Homoclinic tangency and variation of entropy

TL;DR

This paper investigates how homoclinic tangencies influence the variation of topological entropy in dynamical systems, showing that such tangencies can cause entropy discontinuities and variations in different topologies.

Contribution

It proves that homoclinic tangencies associated with maximal entropy hyperbolic sets lead to entropy variation points in the $C^{ abla}$-topology, and provides examples of entropy discontinuity in 3D diffeomorphisms.

Findings

Homoclinic tangency can cause entropy variation in $C^{ abla}$-topology.

Entropy discontinuity can occur among $C^{ abla}$ diffeomorphisms of three-dimensional manifolds.

Results extend to other topologies and tangencies not associated with maximal entropy sets.

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 $C^{\infty}$-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.