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Generalized baker's transformations

Published online by Cambridge University Press:  19 September 2008

Christopher J. Bose
Christopher J. Bose
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA
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Abstract

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A class of automorphisms of the unit square called generalized baker'stransformations (gbt) is defined in such a way that every stationary stochastic process may be represented as the movement of a simple partition of the square under a gbt. This extends the classical example of the representation of independent processes by the well-known baker's transformation.

Every ergodic, positive-entropy automorphism is measurably isomorphic to some gbt (again generalizing the classical result about Bernoulli shifts), and we show that a large class of gbt's satisfying certain continuity restrictions are actually measurably isomorphic to Bernoulli shifts.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 1 , March 1989 , pp. 1 - 17
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[A]Adler, R.. Continued fractions and Bernoulli trials. Ergodic Theory, A Seminar, Courant Institute of Mathematical Sciences. Moser, J., Phillips, E. & Varadhan, S., eds. (1975), Ch. 16, pp 111–120.Google Scholar
[A, Y]Alexander, J. C. & Yorke, J. A.. Fat baker's transformations. Ergod. Th. & Dynam. Sys. 4 (1984), 1–23.CrossRefGoogle Scholar
[Be]Berbee, H. C. P.. Chains with infinite connections: Uniqueness and Markov representation. Centrum Voor Wiskunde en Informatica. Report MS-R8509 (11 1985).Google Scholar
[Bl]Bose, C.. Generalized baker's transformations. Thesis, University of Toronto. (1986).Google Scholar
[B2]Bose, C.. Invariant measures and equilibrium states for piecewise C 1+s mappings of the interval. Trans. Amer. Math. Soc. (1989) to appear.Google Scholar
[B0]Bowen, R.. Bernoulli maps of the interval. Israel J. Math. 28 (1977), 161–168.CrossRefGoogle Scholar
[F, O]Friedman, N. A. & Ornstein, D. S.. On isomorphism of weak-Bernoulli transformations. Adv. Math. 5 (1971), 365–394.CrossRefGoogle Scholar
[H]Halmos, P.. Lectures on Ergodic Theory. Chelsea Publishing Company: New York, 1956.Google Scholar
[K]Krieger, W.. On entropy and generators of measure preserving transformations. Trans. Amer. Math. Soc. 149 (1970), 453–464.CrossRefGoogle Scholar
[L]Ledrappier, F.. Principe variationnel et systèmes dynamiques symboliques. Z. Wahrscheinlichkeitstheorie verw. Gebiete. 30 (1974), 185–202.CrossRefGoogle Scholar
[M]Moser, J.. Stable and random motions in dynamical systems. Hermann Weyl Lectures, Annals of Mathematical Studies Number 77. Princeton University Press: Princeton, 1973.Google Scholar
[O]Ornstein, D. S.. Two Bernoulli shifts with infinite entropy are isomorphic. Adv. Math. 5 (1970), 339–348.CrossRefGoogle Scholar
[R]Rohlin, V. A.. Exact endomorphisms of a Lebesgue space. Transl. Amer. Math. Soc. Series II. 39 (1964), 1–36.CrossRefGoogle Scholar
[W]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153.CrossRefGoogle Scholar