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Entropy-expansiveness and domination for surface diffeomorphisms. (English) Zbl 1151.37032
Summary: Let \(f:M\to M\) be a \(C^r\)-diffeomorphism, \(r\geq 1\), defined on a closed manifold \(M\). We prove that if \(M\) is a surface and \(K\subset M\) is a compact invariant set such that \(T_KM= E\oplus F\) is a dominated splitting then \(f/K\) is entropy expansive. Moreover \(C^1\) generically in any dimension, isolated homoclinic classes \(H(p)\), \(p\) hyperbolic, are entropy expansive.
Conversely, if there exists a \(C^1\) neighborhood \(U\) of a surface diffeomorphism \(f\) and a homoclinic class \(H(p)\), \(p\) hyperbolic, such that for every \(g\in{\mathcal U}\) the continuation \(H(p_g)\) of \(H(p)\) is entropy-expansive then there is a dominated splitting for \(f/H(p)\).

MSC:
37D30 Partially hyperbolic systems and dominated splittings
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
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