Projections of Gibbs States for H\"older Potentials

TL;DR

This paper proves that the projection of a Gibbs state for a H"older potential under a specific factor map retains H"older continuity, simplifying previous proofs and using cone techniques and positive operators.

Contribution

It provides a shorter proof that the projected Gibbs state maintains H"older continuity, improving upon earlier results with a novel approach.

Findings

Projection of Gibbs states preserves H"older continuity.

Uses cone techniques and positive operators for proof.

Simplifies previous complex proofs.

Abstract

In this paper we give a short proof that the projection of a Gibbs state for a H\"older continuous potential on a mixing shift of finite type under a 1-block fiber-wise mixing factor map has a H\"older continuous g function. This improves a number of previous results. The key insight in the proof is to realize the measure of a cylinder set in terms of positive operators and use cone techniques.