In this talk we will study measure-theoretical rigidity and partial rigidity for classes of Cantor dynamical systems including Toeplitz systems and enumeration systems. With the use of Bratteli-Vershik dynamical systems we can control invariant measures. Their structure in the Bratteli diagram leads us to find systems with the desired properties. Among other things, we will analyse different Toeplitz systems for their rigidity and show that there exist Toeplitz systems which have zero entropy and are not partially measure theoretically rigid with respect to any of its invariant measures. Further we show varying rigidity in the family of enumeration systems defined by a linear recursion.

This talk is based on joint work with Henk Bruin, Olena Karpel and Piotr Oprocha.