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On Gibbs measures for almost additive sequences associated to some relative pressure functions

Yuki Yayama

TL;DR

This paper develops a non-additive thermodynamic formalism for weakly almost additive sequences with bounded variation on subshifts, proving that a bounded Borel hat{f} (and in some cases a continuous hat{f}) can reproduce the asymptotic integrals of log f_n across all invariant measures. It then characterizes the Gibbs-equilibrium structure of hat{f} relative to that of the original sequence, and extends these insights to images of Gibbs measures under one-block factor maps via relative pressure sequences. A key contribution is linking almost additivity of relative pressure sequences to the Gibbs property of factor-map images, with explicit criteria in a special finite-type setting that leverage matrix products and Jordan form analysis. These results illuminate when factor maps preserve (weak) Gibbs properties and provide explicit constructions of the target functions hat{g} or hat{h} governing the image measures. Overall, the work bridges non-additive thermodynamic formalism and symbolic dynamics to yield actionable conditions for the Gibbs behavior of projected or reduced systems.

Abstract

Given a weakly almost additive sequence of continuous functions with bounded variation $\mathcal{F}=\{\log f_n\}_{n=1}^{\infty}$ on a subshift $X$ over finitely many symbols, we study properties of a function $f$ on $X$ such that $\lim_{n\to\infty}\frac{1}{n}\int \log f_n dμ=\int f dμ$ for every invariant measure $μ$ on $X$. Under some conditions we construct a function $f$ on $X$ explicitly and study a relation between the property of $\mathcal{F}$ and some particular types of $f$. As applications we study images of Gibbs measures for continuous functions under one-block factor maps. We investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-wise sub-positive mixing property of a factor map. For a special type of one-block factor maps between shifts of finite type, we study necessary and sufficient conditions for the image of a one-step Markov measure to be a Gibbs measure for a continuous function.

On Gibbs measures for almost additive sequences associated to some relative pressure functions

TL;DR

This paper develops a non-additive thermodynamic formalism for weakly almost additive sequences with bounded variation on subshifts, proving that a bounded Borel hat{f} (and in some cases a continuous hat{f}) can reproduce the asymptotic integrals of log f_n across all invariant measures. It then characterizes the Gibbs-equilibrium structure of hat{f} relative to that of the original sequence, and extends these insights to images of Gibbs measures under one-block factor maps via relative pressure sequences. A key contribution is linking almost additivity of relative pressure sequences to the Gibbs property of factor-map images, with explicit criteria in a special finite-type setting that leverage matrix products and Jordan form analysis. These results illuminate when factor maps preserve (weak) Gibbs properties and provide explicit constructions of the target functions hat{g} or hat{h} governing the image measures. Overall, the work bridges non-additive thermodynamic formalism and symbolic dynamics to yield actionable conditions for the Gibbs behavior of projected or reduced systems.

Abstract

Given a weakly almost additive sequence of continuous functions with bounded variation on a subshift over finitely many symbols, we study properties of a function on such that for every invariant measure on . Under some conditions we construct a function on explicitly and study a relation between the property of and some particular types of . As applications we study images of Gibbs measures for continuous functions under one-block factor maps. We investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-wise sub-positive mixing property of a factor map. For a special type of one-block factor maps between shifts of finite type, we study necessary and sufficient conditions for the image of a one-step Markov measure to be a Gibbs measure for a continuous function.
Paper Structure (10 sections, 22 theorems, 53 equations)

This paper contains 10 sections, 22 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Shift spaces
  4. Sequences of continuous functions.
  5. Gibbs measures for weakly almost additive sequences
  6. Applications
  7. Relative pressure functions and images of Gibbs measures for continuous functions
  8. The fiber-wise sub-positive mixing property of factor maps and almost additivity of subadditive sequences associated to relative pressure functions
  9. Proof of Theorem \ref{['phd1']}
  10. Examples

Key Result

Lemma 3.1

Let $(X,\sigma_X)$ be a subshift with the weak specification property and ${\mathcal{F}}=\{\log f_n\}_{n=1}^{\infty}$ be a sequence of continuous functions on $X$. Suppose that there exist a Borel measurable function $f$ on $X$ and a sequence of positive real numbers $\{A_k\}_{k=1}^{\infty}$ satisfy Then the following hold.

Theorems & Definitions (49)

  • Lemma 3.1
  • proof
  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.2
  • proof
  • Corollary 4.1
  • Remark 4.1
  • ...and 39 more