We consider an infinite-to-one factor map from an irreducible shift of finite
type X to a sofic shift Y. A compensation function relates equilibrium states
on X to equilibrium states on Y. The p-Dini condition is given as a way of
measuring the smoothness of a continuous function, with 1-Dini corresponding to
functions with summable variation. Two types of compensation functions are
defined in terms of this condition. We show that the relative equilibrium
states of a 1-Dini function f over a fully supported invariant measure on Y are
themselves fully supported, and have positive relative entropy. We then show
that there exists a compensation function which is p-Dini for all p > 1 which
has relative equilibrium states supported on a finite-to-one subfactor.Comment: 20 pages, 1 figure submitted to Ergodic Theory Dyn. Sys
Abstract. Given a factor code π from a shift of finite type X onto a sofic shift Y , an ergodic measure ν on Y , and a function V on X with summable variation, we prove an invariant upper bound on the number of ergodic measures on X which project to ν and maximize h(µ) + V dµ among all measures in the fiber π −1 (ν). If ν is fully supported, this bound is the class degree of π. This generalizes a previous result for the special case of V = 0.
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
$\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 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.
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