Existence of horseshoe sets with nondegenerate one-sided homoclinic tangencies in ${\mathbb R}^{3}$

Abstract

In this paper, we present some class of three dimensional $C^{\infty}$ diffeomorphisms with nondegenerate one-sided homoclinic tangencies $q$ associated with hyperbolic fixed points $p$ each of which exhibits a horseshoe set. A key point in the proof is the existence of a transverse homoclinic point arbitrarily close to $q$. This result together with Birkhoff-Smale Theorem implies the existence of a horseshoe set arbitrarily close to $q$.