Distal points for Borel measures

Abstract

We introduce the notion of distal point for a Borel measure with respect to a given homeomorphism. We prove that a point is distal for every nonatomic Borel probability measure if and only if it is countably-distal for the given homeomorphism. We prove that the distal point set of a measure is a Borel set. We study the distal measures (i.e. measures for which every point is distal) and prove that they are approximated with respect to the weak* topology by ones with invariant support. Furthermore, the distal measures are dense in the space of measures just when the ones with full support are. Afterwards, we consider the almost distal measures (i.e. measures for which almost every point is distal) and exhibit one which is not distal. Moreover, a circle homeomorphism has an almost distal measure with full support if and only if it is distal. In particular, every countably-distal circle homeomorphism is distal. We prove for circle homeomorphisms with rational rotation number that the almost distal measures are precisely the distal ones. Finally, we prove that every homeomorphism with distal measures has uncountably many almost periodic points and those with almost distal measures on compact spaces have infinitely many nonwandering points.