American Institute of Mathematical Sciences

July  2008, 20(3): 713-724. doi: 10.3934/dcds.2008.20.713

A continuous Bowen-Mane type phenomenon

 1 Departamento de Matemática, Universidad Católica del Norte, Av. Angamos 0610, Antofagasta, Chile, Chile 2 Departamento de Matemática, Fac. de Ciencias, Universidad de Santiago, Alameda 3363, Santiago, Chile 3 Instituto Nacional de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

Received  January 2007 Revised  August 2007 Published  December 2007

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.
Citation: Esteban Muñoz-Young, Andrés Navas, Enrique Pujals, Carlos H. Vásquez. A continuous Bowen-Mane type phenomenon. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 713-724. doi: 10.3934/dcds.2008.20.713
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