On parameter loci of the H\'enon family. (arXiv:1501.01368v3 [math.DS] UPDATED)

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 pre 查看全文>>