INFINITELY MANY COEXISTING SINKS FROM DEGENERATE HOMOCLINIC TANGENCIES

The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families {F(t)} of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, P(t); F(t)(n)(P(t)) = P(t), and \ det DF(t)(n)(P(t)) < 1. We also require the stable and unstable manifolds of P(t) to form homoclinic tangencies as the parameter t varies through t(0). Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values t near t(0). We show that there are parameter values t near t(0) at which F(t) has infinitely many co-existing periodic sinks.