Dynamical simplices and Borel complexity of orbit equivalence

TL;DR

This paper establishes a connection between divisible dynamical simplices and Toeplitz subshifts, showing that orbit equivalence of these subshifts has a complex Borel structure comparable to universal nonarchimedean Polish group actions.

Contribution

It constructs Toeplitz subshifts from any divisible dynamical simplex and analyzes the Borel complexity of their orbit equivalence relation.

Findings

Any divisible dynamical simplex can be realized as invariant measures of a Toeplitz subshift.

Orbit equivalence of Toeplitz subshifts is Borel bireducible to a universal nonarchimedean Polish group action.

The work links dynamical systems with descriptive set theory through Borel complexity analysis.

Abstract

We prove that any divisible dynamical simplex is the set of invariant measures of some Toeplitz subshift. We apply our construction to prove that orbit equivalence of Toeplitz subshifts is Borel bireducible to the universal equivalence relation induced by a Borel action of a nonarchimedean Polish group.