Rigidity and Toeplitz systems

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 exists a Toeplitz system with zero entropy that is not partially measure-theoretically rigid with respect to its (unique) invariant measure. 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒮</m:mi> </m:math> {\mathcal{S}} -adic Toeplitz subshift which has countably infinitely many ergodic invariant probability measures which are rigid for the same rigidity sequence.