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On $C^1$-persistently expansive homoclinic classes

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  • Let $f: M \to M$ be a diffeomorphism defined in a $d$-dimensional compact boundary-less manifold $M$. We prove that $C^1$-persistently expansive homoclinic classes $H(p)$, $p$ an $f$-hyperbolic periodic point, have a dominated splitting $E\oplus F$, $\dim(E)=\mbox{index}(p)$. Moreover, we prove that if the $H(p)$-germ of $f$ is expansive (in particular if $H(p)$ is an attractor, repeller or maximal invariant) then it is hyperbolic.
    Mathematics Subject Classification: Primary: 37D30; Secondary: 37D20, 37C29.

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