A heteroclinic bifurcation of Anosov diffeomorphisms

Abstract

We study some diffeomorphisms in the boundary of the set of Anosov diffeomorphisms mainly from the ergodic viewpoint. We prove that these diffeomorphisms, obtained by isotopy from an Anosov $f:M \mapsto M$ through a heteroclinic tangency, determine a manifold ${\cal M}$ of finite codimension in the set of $C^r$ diffeomorphisms. We prove that any diffeomorphism $F$ in ${\cal M}$ is conjugate to $f$; moreover, there exists a unique SRB measure for $F$, and $F$ is Bernoulli with respect to this measure. In particular, if the dimension of $M$ is two, and $\mu $ is a volume element, we prove that the isotopy can be taken such that the measure is preserved.