Exchangeable, Gibbs and equilibrium measures for Markov subshifts

Abstract

We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of 'symmetric measure': exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one-dimensional case) exhibit a variety of spectral properties.

Gibbs and equilibrium measures for some families of subshifts

“…Let T : X → X be a homeomorphism of a compact metric space. The Gibbs relation of (X, T ) (also called homoclinic relation, or double-tail relation [1,16]) is defined as the pairs of points in X which have asymptotically converging orbits:…”

“…In the following we recall some terminology on conformal measures. For further details and references see [1,5,16].…”

Thermodynamic Formalism for Amenable Groups and Countable State Spaces

“…see [38,Definition 5.2.1]). This kind of measures fits in the more general context of (Ψ,R)-conformal measures explored in [1], where R is a Borel equivalence relation and Ψ : R → R + is a measurable function. Then, Capocaccia's measures, that we simply call conformal measures, can be recovered by taking a function Ψ related to the given potential and R the tail relation in the space of configurations.…”

“…Let φ : X → R be an exp-summable and uniformly continuous potential with finite oscillation. Then, for every 1 2 > > 0, there exist A ∈ F(N), K ∈ F(G), and δ > 0, such that for every (K,δ)-invariant set F ∈ F(G), it holds that = 0, by Lemma 2.4. Therefore, there exist K ∈ F(G) and δ > 0, such that Δ F (φ) < |F | for every finite (K ,δ )-invariant set F.…”

“…Proof. By Proposition 3.8, for every 1 2 > > 0, there exist A ∈ F(N), K ∈ F(G), and δ > 0, such that for every (K,δ)-invariant set F ∈ F(G),…”