## Abstract

In this work we exhibit a one-parameter family of C^{1}- diffeomorphisms F_{α} of the 2-sphere, where α > 1, such that the equator S^{1} is an attracting set for every F _{α} and F_{α}|S^{1} is the identity. For α > 2 the Lebesgue measure on the equator is a non ergodic physical measure having uncountably many ergodic components. On the other hand, for 1 < α < 2 there is no physical measure for F_{α}. If α < 2 this follows directly from the fact that the ω-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 α = 2, and the non existence of physical measure in this critical case is a more subtle issue.

Original language | English |
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Pages (from-to) | 713-724 |

Number of pages | 12 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 20 |

Issue number | 3 |

State | Published - 1 Mar 2008 |

## Keywords

- Ergodic components
- Physical measures