Rigidity and Toeplitz systems
Henk Bruin, Olena Karpel, Piotr Oprocha, Silvia Radinger
TL;DR
The paper addresses when Cantor dynamical systems exhibit measure-theoretic rigidity, focusing on Toeplitz and enumeration systems. It develops a Bratteli-Vershik toolkit to control invariant measures and prove rigidity for certain extensions from odometers, while also constructing counterexamples (e.g., a zero-entropy Toeplitz system not partially rigid) and identifying classes (such as some enumeration systems) that are partially rigid. A notable result is the construction of a minimal $S$-adic Toeplitz subshift with countably many ergodic invariant measures rigid for the same sequence, illustrating nuanced rigidity phenomena across systems. Overall, the work clarifies how rigidity properties interact with entropy, mixing, and the Bratteli-Vershik structure, guiding the design of Cantor systems with prescribed rigidity behavior.
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.