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Symbolic extensions and continuity properties of the entropy. (English) Zbl 1235.37007
The first part of the article is a short presentation of the entropy theory of symbolic extensions. The author states the theorem of symbolic extension entropy due to M. Boyle and T. Downarowicz [Invent. Math. 156, No. 1, 119–161 (2004; Zbl 1216.37004)]. The second and main part of the paper focuses on certain continuity properties of the entropy which are deduced from the main results of that theory. Three main results are obtained:
First, the author proves that for generic continuous dynamical systems in the interval, the entropy (as a function of the invariant measure) is nowhere continuous. The result follows from the complete characterization of possible entropy functions of continuous dynamical systems, due to T. Downarowicz and J. Serafin [Isr. J. Math. 135, 221–250 (2003; Zbl 1054.37004)].
Second, the author deduces a nowhere continuity property of the entropy function for the \(C^1\)-examples of volume-preserving surface diffeomorphisms due to T. Downarowicz and S. Newhouse [Invent. Math. 160, No. 3, 453–499 (2005; Zbl 1067.37018)]. The entropy in these examples is nowhere continuous when restricted to a nonempty weak\(^*\)-compact subset of invariant measures.
Third, and last, the author shows a way to estimate if a nonnegative function defined on a compact metric space (which, in this case, is the entropy \(h\) as a function of the invariant measure) fails to be upper semicontinuous. He proposes to investigate whether it is a difference of non-nengative upper semicontinuous functions. In fact, for \(C^r\) interval maps and \(C^2\) surface diffeomorphisms, \(h\) is a difference of such functions, and so the set of its continuity points is a dense \({\mathcal G}_{\delta}\)-set of invariant measures.

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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