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arXiv · 0909.4250

Weighted equilibrium states for factor maps between subshifts

De-Jun Feng
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Abstract

Let $π:X\to Y$ be a factor map, where $(X,σ_X)$ and $(Y,σ_Y)$ 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 $a_2\geq 0$. 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 $μ$ on $X$ that maximizes $μ(f)+a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y)$. In particular, taking $f\equiv 0$ we see that there is a unique invariant measure $μ$ on $X$ that maximizes the weighted entropy $a_1h_μ(σ_X)+ a_2h_{μ\circ π^{-1}}(σ_Y)$. 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.

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De-Jun Feng. 2009-09-23. Weighted equilibrium states for factor maps between subshifts. https://arxiv.org/abs/0909.4250

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