A multiplicative ergodic theoretic characterization of relative equilibrium states

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.