On parameter loci of the H\'enon family
Zin Arai, Yutaka Ishii

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 , where is the parameter~\cite{H}. We are interested in certain geometric and topological structures of two loci of parameters for which share common dynamical properties; one is the \textit{hyperbolic horseshoe locus} where the restriction of 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 attains the maximal value 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 -axis to the -axis…
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Taxonomy
TopicsMathematical Dynamics and Fractals · Chaos control and synchronization · Quantum chaos and dynamical systems
