Weighted equilibrium states for factor maps between subshifts

Abstract

Let $\pi :X\to Y$ be a factor map, where $(X,{\sigma }_X)$ and $(Y,{\sigma }_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\ge 0$. Let $f$ be a continuous function on $X$ with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure $\mu$ on $X$ that maximizes $\mu (f)+a_1{h}_{\mu }({\sigma }_X)+a_2{h}_{\mu \circ {\pi }^{-1}}({\sigma }_Y)$. In particular, taking $f\equiv 0$ we see that there is a unique invariant measure $\mu$ on $X$ that maximizes the weighted entropy $a_1{h}_{\mu }({\sigma }_X)+a_2{h}_{\mu \circ {\pi }^{-1}}({\sigma }_Y)$. This answers an open question raised by Gatzouras and Peres in \cite{GaPe96}. An extension is also given to high dimensional cases. As an application, we show the uniqueness of invariant measures with full Hausdorff dimension for certain affine invariant sets on the $k$-torus under a diagonal endomorphism.

30 pages

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• Weighted thermodynamic formalism and applications

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• Weighted thermodynamic formalism and applications