Ergodic quasi-invariant measures on topologically mixing subshifts are isomorphic to Bernoulli shifts

TL;DR

This paper demonstrates that ergodic quasi-invariant measures on topologically mixing subshifts are isomorphic to Bernoulli shifts, and shows Gibbs measures on such subshifts are quasi-invariant, advancing understanding of measure-theoretic dynamics.

Contribution

It establishes a new isomorphism result for ergodic measures under quasi-invariance conditions and characterizes Gibbs measures as quasi-invariant on mixing subshifts.

Findings

Ergodic quasi-invariant measures are isomorphic to Bernoulli shifts.

Gibbs measures on topologically mixing subshifts are quasi-invariant.

Provides a link between quasi-invariance and Bernoulli properties in dynamical systems.

Abstract

We prove that a shift ergodic measure on a topologically mixing sub-shift is isomorphic to a Bernoulli shift whenever it is quasi invariant under permutations of finite number of coordinates. We prove also that Gibbs measures on topologically mixing subshift of finite type are quasi invariant.