On Parameter Loci of the Hénon Family

Abstract

The purpose of the current article is to investigate the dynamics of the Hénon family f a,b : (x, y) → (x 2 − a − by, x), where (a, b) ∈ R × R × is the parameter [H]. We are interested in certain geometric and topological structures of two loci of parameters (a, b) ∈ R × R × for which f a,b share common dynamical properties; one is the 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 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 R × R × , which extends in full generality the previous result of Bedford and Smillie [BS2] 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) → htop(f a,b ) at its maximal value.The proof consists of both theoretical and computational parts. In the theoretical part we extend both the dynamical and the parameter spaces over C, investigate their complex dynamical and complex analytic properties, and reduce them to obtain the conclusion over R as in [BS2]. One of our new ingredients is to employ a flexible family of "boxes" in C 2 that are intrinsically two-dimensional and works for all values of b. In the computational part we use interval arithmetic together with some numerical algorithms such as set-oriented computations and the interval Krawczyk method to verify certain numerical criteria which imply analytic, combinatorial and dynamical consequences.

Dynamics of Polynomial Diffeomorphisms of $$\mathbb {C}^2$$ C 2 : Combinatorial and Topological Aspects

“…We note that, when a = a tgc (b), the map f a,b has exactly one orbit of either homoclinic (b > 0) or heteroclinic (b < 0) tangencies of stable and unstable manifolds of suitable saddle fixed points [BS R 1]. The strategy of [BS R 2,AI] is first to extend the dynamical and the parameter spaces over C, investigate their complex dynamical and complex analytic properties, and then reduce them to obtain conclusions over R. In the article [AI] we also employ interval arithmetic together with some numerical algorithms such as set-oriented computations and the interval Krawczyk method to verify certain numerical criteria which imply analytic, combinatorial and dynamical consequences. 23 is obtained by joining two figures in the numerical work of El Hamouly and Mira [EM] and turning it upside down.…”