Recurrence and the minimal center of attraction with respect to a Følner sequence

Abstract

Let (X,G) be a G-system, where X is a compact metric space and G a countable discrete infinite amenable group. In this paper, we study the recurrent points with positive upper {Fn}-density (resp. lower {Fn}-density) for a Følner sequence {Fn} and its relation to the minimal {Fn}-center of attraction. In particular, we show that for a given G-system, the minimal {Fn}-center of attraction is independent of the selection of Følner sequences and can be characterized by the set of all recurrent points with positive upper {Fn}-density (resp. lower {Fn}-density).