A Continuous Bowen-Mañé Type Phenomenon
Carlos Vásquez | Muñoz-Young, Esteban | Navas, Andrés | Pujals, Enrique
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.