Sistemas quase-Anosov

Abstract

In 1970, Hirsch conjectured that given a diffeomorphism $f: M \to M$ in a differentiable manifold M, if $ N \subset M $ is a compact invariant submanifold with a hyperbolic structure as a subset of $M$, then $f$ restricted to $N$ would be a Anosov diffeomorphism. In this work we present a counterexample to this conjecture published in 1976 by Franks and Robinson. Next we present a result of Zeghib showing that $(\phi, M)$ is an Anosov system (flow or diffeomorphism) with splitting property, given a closed invariant submanifold $N$ of $M$, then $(\phi, N)$ is a transitive Anosov system.