Abstract
Zero entropy processes are known to be deterministic—the past determines the present. We show that each is isomorphic, as a system, to a finitarily deterministic one, i.e., one in which to determine the present from the past it suffices to scan a finite (of random length) portion of the past. In fact we show more: the finitary scanning can be done even if the scanner is noisy and passes only a small fraction of the readings, provided the noise is independent of our system.
The main application we present here is that any zero entropy system can be extended to a random Markov process (namely one in which the conditional distribution of the present given the past is a mixture of finite state Markov chains). This allows one to study zero entropy transformations using a procedure completely different from the usual cutting and stacking.
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Kalikow, S., Katznelson, Y. & Weiss, B. Finitarily deterministic generators for zero entropy systems. Israel J. Math. 79, 33–45 (1992). https://doi.org/10.1007/BF02764801
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DOI: https://doi.org/10.1007/BF02764801