Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents
Carlos H. Vásquez
We establish stable ergodicity for diffeomorphisms with partially hyperbolic attractors whose Lyapunov exponents along the center direction are all positive with respect to SRB measures.
keywords: stable ergodicity. Partial hyperbolicity SRB measures Lyapunov exponents
A continuous Bowen-Mane type phenomenon
Esteban Muñoz-Young Andrés Navas Enrique Pujals Carlos H. Vásquez
In this work we exhibit a one-parameter family of $C^1$-diffeomorphisms $F_\alpha$ of the 2-sphere, where $\alpha>1$, such that the equator $\S^1$ is an attracting set for every $F_\alpha$ and $F_\alpha|_{\S^1}$ is the identity. For $\alpha>2$ the Lebesgue measure on the equator is a non ergodic physical measure having uncountably many ergodic components. On the other hand, for $1<\alpha\leq 2$ there is no physical measure for $F_\alpha$. If $\alpha<2$ this follows directly from the fact that the $\omega$-limit of almost every point is a single point on the equator (and the basin of each of these points has zero Lebesgue measure). This is no longer true for $\alpha=2$, and the non existence of physical measure in this critical case is a more subtle issue.
keywords: ergodic components. Physical measures

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