Hausdorff dimension, strong hyperbolicity and complex dynamics

Abstract

In the first part of the paper we show that a hyperbolic area preserving Hénon map has a unique Gibbs measure whose Hausdorff dimension is equal to the Hausdorff dimension of its nonwandering (Julia) set. In the second part we introduce the notion of strong hyperbolicity for diffeomorphisms of compact manifolds. It is a foliation of the tangent space over a hyperbolic set to one dimensional contracting and expanding subspaces with different rates of contractions and expansions. We show that strong hyperbolicity is structurally stable. For a Strong Axiom A diffeomorphism $f$ we state a conjectured variational characterization of the Hausdorff dimension of the nonwandering set of $f$. In the third part we study the dynamics of polynomial maps $f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ which lift to holomorphic maps of $\mathbb C\mathbb P^2$. Let $J(f)$ be the closure of repelling periodic points of $f$. Using the structural stability results we exhibit open set of $f$ for which $J(f)$ behaves like the Julia set of one dimensional polynomial map.