The concept of bounded mean motion for toral homeomorphisms

TL;DR

This paper investigates the property of bounded mean motion in toral homeomorphisms, demonstrating its significance for semi-conjugacy to irrational rotations and exploring its limitations in non-conservative cases.

Contribution

It extends the understanding of bounded mean motion by providing examples and analyzing cases where this property fails, especially in non-conservative settings.

Findings

Bounded mean motion characterizes semi-conjugacy in conservative pseudo-rotations.

Counterexamples show the property does not hold in non-conservative cases.

Unbounded mean motion leads to sensitive dependence on initial conditions.

Abstract

A conservative irrational pseudo-rotation of the two-torus is semi-conjugate to the irrational rotation if and only if it has the property of bounded mean motion [10]. (Here 'irrational pseudo-rotation' means a toral homeomorphism with uniquely defined, totally irrational rotation vector.) The aim of this note is to explore this concept some further. For instance, we provide an example which shows that the preceding statement does not hold in the non-conservative case. Further, we collect a number of observations concerning the case where the bounded mean motion property fails. In particular, we show that a non-wandering irrational pseudo-rotation of the two-torus with unbounded mean motion has sensitive dependence on initial conditions.