A multiplicative ergodic theoretic characterization of relative equilibrium states
In this article, we continue the structural study of factor maps betweeen 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 $\nu$. We study relative equilibrium states, that is shift-invariant measures on $X$ that push forward under the factor map to $\nu$ which maximize the relative pressure: the relative entropy plus the integral of $\phi$. In the non-relative case (where $Z$ is the one point shift and the factor map is trivial), these measures have a very broad range of application: in hyperbolic dynamics, information theory, geometry, Teichm\"uller theory and elsewhere). Relative equilibrium states have also been shown to arise naturally in some contexts in geometric measure theory as a description of measures achieving the Hausdorff dimension in ambient spaces. Previous articles have identified relative versions of well-known notions of degree appearing in one-dimensional symbolic settings, and established bounds in terms of these on the number of ergodic relative equilibrium states. 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.