A Symbolic Characterization of the Horseshoe Locus in the H\'enon Family

TL;DR

This paper characterizes the horseshoe locus in the Hénon family of quadratic diffeomorphisms using symbolic coding and complex techniques, providing criteria and topological descriptions of boundary maps.

Contribution

It introduces a symbolic criterion for identifying horseshoes in the Hénon family and describes the topological conjugacy types at the boundary of the horseshoe locus.

Findings

Provides a symbolic coding criterion for horseshoes.

Describes the boundary topological conjugacy types.

Uses complex techniques in a 2D parameter region.

Abstract

We consider the family of quadratic H\'enon diffeomorphisms of the plane ${\bf R}^2$. A map will be said to be a "horseshoe" if its restriction to the nonwandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for being a horseshoe based on an auxiliary coding which describes positions of points relative to the stable manifold of one of the fixed points. In addition we describe the topological conjugacy type of maps on the boundary of the horseshoe locus. We use complex techniques and we work with maps in a parameter region which is a 2-D analog of the familiar "${1\over 2}$-wake" for the quadratic family $p_c(z) = z^2$.