Weighted equilibrium states for factor maps between subshifts

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 $\text{ba}=(a_1,a_2)\in {\mathbb{R}}^2$ with $a_1>0$ and $a_2\ge 0$. Let $f$ be a continuous function on $X$ with sufficient regularity (H"{o}lder continuity, for instance). We show that there is a unique shift invariant measure $\mu$ on $X$ that maximizes $\mu(f)+a_1h_\mu(\sigma_X)+ a_2h_{\mu\circ \pi^{-1}}(\sigma_Y)$\mathrm{.}Inparticular,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.