On processes which cannot be distinguished by finite observation |
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Authors: | Yonatan Gutman Michael Hochman |
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Institution: | (1) Departmemt of Mathematics, Edmond J. Safra Campus Hebrew University of Jerusalem, Jerusalem, 91904, Israel;(2) Department of Mathematics, Fine Hall, Washington Rd., Princeton, NJ 08544, USA |
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Abstract: | A function J defined on a family C of stationary processes is finitely observable if there is a sequence of functions s
n
such that s
n
(x
1,…, x
n
) → J(X) in probability for every process X=(x
n
) ∈ C. Recently, Ornstein and Weiss proved the striking result that if C is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant defined
on C is entropy 8]. We sharpen this in several ways. Our main result is that if X → Y is a zero-entropy extension of finite entropy ergodic systems and C is the family of processes arising from generating partitions of X and Y, then every finitely observable function on C is constant. This implies Ornstein and Weiss’ result, and extends it to many other families of processes, e.g., it follows
that there are no nontrivial finitely observable isomorphism invariants for processes arising from the class of Kronecker
systems, the class of mild mixing zero entropy systems, or the class of strong mixing zero entropy systems. It also follows
that for the class of processes arising from irrational rotations, every finitely observable isomorphism invariant must be
constant for rotations belonging to a set of full Lebesgue measure.
This research was supported by the Israel Science Foundation (grant No. 1333/04) |
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Keywords: | |
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