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Infinite partitions and Rokhlin towers

Published online by Cambridge University Press:  16 September 2011

STEVEN KALIKOW
STEVEN KALIKOW*
Affiliation:
Department of Mathematics, University of Memphis, 3725 Norriswood, Memphis, TN 38152, USA (email: skalikow@memphis.edu)
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Abstract

We find a countable partition P on a Lebesgue space, labeled {1,2,3,…}, for any non-periodic measure-preserving transformation T such that P generates T and, for the T,P process, if you see an n on time −1 then you only have to look at times −n,1−n,…−1 to know the positive integer i to put at time 0 . We alter that proof to extend every non-periodic T to a uniform martingale (i.e. continuous g function) on an infinite alphabet. If T has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an independent identically distributed process on ℤ2 when you read from right to left but where each column determines the next if you read left to right.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 707 - 738
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Alpern, S.. Return times and conjugates of an antiperiodic transformation. Ergod. Th. & Dynam. Sys. 1(2) (1981), 135–143.CrossRefGoogle Scholar
[2]Berbee, H.. Chains with infinite connections: uniqueness and Markov representation. Probab. Theory Related Fields 76 (1987), 243–253.CrossRefGoogle Scholar
[3]Doeblin, W. and Fortet, R.. Sur des chaǐ nes à liaisons complètes. Bull. Soc. Math. France 65 (1937), 132–148.CrossRefGoogle Scholar
[4]Kalikow, S.. Random Markov processes and uniform martingales. Israel J. Math. 71(1) (1990), 33–54.CrossRefGoogle Scholar
[5]Kalikow, S., Katznelson, Y. and Weiss, B.. Finitarily deterministic generators for zero entropy systems. Israel J. Math. 79 (1992), 33–45.CrossRefGoogle Scholar
[6]Keane, M.. Strongly mixing g-measures. Invent. Math. 16 (1972), 309–324.CrossRefGoogle Scholar
[7]King, J. L.. Dilemma of the sleeping stockbroker. Amer. Math. Monthly 99(4) (1992), 335–338.CrossRefGoogle Scholar
[8]Lehrer, E. and Weiss, B.. An ε-free Rokhlin lemma. Ergod. Th. & Dynam. Sys. 2(1) (1982), 45–48.CrossRefGoogle Scholar
[9]Ornstein, D.. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, New Haven, CT, 1974, Theorem 5, p. 53.Google Scholar
[10]Ornstein, D.. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, New Haven, CT, 1974, Corollary 3, p. 44.Google Scholar
[11]Parry, W.. Entropy and Generators in Ergodic Theory (Mathematical Lecture Note Series). Benjamin, Inc., New York–Amsterdam, 1968, xii + 124 pp.Google Scholar
[12]Petit, B.. Schemes de Bernoulli et g-measure. C. R. Acad. Sci. Paris Sér. A 280 (1975), 17–20.Google Scholar
[13]Prikhod́ko, A. A.. Partitions of the phase space of a measure-preserving Zd-action into towers. Mat. Zametki 65(5) (1999), 712–725.Google Scholar
[14]Rokhlin, V.. Generators in ergodic theory. Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron. 18(1) (1963), 26–32.Google Scholar
[15]Rudolph, D.. A two-valued step coding for ergodic flows. Math. Z. 150(3) (1976), 201–220.CrossRefGoogle Scholar
[16]Ryzhikov, V. V.. The Rohlin–Halmos property without ϵ does not hold for the actions of the group Z 2. Mat. Zametki 44(2) (1988), 208–215, 287 (in Russian), translation in Math. Notes 44(1–2) (1988), 596–600 (1989).Google Scholar
[17]Şahin, A. A.. The ℤd Alpern multi-tower theorem for rectangles: a tiling approach. Dyn. Syst. 24(4) (2009), 485–499.CrossRefGoogle Scholar