Two remarks on the Shrinking Target Property

TL;DR

This paper characterizes when translations on the torus have the shrinking target property and constructs examples of mixing maps that lack this property, highlighting nuanced dynamical behaviors.

Contribution

It establishes a precise criterion for the shrinking target property in toral translations and constructs mixing maps without this property using reparametrizations.

Findings

Translations with vectors of constant type have the monotone shrinking target property.

Existence of mixing area-preserving maps on the three torus without the shrinking target property.

Reparametrizations can produce mixing flows that lack the shrinking target property.

Abstract

We show that a translation of vector V on the torus has the monotone shrinking target property if and only if the vector V is of constant type. Then, using reparametrizations of linear flows, we show that there exist area preserving real analytic maps of the three torus that are mixing of all orders and do not enjoy the monotone shrinking target property.