Finitarily deterministic generators for zero entropy systems |
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Authors: | Steven Kalikow Yitzhak Katznelson Benjamin Weiss |
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Institution: | (1) Department of Mathematics, Cornell University, Ithaca, N.Y., U.S.A.;(2) Department of Mathematics, Stanford University, 94305-2125 Stanford, CD, U.S.A.;(3) Department of Mathematics, Hebrew University, Jerusalem, Israel |
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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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Keywords: | |
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