In this work we exhibit a one-parameter family of C-1-diffeomorphisms F-alpha of the 2-sphere, wher alpha > 1, such that the equator S-1 is an attracting set for every F-alpha and F-alpha vertical bar(1)(S) is the identity. For alpha > 2 the Lebesgue measure on the equtor is a non ergodic physical measure having uncountably many ergoidc components. On the other hand, for 1 < alpha <= 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 in this critical case is a more subtle issue.