On parameter loci of the H\'enon family

Abstract

We characterize the hyperbolic horseshoe locus and the maximal entropy locus of the H\'enon family defined on \(\mathbb{R}^2\). More specifically, we show that (i) the two parameter loci are both connected and simply connected (by adding their corresponding one-dimensional loci), (ii) the closure of the hyperbolic horseshoe locus coincides with the maximal entropy locus, and (iii) their boundaries are identical and piecewise real analytic with two analytic pieces. The strategy of our proof is first to extend the dynamical and the parameter spaces over \(\mathbb{C}\), investigate their complex dynamical and complex analytic properties, and then reduce them to obtain the conclusion over \(\mathbb{R}\). 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.