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 ) ↦ ( x 2 − a − by , x ), where ( a , b ) ∈ R × R × is the parameter (Hénon in Commun Math Phys 50(1): 69–77, 1976 ). 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 (Small Jacobian Ergod Theory Dyn Syst 26(5): 1259–1283, 2006 ) 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 ) ↦ h top ( f a , b ) at its maximal value. The proof consists of 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 Bedford and Smillie ( 2006 ). One of our new ingredients is to employ a flexible family of “boxes” in C 2 which is 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.