Weak Gibbs and Equilibrium Measures for Shift Spaces

TL;DR

This paper proves that for a broad class of irreducible shift spaces and summable potentials, the equilibrium measures are weak Gibbs measures, extending known results to higher dimensions and specific shift types.

Contribution

It establishes the weak Gibbs property of equilibrium measures for irreducible shift spaces with absolutely summable potentials, including sofic shifts in one dimension.

Findings

Equilibrium measures are weak Gibbs measures for a large class of shift spaces.

The result applies to higher-dimensional shift spaces and one-dimensional sofic shifts.

The paper broadens the understanding of measure properties in symbolic dynamics.

Abstract

For a large class of irreducible shift spaces $X\subset\tA^{\Z^d}$, with $\tA$ a finite alphabet, and for absolutely summable potentials $\Phi$, we prove that equilibrium measures for $\Phi$ are weak Gibbs measures. In particular, for $d=1$, the result holds for irreducible sofic shifts.