Homoclinic tangency and variation of entropy

Abstract

In this paper we study the effect 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 ∞ -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 ∞ diffeomorphisms of three dimensional manifolds.

Invariance of entropy for maps isotopic to Anosov *

“…From now on, let us denote E ws f 0 by E c f 0 . Now, around p we deform isotopically f 0 'a la Tahzibi-Bronzi' (see [3]) into a diffeomorphism f having the following properties:…”

“…For r > 0 denote by D r the r disc in R 2 . As in [3] we construct an isotopy {h t : D 1 → D 1 } t∈[0,1] satisfying the following: (a) h t (z) = λ ws f 0 z, for all t ∈ [0, 1/2], |z| 1/2; (b) h t (0) = 0, for all t ∈ [0, 1]; (c) h 0 is the Smale's horseshoe map (hence, h top (h 0 ) = log 2); (d) h t (z) = λ ws f 0 z, for all t ∈ [1/2, 1], z ∈ D 1 . See figure 2 for a view of h t on the disc D 1 and figure 3 for a zoom inside the disc D 1/2 .…”

Invariance of entropy for maps isotopic to Anosov

“…For r > 0 denote by D r the r disc in R 2 . As in [3] we construct an isotopy {h t : D 1 → D 1 } t∈[0,1] satisfying the following (1) h t (z) = λ ws f 0 z, for all t ∈ [0, 1/2], |z| ≥ 1/2; (2) h t (0) = 0, for all t ∈ [0, 1];…”