Measure of full dimension for some nonconformal repellers

Given $(X,T)$ and $(Y,S)$ mixing subshifts of finite type such that $(Y,S)$ is a factor of $(X,T)$ with factor map $\pi$:$\ X\to Y$, and positive Hölder continuous functions $\varphi$:$\ X\to \mathbb{R}$ and $\psi$:$\ Y\to \mathbb{R}$, we prove that the maximum of

$\frac{h_{\mu\circ \pi^{-1}}(S)}{\int \psi\circ\pi\d\mu}+ \frac{h_\mu(T)-h_{\mu\circ \pi^{-1}}(S)}{\int \varphi\d\mu}$

over all $T$-invariant Borel probability measures $\mu$ on $X$ is attained on the subset of ergodic measures. Here $h_\mu(T)$ stands for the metric entropy of $T$ with respect to $\mu$. 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.