Pruning fronts and the formation of horseshoes

Abstract

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a homeomorphism of the plane. We define open sets $P \subset \mathbb{R}^{2}$, called pruning fronts after the work of Cvitanović et al. in 1988, for which it is possible to construct an isotopy $H: \mathbb{R}^{2} \times [0,1] \rightarrow \mathbb{R}^{2}$ with open support contained in $\bigcup _{n \in {\mathbb{Z}} } f^{n} (P)$ such that $H(\cdot, 0 ) = f(\cdot)$ and $H(\cdot, 1) = f_{P} (\cdot)$, where $f_P$ is a homeomorphism under which every point of $P$ is wandering. Applying this construction when $f$ is Smale's horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behavior. This family is a two-dimensional analog of a one-dimensional universal family.