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.