On parameter loci of the H\'enon family

TL;DR

This paper characterizes the parameter regions of the Hénon family where the dynamics are hyperbolic and of maximal entropy, showing these regions are connected, piecewise analytic, and related by a real analytic function.

Contribution

It extends previous results by providing a full characterization of hyperbolic and maximal entropy loci as graphs of real analytic functions for all parameter ranges.

Findings

The hyperbolic horseshoe and maximal entropy loci are both connected and simply connected.

The closure of the hyperbolic horseshoe locus coincides with the maximal entropy locus.

The boundaries of these loci are identical and piecewise analytic.

Abstract

The purpose of the current article is to investigate the dynamics of the H\'enon family $f_{a, b} : (x, y) \mapsto (x^2-a-by, x)$, where $(a, b)\in \mathbb{R}\times\mathbb{R}^{\times}$ is the parameter~\cite{H}. We are interested in certain geometric and topological structures of two loci of parameters $(a, b)\in\mathbb{R}\times\mathbb{R}^{\times}$ for which $f_{a, b}$ share common dynamical properties; one is the \textit{hyperbolic horseshoe locus} where the restriction of $f_{a, b}$ to its non-wandering set is hyperbolic and topologically conjugate to the full shift with two symbols, and the other is the \textit{maximal entropy locus} where the topological entropy of $f_{a, b}$ attains the maximal value $\log 2$ among all H\'enon maps. The main result of this paper states that these two loci are characterized by the graph of a real analytic function from the $b$-axis to the $a$-axis of the parameter space $\mathbb{R}\times\mathbb{R}^{\times}$, which extends in full generality the previous result of Bedford and Smillie for $|b|<0.06$. As consequences of this result, we show that (i) the two loci are both connected and simply connected in $\{b>0\}$ and in $\{b<0\}$, (ii) the closure of the hyperbolic horseshoe locus coincides with the maximal entropy locus, (iii) the boundaries of both loci are identical and piecewise analytic with two analytic pieces. Among others, the consequence (i) indicates a weak form of monotonicity of the topological entropy as a function of the parameter $(a, b)\mapsto h_\mathrm{top}(f_{a, b})$ at its maximal value.