arXiv subjects

Zin Arai

Publications and source records attributed to Zin Arai.

4 recordsLinked to original sources

arXiv2025

More on the Concept of Anti-integrability for Hénon Maps

For the family of Hénon maps $(x,y)\mapsto (\sqrt{a}(1-x^2)-b y,x)$ of $\mathbb{R}^2$, the so-called anti-integrable (AI) limit concerns the limit $a\to\infty$ with fixed Jacobian $b$. At the AI limit, the dynamics reduces to a subshift of finite type. There is a one-to-one correspondence between sequences allowed by the subshift and the AI orbits. The theory of anti-integrability says that each AI orbit can be continued to becoming a genuine orbit of the Hénon map for $a$ sufficiently large (and fixed Jacobian). In this paper, we assume $b$ is a smooth function of $a$ and show that the theory can be extended to investigating the limit $\lim_{a\to\infty} b/\sqrt{a}=\hat{r}$ for any $\hat{r}>0$ provided that the one dimensional quadratic map $x\mapsto \displaystyle\frac{1}{\hat{r}}(1-x^2)$ is hyperbolic.

math.DS↗

arXiv2018

Boundary of the horseshoe locus for the Hénon family

The purpose of this article is to investigate geometric properties of the parameter locus of the Hénon family where the uniform hyperbolicity of a horseshoe breaks down. As an application, we obtain a variational characterization of equilibrium measures "at temperature zero" for the corresponding non-uniformly hyperbolic Hénon maps. The method of the proof also yields that the boundary of the hyperbolic horseshoe locus in the parameter space consists of two monotone pieces, which confirms a conjecture in [AI]. The proofs of these results are based on the machinery developed in [AI] which employs the complexification of both the dynamical and the parameter spaces of the Hénon family together with computer assistance.

math.DS↗

arXiv2018

On parameter loci of the Hénon family

The purpose of the current article is to investigate the dynamics of the Hénon 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 Hénon 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.

math.DS↗

arXiv2007

On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies

We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Hénon map. Indeed, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Our main tool is a rigorous computational algorithm for verifying the uniform hyperbolicity of chain recurrent sets. In addition, we show that the dynamics of the real Hénon map is completely determined by the monodromy of a certain loop, providing the parameter of the map is contained in the hyperbolic horseshoe locus of the complex Hénon map.

math.DS↗