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On a Class of Generalized Baker's Transformations

Published online by Cambridge University Press:  20 November 2018

M. Rahe*
Affiliation:
Texas A & M University, College Station, Texas, U.S.A.
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Abstract

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Let f define a baker's transformation (Tf, Pf). We find necessary and sufficient conditions on f for (Tf, Pf) to be an N(ω)-step random Markov chain. Using this result, we give a simplified proof of Bose's results on Holder continuous baker's transformations where f is bounded away from zero and one. We extend Bose's results to show that, for the class of baker's transformations which are random Markov chains where TV has finite expectation, a sufficient condition for weak Bernoullicity is that the Lebesgue measure λ{x f(x) = 0 or f(x) = 1} = 0. We also examine random Markov chains satisfying a strictly weaker condition, those for which the differences between the entropy of the process and the conditional entropy given the past to time n form a summable sequence; and we show that a similar result holds. A condition is given on/ which is weaker than Holder continuity, but which implies that the entropy difference sequence is summable. Finally, a particular baker's transformation is exhibited as an easy example of a weakly Bernoulli transformation on which the supremum of the measure of atoms indexed by n-strings decays only as the reciprocal of n.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Bose, C. J., Generalized baker's transformations, Ergodic Theory and Dynamical Systems 9(1989), 117.Google Scholar
2. Kalikow, S., Random Markov Processes and Uniform Martingales, Isreal Journal of Mathematics 71(1990), 3354.Google Scholar
3. del Junco, A. and Rahe, M., Finitary codings and weak Bernoulli partitions, Proceedings of the American Mathematical Society 75(1979), 259264.Google Scholar