Preprint A571/2007
A Continuous Bowen-Mañé Type Phenomenon

Carlos Vásquez | Muñoz-Young, Esteban | Navas, Andrés | Pujals, Enrique

**Keywords: **

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