Homoclinic tangency and variation of entropy

Abstract

In this paper we study the e¤ect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of entropy variation in the C^{\infty} -topology. We also prove results about variation of entropy in other topologies and when the tangency does not correspond to a basic set with maximal entropy. We also show an example of discontinuity of the entropy among C^{\infty} diffeomorphisms of three dimensional manifolds.

A consequence of the growth of rotation sets for families of diffeomorphisms of the torus

“…Our main theorem says that every time the rotation set locally grows near a rational point, then nearby maps must have tangencies, which generically unfold as the parameter changes. And this is a phenomenon which is associated to the growth of topological entropy, see [5] and [23]. More precisely, in the two previous papers it is proved that generically, if a surface diffeomorphism f has arbitrarily close neighbors with larger topological entropy, then f has a periodic saddle point with a homoclinic tangency.…”