Invariant measures and orbit equivalence for generalized Toeplitz subshifts

TL;DR

This paper constructs Toeplitz subshifts with prescribed invariant measure structures for certain groups and demonstrates their orbit equivalence to given Toeplitz flows, advancing understanding of dynamical systems and measure theory.

Contribution

It introduces a method to realize any Choquet simplex as the invariant measure set for Toeplitz G-subshifts and shows orbit equivalence between Toeplitz flows and higher-dimensional Toeplitz subshifts.

Findings

Existence of Toeplitz G-subshifts with prescribed invariant measures

Construction of Toeplitz ${f Z}^d$-subshifts orbit equivalent to given Toeplitz flows

Extension of Toeplitz subshifts to higher dimensions with preserved orbit structure

Abstract

We show that for every metrizable Choquet simplex $K$ and for every group $G$, which is infinite, countable, amenable and residually finite, there exists a Toeplitz $G$-subshift whose set of shift-invariant probability measures is affine homeomorphic to $K$. Furthermore, we get that for every integer $d\geq 1$ and every Toeplitz flow $(X,T)$, there exists a Toeplitz ${\mathbb Z}^d$-subshift which is topologically orbit equivalent to $(X,T)$.