Topological complexity, minimality and systems of order two on torus

TL;DR

This paper investigates the properties of certain dynamical systems on the torus, providing criteria for minimality, system order, and maximal equicontinuous factors, and computes their topological complexity, addressing an open question.

Contribution

It offers new criteria for minimality and system order in torus extensions and computes their topological complexity, answering an open question by Host-Kra-Maass.

Findings

Criteria for minimality and order in torus extensions established

Topological complexity of the systems computed

Negative answer to an open question by Host-Kra-Maass

Abstract

The dynamical system on T^2 which is a group extension over an irrational rotation on T^1 is investigated. The criterion when the extension is minimal, a system of order 2 and when the maximal equicontinuous factor is the irrational rotation is given. The topological complexity of the extension is computed, and a negative answer to the latter part of an open question raised by Host-Kra-Maass is obtained.