Thermodynamic Formalism Out of Equilibrium Part II: Semi-Ruelle Operator, Conformal Measures, and Effective Expansion and Quasi-Compactness
Abstract
We introduce a general machinery to study {\em thermodynamic formalism out of equilibrium}: The thermodynamics of a topological Markov shift (denoted by Σ-) where the potential is given by a random walk on a compact metric space X (and the randomness is driven by a Gibbs process). We introduce the {\em semi-Ruelle operator}, which acts on C(Σ-× X). We construct conformal measures and harmonic functions for the semi-Ruelle operator. We present a few applications: (1) We provide a new proof to the POE variational principle (POE stands for the {\em pressure out of equilibrium} which is associated with the process), and we show that maximizing measures in the POE variational principle admit positive {\em entropy out of equilibrium}, and satisfy {\em semi-Gibbs estimates}. (2) In the setting where the random walk on the fiber X is given by C1+ diffeomorphisms (which are allowed to be very dissipative), and it satisfies the open condition of {\em effective expansion on average}, we show that the {\em averaged semi-Ruelle operator} is quasi-compact when acting on a Sobolev function space. An application includes proving a spectral gap when assuming volume decay of correlations, and proving bounds on the dimension of stationary measure in terms of similarity dimension.
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BibTeX
@misc{ovadia2026thermodynamic,
title = {{Thermodynamic Formalism Out of Equilibrium Part II: Semi-Ruelle Operator, Conformal Measures, and Effective Expansion and Quasi-Compactness}},
author = {Snir Ben Ovadia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12068},
eprint = {2609.12068},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12068},
}
RIS
TY - UNPB
TI - Thermodynamic Formalism Out of Equilibrium Part II: Semi-Ruelle Operator, Conformal Measures, and Effective Expansion and Quasi-Compactness
AU - Snir Ben Ovadia
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12068
UR - https://arxiv.org/abs/2609.12068
AB - We introduce a general machinery to study {\em thermodynamic formalism out of equilibrium}: The thermodynamics of a topological Markov shift (denoted by Σ^-) where the potential is given by a random walk on a compact metric space X (and the randomness is driven by a Gibbs process). We introduce the {\em semi-Ruelle operator}, which acts on C(Σ^-× X). We construct conformal measures and harmonic functions for the semi-Ruelle operator. We present a few applications: (1) We provide a new proof to the POE variational principle (POE stands for the {\em pressure out of equilibrium} which is associated with the process), and we show that maximizing measures in the POE variational principle admit positive {\em entropy out of equilibrium}, and satisfy {\em semi-Gibbs estimates}. (2) In the setting where the random walk on the fiber X is given by C^1+ diffeomorphisms (which are allowed to be very dissipative), and it satisfies the open condition of {\em effective expansion on average}, we show that the {\em averaged semi-Ruelle operator} is quasi-compact when acting on a Sobolev function space. An application includes proving a spectral gap when assuming volume decay of correlations, and proving bounds on the dimension of stationary measure in terms of similarity dimension.
ER -