On a Class of Generalized Baker's Transformations

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.