Entropy Versus Speed in Ergodic Markov Maps

Abstract

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Let $f = (f_1 , f_2 , \cdots ,f_n )$ be a sequence of positive numbers. We construct a class of piecewise linear, ergodic Markov maps $\mathcal{C}_f $ such that for each $\tau \in \mathcal{C}_f $, there exist an interval I and a partition $\{ I_i \} _{i = 1}^n $ of I with the property that $\mu (I_i ) = f_i $, where $\mu $ is the unique measure invariant under $\tau $. In $\mathcal{C}_f $ the trade-off between entropy (randomness) and the speed of numerically computing the orbit $\{ \tau ^j (x)\} _{j\geqq 0} $ can be assessed.