and are studied in detail when new properties are presented. The second part of the text focuses on the ergodic theory of continuous transformations of compact metrizable spaces. The family of invariant probability measures for such a transformation is studied and related to properties of the transformation such as topological traitivity, minimality, the size of the non-wandering set, and existence of periodic points. Topological entropy is introduced and related to measure-theoretic entropy. Topological pressure and equilibrium states are discussed, and a proof is given of the variational principle that relates pressure to measure-theoretic entropies. Several examples are studied in detail. The final chapter outlines significant results and some applications of ergodic theory to other branches of mathematics.
- GenresMathematics
250 pages, Hardcover
First published December 16, 1981
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December 23, 2022
In view of how immense the corpus of modern mathematics has by now become, a pertinent issue the eager student will frequently encounter is the crucial difference between having heard of something and actually knowing it well. For it can indeed assist comprehension to accept on faith the result of a theorem without having gone through its proof, as so often happens in the introductory segment of a talk at a colloquium. To be sure, attending colloquia and listening to the high-level overview the speakers give of their reasons for engaging in research on a given topic can be an apt means by which the beginning graduate student can find his bearings in a new field. But one will never become a good mathematician unless sooner or later he descends from the Olympian heights and gets down to the painstaking grind of following an argument in detail and experimenting with typical examples.
This elementary, if unfortunately often unheeded observation is why Peter Walters’ old textbook in the Springer GTM series from 1982, An Introduction to Ergodic Theory, is so commendable. For one will find in this text not only a sure guide to the elementary concepts of ergodic theory but a steady development of them all the way up to advanced results on the frontier of knowledge. Most all statements are proved in full with the exception of a few from Royden’s Real Analysis (see our review here) quoted in chapter two and a handful of deep results like theorem 4.28. Proofs are of manageable length, normally not more than a page (hence, he can’t include the KAM theorem and such). Walters’ running commentary proves to be very helpful in giving insight into the meaning of what is about to be proved or has just been proved, usually by suggesting pictorially why the derivation succeeds or how a statement could be strengthened. The exposition in Walters is noticeably slower paced and more methodical than in Arnold and Avez’ Ergodic Problems of Classical Mechanics (cf. our immediately preceding review, here); hence one gets a rounded picture of the subject rather than just a collection of highlights.
Contents: what is nice about ergodic theory for the student not necessarily wanting to become a specialist is that it can serve as a proving ground for one’s understanding of analysis and related fields. Just to name a couple instances, among other things the Lebesgue covering lemma plays an important role throughout and convexity of the pressure is derived from Hölder’s inequality. Moreover, Walters assumes as known background on compact abelian groups (not necessarily finite-dimensional).
The first major result is Birkhoff’s ergodic theorem in chapter one [pp. 34-39]. Compare with the finite-dimensional case in Halmos (see our review of the latter’s Finite-dimensional Vector Spaces, here). The proof here employs the dominated convergence theorem. Walters’ understated expository style does pay attention to aesthetics; note how much more efficient proof of theorem 1.31 characterizing finite Markov chains is than in Shiryaev (see our review of the latter’s fine textbook entitled Probability, here). The level of craftsmanship bears comparison with Walter Rudin in Principles of Mathematical Analysis (see our review, here).
Walters’ next move is to employ spectral methods in the study of measure-preserving transformations. The idea going back to Koopman is quite ingenious: given the transformation T, define its action on functions via Uf = f ○ T, thereby linearizing the problem and rendering it susceptible to operator-theoretic techniques. For the original reference, see G.D. Birkhoff and B.O. Koopman, Recent contributions to the ergodic theory, Proceedings of the National Academy of Sciences 18(3), 279–282 (1932). As Walters explains, the spectrum of the linear operator U is an isomorphism invariant and – what is surprising – sensitive to the ergodic properties of T. For instance, T is weak mixing iff it has continuous spectrum (theorem 1.26). In fact, the representation theorem 3.6 completely characterizes ergodic transformations with discrete spectrum up to conjugacy [p. 73].
In chapter four we finally get to another important invariant, the so-called entropy. Computations are written out in satisfying detail (though for the expert they would be routine). The entropy can be rather hard to calculate in principle unless one has a condition that would show h=h(A) for some partition A or at least h=lim h(Aₙ) : §4.6, examples in which the technique is applied in §§4.7, 4.9, namely Bernoulli shifts and Kolmogorov automorphisms such as the Markov shift, if its transition matrix is irreducible and aperiodic [p. 111].
Chapter five switches gears and discusses topological dynamics, a rewarding topic because its concepts relating to the asymptotic behavior are fairly intuitive (the wandering and non-wandering sets) yet one can prove a lot about them even without having any measure around. This constitutes an emerging theme for the remainder of the book, viz., to investigate the relations between measure-theoretic and topological properties of a mapping (in the case it has both). To this reviewer, however, the most exciting section of the present text comes in chapter six on invariant measures for continuous transformations. Let X be a compact metrizable space and define the space M(X) of probability measures on it. Due to embedding established by the Riesz representation theorem, one may regard a probability measure as a positive linear functional on the space of continuous functions C(X) and M(X) becomes a collection in the dual C(X)*. Of course, a transformation T acts on M(X) in the obvious way. The reason this device is so appealing is that it lends non-trivial content to the Riesz representation theorem which remains purely abstract in Royden for instance, and makes relevant functional-analytic concepts such as the weak* topology, convexity, extremality and so on in a context in which they can be brought to bear upon intuitive geometrical properties of the transformation T, like invariance, ergodicity, mixing, periodicity and wandering. Here we have a very attractive mix between the abstract and concrete, since many continuous transformations T of compact metric spaces X are known and fairly concrete. The existence of invariant measures is obtained by direct argument although one could also use the Markov-Kakutani fixed-point theorem [p. 151]. Theorem 6.2 provides a nice characterization of ergodicity in terms of M(X,T), [p. 152f], and there is as well a good interpretation of ergodicity and mixing in terms of the weak* topology on M(X,T), [p. 154f]. A subtlety as to whether a function lies in L² or L¹ matters here!
Chapter seven introduces a topological version of the entropy, which was worked out very cleverly by Adler, Konheim and McAndrew and later refined by Dinaburg and Bowen. Instead of partitions of a measure space, one has open covers of a topological space and counting the number of sets in a finite subcover takes the place of the measure. Again, topological entropy can be just as difficult to estimate as the measure-theoretic entropy, but Walters is up to the challenge! Theorem 8.6 relating the topological entropy to measure-theoretic entropy is perhaps hardest in book, with a 1 + ½ page proof, intricate but not deep [pp. 188-190]. Theorem 8.7 relating extremality and ergodicity in M(X) is nice too. Theorems 8.10 and 8.11 supply some good illustrations of practical calculations with entropies [p. 194f], while theorem 8.14 on the entropy of a linear transformation A may be not as hard but makes good sense [p. 201]: it shows how if one can visualize how T acts locally one can compute topological entropy via partitions. Since successive application of A will stretch a small cell it is not surprising that the entropy can be related to the logarithms of the eigenvalues of A. Indeed, one can see that for a general diffeomorphism its entropy must be a suitable average over the logarithms of the eigenvalues of its local linearizations. Theorems 8.17, 18 on how the unique measure with maximal entropy describes the distribution of periodic points of a two-sided Markov chain are also fundamental and intuitive.
Chapter nine defines topological pressure as a generalization of entropy. The fundamental result of the chapter relating topological pressure to measure-theoretic entropy, theorem 9.10, has a 4-page computational proof [pp. 218-221] – only the most industrious will venture here! The variational principle characterizes what pressure really is but Walters defers to David Ruelle for the connection to statistical mechanics (thermodynamically, it is minus the free energy, it remains unclear why it was given the name pressure). Theorem 9.12 uses the inverse Legendre transform to get the entropy in terms of the pressure. Then theorem 9.13 implies the existence of equilibrium states corresponding to every distribution f of the internal energy, for expansive homeomorphisms at least.
For a change of pace, the closing chapter ten invites the curious reader to follow up in the literature by quoting many advanced structural results without proof. For ergodic theory crops up everywhere when one has a dynamical system and is interested in characterizing its long-term behavior, where one may replace time translation by the action of a group in general. For instance, a statement of some deep theorems relating ergodic invertible measure-preserving transformations to factors of von Neumann algebras [p. 237]. Another good reference to consult would be a review article by G.W. Mackey on the close connections between ergodic theory and probability, harmonic analysis and group representations [Harmonic analysis as the exploitation of symmetry – a historical survey, Bulletin of the American Mathematical Society 3, 543-697 (1980)].
Summary: Walters complements Arnold and Avez well, being less comprehensive but fuller and more readable in its treatment of what it does contain – with an emphasis on models that can be completely characterized from the mathematical point of view to the practical exclusion of any realistic systems arising in physics. After all, one really needs both in order to arrive at a mature judgment. The exposition of the main ideas of ergodic theory in Walters is notably more connected and better maintains a logical flow. Thus, good for cutting one’s teeth on operator theory, some non-trivial measure theory, a little functional analysis and lastly some statistical mechanics, which would be relevant if concepts like entropy and topological pressure are to have meaning beyond their formal definition. The biggest drawback with the present text is its lack of homework exercises. Five stars: after reading Arnold and Avez one will know about ergodic theory whereas after Walters one will know it!
This elementary, if unfortunately often unheeded observation is why Peter Walters’ old textbook in the Springer GTM series from 1982, An Introduction to Ergodic Theory, is so commendable. For one will find in this text not only a sure guide to the elementary concepts of ergodic theory but a steady development of them all the way up to advanced results on the frontier of knowledge. Most all statements are proved in full with the exception of a few from Royden’s Real Analysis (see our review here) quoted in chapter two and a handful of deep results like theorem 4.28. Proofs are of manageable length, normally not more than a page (hence, he can’t include the KAM theorem and such). Walters’ running commentary proves to be very helpful in giving insight into the meaning of what is about to be proved or has just been proved, usually by suggesting pictorially why the derivation succeeds or how a statement could be strengthened. The exposition in Walters is noticeably slower paced and more methodical than in Arnold and Avez’ Ergodic Problems of Classical Mechanics (cf. our immediately preceding review, here); hence one gets a rounded picture of the subject rather than just a collection of highlights.
Contents: what is nice about ergodic theory for the student not necessarily wanting to become a specialist is that it can serve as a proving ground for one’s understanding of analysis and related fields. Just to name a couple instances, among other things the Lebesgue covering lemma plays an important role throughout and convexity of the pressure is derived from Hölder’s inequality. Moreover, Walters assumes as known background on compact abelian groups (not necessarily finite-dimensional).
The first major result is Birkhoff’s ergodic theorem in chapter one [pp. 34-39]. Compare with the finite-dimensional case in Halmos (see our review of the latter’s Finite-dimensional Vector Spaces, here). The proof here employs the dominated convergence theorem. Walters’ understated expository style does pay attention to aesthetics; note how much more efficient proof of theorem 1.31 characterizing finite Markov chains is than in Shiryaev (see our review of the latter’s fine textbook entitled Probability, here). The level of craftsmanship bears comparison with Walter Rudin in Principles of Mathematical Analysis (see our review, here).
Walters’ next move is to employ spectral methods in the study of measure-preserving transformations. The idea going back to Koopman is quite ingenious: given the transformation T, define its action on functions via Uf = f ○ T, thereby linearizing the problem and rendering it susceptible to operator-theoretic techniques. For the original reference, see G.D. Birkhoff and B.O. Koopman, Recent contributions to the ergodic theory, Proceedings of the National Academy of Sciences 18(3), 279–282 (1932). As Walters explains, the spectrum of the linear operator U is an isomorphism invariant and – what is surprising – sensitive to the ergodic properties of T. For instance, T is weak mixing iff it has continuous spectrum (theorem 1.26). In fact, the representation theorem 3.6 completely characterizes ergodic transformations with discrete spectrum up to conjugacy [p. 73].
In chapter four we finally get to another important invariant, the so-called entropy. Computations are written out in satisfying detail (though for the expert they would be routine). The entropy can be rather hard to calculate in principle unless one has a condition that would show h=h(A) for some partition A or at least h=lim h(Aₙ) : §4.6, examples in which the technique is applied in §§4.7, 4.9, namely Bernoulli shifts and Kolmogorov automorphisms such as the Markov shift, if its transition matrix is irreducible and aperiodic [p. 111].
Chapter five switches gears and discusses topological dynamics, a rewarding topic because its concepts relating to the asymptotic behavior are fairly intuitive (the wandering and non-wandering sets) yet one can prove a lot about them even without having any measure around. This constitutes an emerging theme for the remainder of the book, viz., to investigate the relations between measure-theoretic and topological properties of a mapping (in the case it has both). To this reviewer, however, the most exciting section of the present text comes in chapter six on invariant measures for continuous transformations. Let X be a compact metrizable space and define the space M(X) of probability measures on it. Due to embedding established by the Riesz representation theorem, one may regard a probability measure as a positive linear functional on the space of continuous functions C(X) and M(X) becomes a collection in the dual C(X)*. Of course, a transformation T acts on M(X) in the obvious way. The reason this device is so appealing is that it lends non-trivial content to the Riesz representation theorem which remains purely abstract in Royden for instance, and makes relevant functional-analytic concepts such as the weak* topology, convexity, extremality and so on in a context in which they can be brought to bear upon intuitive geometrical properties of the transformation T, like invariance, ergodicity, mixing, periodicity and wandering. Here we have a very attractive mix between the abstract and concrete, since many continuous transformations T of compact metric spaces X are known and fairly concrete. The existence of invariant measures is obtained by direct argument although one could also use the Markov-Kakutani fixed-point theorem [p. 151]. Theorem 6.2 provides a nice characterization of ergodicity in terms of M(X,T), [p. 152f], and there is as well a good interpretation of ergodicity and mixing in terms of the weak* topology on M(X,T), [p. 154f]. A subtlety as to whether a function lies in L² or L¹ matters here!
Chapter seven introduces a topological version of the entropy, which was worked out very cleverly by Adler, Konheim and McAndrew and later refined by Dinaburg and Bowen. Instead of partitions of a measure space, one has open covers of a topological space and counting the number of sets in a finite subcover takes the place of the measure. Again, topological entropy can be just as difficult to estimate as the measure-theoretic entropy, but Walters is up to the challenge! Theorem 8.6 relating the topological entropy to measure-theoretic entropy is perhaps hardest in book, with a 1 + ½ page proof, intricate but not deep [pp. 188-190]. Theorem 8.7 relating extremality and ergodicity in M(X) is nice too. Theorems 8.10 and 8.11 supply some good illustrations of practical calculations with entropies [p. 194f], while theorem 8.14 on the entropy of a linear transformation A may be not as hard but makes good sense [p. 201]: it shows how if one can visualize how T acts locally one can compute topological entropy via partitions. Since successive application of A will stretch a small cell it is not surprising that the entropy can be related to the logarithms of the eigenvalues of A. Indeed, one can see that for a general diffeomorphism its entropy must be a suitable average over the logarithms of the eigenvalues of its local linearizations. Theorems 8.17, 18 on how the unique measure with maximal entropy describes the distribution of periodic points of a two-sided Markov chain are also fundamental and intuitive.
Chapter nine defines topological pressure as a generalization of entropy. The fundamental result of the chapter relating topological pressure to measure-theoretic entropy, theorem 9.10, has a 4-page computational proof [pp. 218-221] – only the most industrious will venture here! The variational principle characterizes what pressure really is but Walters defers to David Ruelle for the connection to statistical mechanics (thermodynamically, it is minus the free energy, it remains unclear why it was given the name pressure). Theorem 9.12 uses the inverse Legendre transform to get the entropy in terms of the pressure. Then theorem 9.13 implies the existence of equilibrium states corresponding to every distribution f of the internal energy, for expansive homeomorphisms at least.
For a change of pace, the closing chapter ten invites the curious reader to follow up in the literature by quoting many advanced structural results without proof. For ergodic theory crops up everywhere when one has a dynamical system and is interested in characterizing its long-term behavior, where one may replace time translation by the action of a group in general. For instance, a statement of some deep theorems relating ergodic invertible measure-preserving transformations to factors of von Neumann algebras [p. 237]. Another good reference to consult would be a review article by G.W. Mackey on the close connections between ergodic theory and probability, harmonic analysis and group representations [Harmonic analysis as the exploitation of symmetry – a historical survey, Bulletin of the American Mathematical Society 3, 543-697 (1980)].
Summary: Walters complements Arnold and Avez well, being less comprehensive but fuller and more readable in its treatment of what it does contain – with an emphasis on models that can be completely characterized from the mathematical point of view to the practical exclusion of any realistic systems arising in physics. After all, one really needs both in order to arrive at a mature judgment. The exposition of the main ideas of ergodic theory in Walters is notably more connected and better maintains a logical flow. Thus, good for cutting one’s teeth on operator theory, some non-trivial measure theory, a little functional analysis and lastly some statistical mechanics, which would be relevant if concepts like entropy and topological pressure are to have meaning beyond their formal definition. The biggest drawback with the present text is its lack of homework exercises. Five stars: after reading Arnold and Avez one will know about ergodic theory whereas after Walters one will know it!
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