Geometric and Measure-Theoretic Shrinking Targets in Dynamical Systems

TL;DR

This paper explores the conditions under which shrinking targets in ergodic dynamical systems are detectable or not, using geometric and measure-theoretic approaches, and presents related theorems and constructions.

Contribution

It introduces new Baire category theorems and constructions for fixed maps, advancing understanding of shrinking targets in ergodic theory.

Findings

Proved Baire category theorems related to shrinking targets

Constructed examples of fixed maps with specific shrinking target properties

Discussed open questions in the field of shrinking targets

Abstract

We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.