# American Institute of Mathematical Sciences

July  2006, 14(3): 465-481. doi: 10.3934/dcds.2006.14.465

## On $C^1$-persistently expansive homoclinic classes

 1 Facultad de Ciencias, Iguá 4225 y Mataojo, Universidad de la República, Montevideo, Uruguay 2 Facultad deIngeniería, Herrera y Reissig 565, Universidad de la República, Montevideo, Uruguay

Received  October 2004 Revised  June 2005 Published  December 2005

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.
Citation: Martín Sambarino, José L. Vieitez. On $C^1$-persistently expansive homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 465-481. doi: 10.3934/dcds.2006.14.465
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