Diffeomorphisms with infinitely many sinks

Abstract

LET M be a compact C” manifold without boundary. and let Dir(M) denote the space of C’diffeomorphisms of I that is, f’(p) = p but fk(p) # p for 0 < k < V. We say that p is a sink if each eigenvalue of the derivativef’ at p has absolute valuelessthan one. S. Smale suggested in [12] that most diffeomorphisms on S2 with the C’ topology ought to have only finitely many sinks. We show here that this is not the case for r 2 2 by proving the following