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Symbolic extensions for nonuniformly entropy expanding maps. (English) Zbl 1277.37049
Symbolic extensions of a smooth dynamical system, if they exist, give some coding information of the dynamical behaviors. There are \(C^1\) examples for which there is no symbolic extension. It is proved that all \(C^\infty\) diffeomorphisms are asymptotically \(h\)-expansive [J. Buzzi, Israel J. Math. 100, 125–161 (1997; Zbl 0889.28009)], and all asymptotically \(h\)-expansive systems with finite entropy admit principal symbolic extensions [M. Boyle et al., Forum Math. 14, No. 5, 713–757 (2002; Zbl 1030.37012)]. So the results are quite satisfying in the \(C^\infty\)-category. On the other hand, it is much more involved in the finite regularity cases. Nonetheless, it is conjectured that symbolic extensions do exist in the \(C^r\)-regularity (for all \(r>1\), see [T. Downarowicz and S. Newhouse, Invent. Math. 160, 453–499 (2005; Zbl 1067.37018)] for a conjectured upper bound of the symbolic extension entropy). The 1D case has been settled in [T. Downarowicz and A. Maass, Invent. Math. 176, No. 3, 617–636 (2009; Zbl 1185.37100)]. The author of the paper under review had proved the conjecture for \(C^2\) surface local diffeomorphisms [the author, Ann. Sci. Éc. Norm. Supér. (4) 45, No. 2, 337–362 (2012; Zbl 1282.37015)].
The paper under review gives another advance of the symbolic extension conjecture for nonuniformly entropy-expanding maps, the systems for which all ergodic measures with positive entropy have only positive Lyapunov exponents. More precisely, it is proved that, there exists some symbolic extension, and there is an upper bound (close to optimal) for the symbolic extension entropy in terms of Lyapunov exponents.

37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
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