A continuous bowen-mañé type phenomenon

In this work we exhibit a one-parameter family of C¹- diffeomorphisms F_α of the 2-sphere, where α > 1, such that the equator S¹ is an attracting set for every F _α and F_α|S¹ 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.