A multiplicative ergodic theoretic characterization of relative equilibrium states




Antonioli, John
Hong, Soonjo
Quas, Anthony

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Ergodic Theory and Dynamical Systems


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.



relative thermodynamic formalism, multiplicative ergodic theory, transfer operators


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