Remarks on shrinking target properties

TL;DR

This paper explores various notions of shrinking target properties in dynamical systems, demonstrating that certain expanding maps possess these properties while some isometries do not.

Contribution

It introduces and analyzes different shrinking target properties, establishing their presence or absence in specific classes of dynamical systems.

Findings

Expanding circle maps have shrinking target properties.

Circle homeomorphisms lack shrinking target properties.

Isometries of complete, separable metric spaces do not have shrinking target properties.

Abstract

This paper defines and describes a few (related) notions of shrinking target property. We show that simultaneous expanding circle maps have a certain shrinking target property, but that circle homeomorphisms and isometries of complete, separable metric spaces do not have any shrinking target property.