The concept of bounded mean motion for toral homeomorphisms

摘要

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 [T. Jaeger, Linearisation of conservative toral homeomorphisms, Invent. Math., published online 9 December 2008, DOI: 10.1007/s00222-088-0171-5]. (Here 'irrational pseudo-rotation' means a toral homeomorphism with a unique and totally irrational rotation vector). The aim of this note is to explore this concept 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.