Exchangeable measures for subshifts

Abstract

Let $\Om$ be a Borel subset of $S^\Bbb N$ where $S$ is countable. A measure is called exchangeable on $\Om$, if it is supported on $\Om$ and is invariant under every Borel automorphism of $\Om$ which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when $\Om=S^\Bbb N$. We apply the ergodic theory of equivalence relations to study the case $\Om\neq S^\Bbb N$, and obtain versions of this theorem when $\Om$ is a countable state Markov shift, and when $\Om$ is the collection of beta expansions of real numbers in $[0,1]$ (a non-Markovian constraint)

On manifolds admitting stable type Ⅲ$_{\textbf1}$ Anosov diffeomorphisms

“…In the proof of Lemma 15, for all D ∈ C * t , we specified π ∈ S ∞ such that V D = π| D and thus V is locally constant and chosen from the orbit of S ∞ on Σ + . Since µ + = M π Q , Q is a Gibbs measure, in fact the measure of maximal entropy on Σ + , it is an invariant measure for the action of S ∞ , see [3]. By this,φ…”

“…This is done by showing the ergodicity of the tail relation of S log T using its identification as T (S)φ whereφ is the tail relation cocycle arising from the function ϕ = log T . 3 Denote by…”

Exchangeable, Gibbs and equilibrium measures for Markov subshifts

“…Next, as in step 1 of the proof of [ANS,Theorem 5.0], p is the restriction of an irreducible, shift-invariant Markov measure µ ∈ P(X) to some clopen set in X. We claim first that for each s ∈ S, either N s (x) := n∈Z δ x n ,s = 0 p-almost surely, or N s (x) = ∞ p-almost surely.…”

“…By the well-known cohomology lemma (see for example [ANS,Lemma 4.3]), g = a + h − h • T +ḡ where a ∈ G,ḡ : X + → G g := {g n (x)−g n (x ) : n ≥ 1, T n x = x, T n x = x } and h : X → G are both generated by site functions such thatḡ : X → G g is aperiodic (in the sense that Gḡ = G g ). By the well-known cohomology lemma (see for example [ANS,Lemma 4.3]), g = a + h − h • T +ḡ where a ∈ G,ḡ : X + → G g := {g n (x)−g n (x ) : n ≥ 1, T n x = x, T n x = x } and h : X → G are both generated by site functions such thatḡ : X → G g is aperiodic (in the sense that Gḡ = G g ).…”

Tail invariant measures of the Dyck shift

“…There are known results for the case of the one sided tail of (mixing) SFT's [3]. Also, for the case of the β-shift it is known that there exists a unique tail-invariant measure [1]. In all of these examples the tail-invariant measure is also equivalent to a unique shift invariant measure of maximal entropy.…”