Rigidity and Toeplitz systems

TL;DR

This paper investigates measure-theoretic rigidity in Cantor dynamical systems, especially Toeplitz and enumeration systems, revealing new examples and properties related to rigidity and entropy.

Contribution

It introduces new constructions of Toeplitz and enumeration systems with specific rigidity and entropy properties, expanding understanding of measure-theoretic rigidity in these systems.

Findings

Existence of zero-entropy Toeplitz systems that are not partially measure-theoretically rigid.

All enumeration systems defined by linear recursion are partially rigid.

Constructed a Toeplitz subshift with infinitely many ergodic measures sharing the same rigidity sequence.

Abstract

The aim of this paper is to study measure-theoretical rigidity and partial rigidity for classes of Cantor dynamical systems including Toeplitz systems and enumeration systems. We use Bratteli diagrams to control invariant measures that are produced in our constructions. This leads to systems with desired properties. Among other things, we show that there exist Toeplitz systems with zero entropy which are not partially measure-theoretically rigid with respect to any of its invariant measures. We investigate enumeration systems defined by a linear recursion, prove that all such systems are partially rigid and present an example of an enumeration system which is not measure-theoretically rigid. We construct a minimal $\mathcal{S}$-adic Toeplitz subshift which has countably infinitely many ergodic invariant probability measures which are rigid for the same rigidity sequence.