Homoclinic tangency and variation of entropy

Abstract

Topological entropy is one of the most important invariants of topological conjugacy in dynamical systems. By the Ω−stability of Axiom A diffeomorphisms with no cycle condition, it comes out that the entropy is a C−locally constant function among such dynamics. We say that a diffeomorphism f is a point of constancy of topological entropy in C topology if there exists a C−neighborhood U of f such that for any diffeomorphism g ∈ U , h(g) = h(f). We also call a diffeomorphism as a point of variation of entropy if it is not a point of constancy. In [11], Pujals and Sambarino proved that surface diffeomorphisms far from homoclinic tangency are the constancy points of topological entropy in C topology. In this paper we address the reciprocal problem. That is we are interested in the effect of a homoclinic tangency to the variation of the topological entropy for a surface diffeomorphism. Of course after unfolding a homoclinic tangency, new periodic points will emerge, but it is not