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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