A multiplicative ergodic theoretic characterization of relative equilibrium states
Date
2022
Authors
Antonioli, John
Hong, Soonjo
Quas, Anthony
Journal Title
Journal ISSN
Volume Title
Publisher
Ergodic Theory and Dynamical Systems
Abstract
In this article, we continue the structural study of factor maps between symbolic
dynamical systems and the relative thermodynamic formalism. Here, one is studying a
factor map from a shift of finite type X (equipped with a potential function) to a sofic shift
Z, equipped with a shift-invariant measure ν. We study relative equilibrium states, that is,
shift-invariant measures on X that push forward under the factor map to ν which maximize
the relative pressure: the relative entropy plus the integral of φ. In this paper, we establish a
new connection to multiplicative ergodic theory by relating these factor triples to a cocycle
of Ruelle–Perron–Frobenius operators, and showing that the principal Lyapunov exponent
of this cocycle is the relative pressure; and the dimension of the leading Oseledets space
is equal to the number of measures of relative maximal entropy, counted with a previously
identified concept of multiplicity.
Description
Keywords
relative thermodynamic formalism, multiplicative ergodic theory, transfer operators
Citation
Antonioli, J., Hong, S., & Quas, A. (2022). “A multiplicative ergodic theoretic characterization of relative equilibrium states.” Ergodic Theory and Dynamical Systems, 1-16. https://doi.org/10.1017/etds.2022.15