SRB measures of certain almost hyperbolic diffeomorphisms with a tangency

Eleonora Catsigeras,,Instituto De Matemática, Universidad De La República ,Heber Enrich

doi:10.3934/dcds.2001.7.177

Eleonora Catsigeras, ,Instituto De Matemática, Universidad De La República + Show 1 more

Open Access

https://doi.org/10.3934/dcds.2001.7.177

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Abstract

We study topological and ergodic properties of some almost hyperbolic diffeomorphisms on two dimensional manifolds. Under generic conditions, diffeomorphisms obtained from Anosov by an isotopy pushing together the stable and unstable manifolds to be tangent at a fixed point, are conjugate to Anosov. For a finite codimension subset at the boundary of Anosov there exist a SRB measure and an unique ergodic attractor.

Highlights

We consider some C3 diffeomorphisms at the boundary of Anosov, inspired in the examples of Lewowicz [18]
We first prove that they are topologically conjugate to Anosov. We prove that such systems exhibit only one ergodic attractor, as in Palis’conjecture [29], their stable and unstable foliations are not C1 transversal and there is not a uniform separation between positive and negative Lyapounov exponents
The diffeomorphisms we study in this paper are non uniformly hyperbolic examples in dimension two

Summary

Introduction

We consider some C3 diffeomorphisms at the boundary of Anosov, inspired in the examples of Lewowicz [18]. In [8] Carvalho weakens a stable subspace of a fixed point of a n-dimensional Anosov diffeomorphism, but maintaining strong the unstable space She bounds distortion of backward iterates of unstable volume elements, to conclude that there exists a SRB measure. In [14] the authors weaken the unstable direction of a fixed point of a twodimensional Anosov diffeomorphism, maintaining strong the stable direction They prove that the sum of unstable lengths of backward iterates is not bounded, that it does not exist a SRB measure with positive Lyapounov exponents, and that the non hyperbolic fixed point is the unique ergodic attractor. The author proves that under some conditions there exists a SRB measure with positive Lyapounov exponentes, and under the complementary conditions, the unique ergodic attractor is the non hyperbolic fixed point.

We write the quadratic form

We have

We assert that