Class degree and relative maximal entropy

Abstract

Given a factor code π from a one-dimensional shift of finite type X onto an irreducible sofic shift Y , if π is finite-to-one there is an invariant called the degree of π which is defined the number of preimages of a typical point in Y . We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure ν on Y , we find an invariant upper bound on the number of ergodic measures on X which project to ν and have maximal entropy among all measures in the fibre π −1 {ν}. We show that this bound and the class degree of the code agree when ν is ergodic and fully supported. One of the main ingredients of the proof is a uniform distribution property for ergodic measures of relative maximal entropy. * We thank the referee for detailed and helpful comments.

Relative equilibrium states and class degree

“…Petersen, Quas, and Shin proved that the number of ergodic measures of relative maximal entropy is always finite and gave an explicit upper bound [6]. Allahbakhshi and Quas improved the upper bound to a conjugacy-invariant upper bound and introduced the notion of class degree [2]. In the special case of ν with full support, their upper bound is equal to the class degree of the factor code.…”

“…, µ d+1 are distinct ergodic measures of relative maximal entropy over ν where d is the number of letters (for X) that project to a fixed letter b for Y with ν(b) > 0. Form a relatively independent joining of the d + 1 measures over ν. Pigeonhole's principle then forces at least two, say µ 1 , µ 2 , of the d + 1 measures to have the property λ({(x (1) , x (2) ) :…”

“…Since for every (x (1) , x (2) ) in some set of positive measure with respect to λ, there are infinitely many i for which x…”

“…i , one can construct a point x (3) which is the result of splicing parts of x (1) or x (2) depending on the outcome of tossing a fair coin at every i for which x (1) i = x (2) i . The probability distribution of the new point x (3) is a measure µ 3 on X which projects to ν.…”

“…The notion of class degree of a factor code is defined using an equivalence relation within fibers [2]. Given a point y, the fiber π −1 (y) is divided into finitely many components under the following equivalence relation: x, x ′ ∈ π −1 (y) are equivalent if there is x ′′ in the same fiber that agrees with x on (−∞, n] for a given arbitrary n and with x ′ on [m, +∞) for some m > n and vice versa.…”