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Entropy 2016, 18(12), 428; doi:10.3390/e18120428

Fiber-Mixing Codes between Shifts of Finite Type and Factors of Gibbs Measures

Department of Mathematics, Ajou University, 206 Worldcup-ro, Suwon 16499, Korea
Academic Editor: Tomasz Downarowicz
Received: 7 October 2016 / Revised: 20 November 2016 / Accepted: 24 November 2016 / Published: 30 November 2016
(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)
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Abstract

A sliding block code π : X Y between shift spaces is called fiber-mixing if, for every x and x in X with y = π ( x ) = π ( x ) , there is z π - 1 ( y ) which is left asymptotic to x and right asymptotic to x . A fiber-mixing factor code from a shift of finite type is a code of class degree 1 for which each point of Y has exactly one transition class. Given an infinite-to-one factor code between mixing shifts of finite type (of unequal entropies), we show that there is also a fiber-mixing factor code between them. This result may be regarded as an infinite-to-one (unequal entropies) analogue of Ashley’s Replacement Theorem, which states that the existence of an equal entropy factor code between mixing shifts of finite type guarantees the existence of a degree 1 factor code between them. Properties of fiber-mixing codes and applications to factors of Gibbs measures are presented. View Full-Text
Keywords: shift of finite type; entropy of a shift space; infinite-to-one; fiber-mixing; replacement theorem; class degree; Gibbs measure shift of finite type; entropy of a shift space; infinite-to-one; fiber-mixing; replacement theorem; class degree; Gibbs measure
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Jung, U. Fiber-Mixing Codes between Shifts of Finite Type and Factors of Gibbs Measures. Entropy 2016, 18, 428.

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