Transversal Families of Skew-product Axiom a Endomorphisms

We study families of Axiom A skew products with the transversality condition and in particular, the Hausdorff dimension of their fibers, by using thermodynamical formalism. The maps we consider can be non-invertible, and the study of their dynamics is influenced greatly by this fact. We introduce and employ probability measures (constructed from equilibrium measures on the natural extension), which are supported on the fibers of the skew product. A stronger condition, that of Uniform Transversality is then considered in order to obtain a general formula for Hausdorff dimension of fibers for all base points and almost all parameters. In the end we study a large class of examples of transversal Axiom A families which locally depend linearly on the parameters, and also another class of examples related to complex dynamics. 1. Transversal Families of Skew-Product Axiom A Endomorphisms Recall from [6] that a continuous self-map f : X → X of a compact metric space (X, ρ) is called open distance expanding, provided that f is open, Lipschitz continuous, and there are three constants η > 0, γ > 1 and an integer k ≥ 1, such that ρ(f(x), f(z)) ≥ γρ(x, z) whenever ρ(x, z) ≤ η. It is fairly easy to see that changing the metric ρ in a bi-Lipschitz manner, we may assume without loss of generality that k = 1. There is an abundance of open distance expanding maps. We want to bring reader’s attention now to one particular class of them, called expanding repellers. Let U be a bounded open subset of a Euclidean space R with some p ≥ 1. A map f : U → R is called an expanding repeller if and only if the following conditions are satisfied. i) f : U → R is a C endomorphism. ii) X = ⋂∞ n=0 f −n(U) is a compact f -invariant (f(X) = X) subset of U . The map f : X → X is transitive. iii) The map f : X → X is infinitesimally expanding, i.e. there exists k ≥ 1 such that for all x ∈ X and for all v ∈ R, we have ||Dxf(v)|| ≥ 2||v||. Clearly, f : X → X is an open distance (with respect to the Euclidean metric) expanding map. Date: January 2, 2007. 1991 Mathematics Subject Classification. Primary: 37D35, 37D20; Secondary: 37A35.