A continuous Bowen-Mane type phenomenon
Esteban Muñoz-Young Andrés Navas Enrique Pujals Carlos H. Vásquez
Discrete & Continuous Dynamical Systems - A 2008, 20(3): 713-724 doi: 10.3934/dcds.2008.20.713
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
Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents
Carlos H. Vásquez
Journal of Modern Dynamics 2009, 3(2): 233-251 doi: 10.3934/jmd.2009.3.233
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

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