A continuous Bowen-Mane type phenomenon

Esteban Muñoz-Young; Andrés Navas; Enrique Pujals; Carlos H. Vásquez

doi:10.3934/dcds.2008.20.713

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2008, Volume 20, Issue 3: 713-724. Doi: 10.3934/dcds.2008.20.713

This issue Previous Article Multiscale homogenization of monotone operators Next Article On partial regularity for the Navier-Stokes equations

A continuous Bowen-Mane type phenomenon

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

Received: December 31, 2006

Revised: July 31, 2007

Published: August 2008

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Abstract

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:

Physical measures,
ergodic components.

Mathematics Subject Classification: Primary: 37C40. Secondary: 37A10.

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