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The entropy theory of symbolic extensions
Authors:Email author" target="_blank">Mike?BoyleEmail author  Email author" target="_blank">Tomasz?DownarowiczEmail author
Institution:(1) Department of Mathematics, University of Maryland, 20742-4015 College Park, MD, USA;(2) Institute of Mathematics, Technical University, Wybrzeze Wyspianacuteskiego 27, 50-370 Wroclstrokaw, Poland
Abstract:Fix a topological system (X,T), with its space K(X,T) of T-invariant Borel probabilities. If (Y,S) is a symbolic system (subshift) and phiv:(Y,S)rarr(X,T) is a topological extension (factor map), then the function hphivext on K(X,T) which assigns to each mgr the maximal entropy of a measure ngr on Y mapping to mgr is called the extension entropy function of phiv. The infimum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by hsex. In this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on (X,T). The entropy structure hamilt is a sequence of entropy functions hk defined with respect to a refining sequence of partitions of X (or of X×Z, for some auxiliary system (Z,R) with simple dynamics) whose boundaries have measure zero for all the invariant Borel probabilities. We develop the functional analysis and computational techniques to produce many dynamical examples; for instance, we resolve in the negative the question of whether the infimum of the topological entropies of symbolic extensions of (X,T) must always be attained, and we show that the maximum value of hsex need not be achieved at an ergodic measure. We exhibit several characterizations of the asymptotically h-expansive systems of Misiurewicz, which emerge as a fundamental natural class in the context of the entropy structure. The results of this paper are required for the Downarowicz-Newhouse results DN] on smooth dynamical systems. Mathematics Subject Classification (2000) Primary: 37B10; Secondary: 37B40, 37C40, 37C45, 37C99, 37D35
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