The concept of bounded mean motion for toral homeomorphisms
Abstract
Description
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.
Some of the results of the previous version were improved in arXiv:0803.2428 . In order to avoid intersections, the remaining material was reorganised, and a new statement concerning sensitive dependence on initial conditions was added. 17 pages, 2 figures
Some of the results of the previous version were improved in arXiv:0803.2428 . In order to avoid intersections, the remaining material was reorganised, and a new statement concerning sensitive dependence on initial conditions was added. 17 pages, 2 figures