Measure of full dimension for some nonconformal repellers

Abstract

Given (X, T) and (Y, S) mixing subshifts of finite type such that (Y, S) is a factor of (X, T) with factor map : X Y, and positive Holder continuous functions φ: X → R and Ψ : Y → R, we prove that the maximum of hμo π-1(S)/∫ Ψ o π dμ + hμ(T) - hμoπ-1(S)/∫ π dμ over all T-invariant Borel probability measures μon X is attained on the subset of ergodic measures. Here hμ(T) stands for the metric entropy of T with respect to μ As an application, we prove the existence of an ergodic invariant measure with full dimension for a class of transformations treated in [11], and also for the transformations treated in [17], where the author considers nonlinear skew-product perturbations of general Sierpinski carpets. In order to do so we establish a variational principle for the topological pressure of certain noncompact sets.