On a Class of Generalized Baker's Transformations

Rahe, M.

Canadian journal of mathematics, Tome 45 (1993) no. 3, pp. 638-649

Résumé

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{xf(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.

Détail

DOI : 10.4153/CJM-1993-035-0

Mots-clés : 28A65, generalized baker's transformation, random Markov chains, uniform martingales, weak Bernoulli

Bibliographie

[1] 1. Bose, C. J., Generalized baker's transformations, Ergodic Theory and Dynamical Systems 9(1989), 1–17. Google Scholar

[2] 2. Kalikow, S., Random Markov Processes and Uniform Martingales, Isreal Journal of Mathematics 71(1990), 33–54. Google Scholar

[3] 3. del Junco, A. and Rahe, M., Finitary codings and weak Bernoulli partitions, Proceedings of the American Mathematical Society 75(1979), 259–264. Google Scholar