Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map

Abstract

We show how to link topological tools with a local hyperbolic behaviour to prove the existence of homoclinic and heteroclinic trajectories for a map. We apply this technique for the Hénon map h with classical parameter values (a = 1.4,b = 0.3). For this map we give a computer-assisted proof of the existence of an infinite number of homoclinic and heteroclinic trajectories. We also introduce the method for computation of the lower bound of the topological entropy of a map based on the covering relations involving different iterations of the map and we prove that the topological entropy of h is larger than 0.3381.