Variations on topological recurrence

TL;DR

This paper explores recurrence properties in topological dynamics, establishing links between different systems and finite versions, and demonstrating that Bohr recurrence sets suffice for recurrence in nilsystems and affine systems.

Contribution

It introduces new relations between recurrence in various systems and extends recurrence properties from rotations to nilsystems and affine systems.

Findings

Bohr recurrence sets suffice for recurrence in nilsystems

Extension of recurrence properties to multiple recurrence in affine systems

Connections established between recurrence and combinatorial problems

Abstract

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.