Variations on topological recurrence

Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate finite versions of recurrence, and describe connections to combinatorial problems. In particular, we show that sets of Bohr recurrence (meaning sets of recurrence for rotations) suffice for recurrence in nilsystems. Additionally, we prove an extension of this property for multiple recurrence in affine systems

Constant-length substitutions and countable scrambled sets

In this paper we provide examples of topological dynamical systems having either finite or countable scrambled sets. In particular we study conditions for the existence of Li-Yorke, asymptotic and distal pairs in constant--length substitution dynamical systems. Starting from a circle rotation we also construct a dynamical system having Li--Yorke pairs, none of which is recurrent

Eigenvalues of Toeplitz minimal systems of finite topological rank

In this article we characterize measure theoretical eigenvalues of Toeplitz Bratteli-Vershik minimal systems of finite topological rank which are not associated to a continuous eigenfunction. Several examples are provided to illustrate the different situations that can occur