# Weighted equilibrium states for factor maps between subshifts

## TL;DR

This paper proves the existence and uniqueness of equilibrium states for factor maps between subshifts with weak specification, extending to high dimensions and applications in affine invariant sets, answering an open question in the field.

## Contribution

It establishes the uniqueness of weighted equilibrium states for factor maps between subshifts with weak specification, including high-dimensional cases and applications to Hausdorff dimension.

## Key findings

- Unique invariant measure maximizes weighted entropy and potential.
- Extension to high-dimensional subshifts and affine invariant sets.
- Answers an open question by Gatzouras and Peres.

## 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 $a_2\geq 0$. 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 $\mu(f)+a_1h_\mu(\sigma_X)+ a_2h_{\mu\circ \pi^{-1}}(\sigma_Y)$. In particular, taking $f\equiv 0$ we see that there is a unique invariant measure $\mu$ on $X$ that maximizes the weighted entropy $a_1h_\mu(\sigma_X)+ a_2h_{\mu\circ \pi^{-1}}(\sigma_Y)$. 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.