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

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.

We give a subclass $\mathcal{L}$ of *Non-linear Lalley-Gatzouras carpets* and an open set $\mathcal{U}$ in $\mathcal{L}$ such that any carpet in $\mathcal{U}$ has a unique ergodic measure of full dimension. In particular, any Lalley-Gatzouras carpet which is close to a *non-trivial* general Sierpinski carpet has a unique ergodic measure of full dimension.

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