# An introduction to ergodic theory - Walters P.

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{0, l,...,k — 1} and T2 is the two-sided shift on X2 = П-* {0,1.....1 - I}-00

and that m2 are shift invariant measures on Xly X2 respeciively. Suppose ф: X, -*X2 is an invertible measure-preserving transformation with фТг(х) = Т2ф(х) a.e.. We could consider ф to be computable if for almost all x e A, we could compute (ф(х))0, the zero-th coordinate of ф(х), from just a finite number of coordinates of x. This leads to the notion of finitary isomorphism studied by M. Keane and M. Smorodinsky (Keaneand Smorodinsky [1],[2]).

They showed that two Bernoulli shifts with equal entropy are finitary isomorphic, strengthening the theorem of Ornstein (Theoiem 4.28) in this case. They also extended this result to irreducible aperiodic Markov shifts.

A topological version of Ornstein’s theorem has been obtained by R. L. Adler and B. Marcus [1]. They showed that for the class of irreducible aperiodic topological Markov chains topological entropy is a complete invariant for the equivalence relation of “almost topological conjugacy”.

§10.5 Transformations of Intervals

The study of transformations of intervals can arise in many ways from studying other problems. Consideration of billiard problems (spe Sinai [2]) leads to the consideration of maps of intervals, so does the study of the well-known Lorenz differential equation (see Williams [1]), and certain problems

§10.6 Further Reading

239

in complex function theory can be reduced in a similar way (see Bowen [6]). The family of maps Ta(x) = лх(1 — x) mod 1 of [0,1), where a > 0, have received a lot of attention. One of the main problems is to decide which maps preserve a measure which is absolutely continuous relative to Lebesgue measurr and then to determine the ergodic properties relative to this measure. There are many recent results in this direction.

§10.6 Further Reading

We indicate here where some topics not covered in this book can be found.

A good source for the history of ergodic theory and its close connections with probability theory, harmonic analysis and group representations is Mackey’s survey article, Mackey [1].

The modern connections of ergodic theory with statistical mechanics is described in the books of Ruelle (Ruelle [2]) and Israel (Israel [1]). The connections of this theory with the study of diffeomorphisms is given in Ruelle [2] and Bowen [2j, [3]. See Arnold and Avez [1] for some earlier theory.

We have not described the theory of flows of measure-preserving transformations (i.e. measure-preserving actions of R). These are studied in Hopfs book (Hopf [1]) and Sinai’s book (Sinai [2]). Also presented in Sinai’s book is an introduction to the theory of geodesic flows. Geodesic flows are, perhaps, the most important examples of flows. Their importance and smooth structure is descnoed in Abraham and Marsden [1]. Another important class of flows, billiard flows, are also studied in Sinai [2].

The important topic of approximation of measure-preserving transformations by periodic transformations is described in Katok and Stepin [1].

H. Furstenberg has used ergodic theory and topological dynamics to give proofs of some important theorems in number theory. In particular he has given a proof of Szemeredi's theorem (Furstenberg [2]). Several other results are proved in Furstenberg and Weiss [1].

A detailed description of many theorems in ergodic theory known before 1967 appears in Vershik and Yuzviskii [1], and a description of much of the theory discovered between 1967 and 1974 appears in Katok, Sinai and Stepin [1]. Also many references are given there. The notes of Jacobs (Jacobs [2]) give an excdlent account of ergodic theory up to 1962. A detailed account of entropy theory is given in Rohlin [3]. The proofs of seme of the theorems on entropy that we stated without proof are given in Parry [2].

References

R. Abraham and J. E. Marsden [1] Foundations of Mechanics, 2nd ed. Benjamin, Reading, MA, 1978.

L. M. Abramov

[1] Metric automorphisms with quasi-discrete spectrum. Amer. Math. Soc. Translations 2, 39, 37-56 (1964).

R. L. Adler, A. G. Konheim, and М. H. McAndrew [1] Topological entropy, Trans. Amer. Math. Soc. 114, 309-319 (1965).

R. L. Adler and B. Marcus [1] Topological entropy and equivalence of dynamical systems. Memoirs of Amer. Math Soc. No. 219, 1979.

V. 1. Arnold and A Avcz

СП Hriioiitc problems of Classical Mechanics benjamin. New York, 1968.

L. Ausfander, L. W. Green and F. J. Hahn [1] Flows on homogeneous spaces. Princeton Univ. Press, Ann. Math Studies No. 53, 1963.

K. R. Berg

[]] Convolution of invariant measures, maximal entropy. Math. Systems Theory 3, 146-150(1969).

P. Billingsley

[1] Ergodic Theory and Information, Wiley, New York, 1965.

G. D. BirkolT

[I] Dynamical systems. Amer. Math. Soc. Colloqium Publications Vol IX, 1927.

J. R. Blum and D. L. Hanson

[1] On the isomorphism problem for Bernoulli schemes. Bull. Amer. Math. Soc. 69, 221-223(1963).

[2] On the mean ergodic theorem for subsequences. Bull Amer. Math. Soc. 66, 308-311 (1960).

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