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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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