Homoclinic Points in the Composition of Two Reflections

Abstract

We examine how two simple maps can be combined to produce a map with chaotic behaviour. To be more precise, let $f,g:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be $C^1$ functions with domain all of ${\mathbb {R}}$. Let $F:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2$ denote a horizontal reflection in the curve $x=-f(y)$, and let $G:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2$ denote a vertical reflection in the curve $y=g(x)$. We consider maps of the form $T=G\circ F$ and show that a simple geometric condition on the fixed point sets of F and G leads to the existence of a homoclinic point for T.