Exchangeable measures for subshifts

Abstract

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