Rotation and dynamics for simple solenoidal maps of Tori

Abstract

The rotation number for a circle map has provided a complete and useful classification for that class of maps. In higher dimensions there is still progress to be made towards obtaining a more complete understanding of the relationship between the map and its average rotation. In this dissertation, we explore a class of homeomorphisms on the d dimensional torus T d that preserve each leaf of a foliation of the torus into parallel lines densely winding on T d. First the rotation sets of such maps are explored, with particular emphasis on those maps that have a single fixed point; zero is necessarily an element of those rotation sets. Conditions are found that show when these maps have a non-trivial rotation set. When such maps, with non trivial rotation sets, are created as the time-one map of a flow it is shown that the existence of merely two, or infinitely many ergodic measures is connected to the solvability of a cohomological equation. An example, of the infinitely many ergodic measure case, is provided. Finally, we explore on T ² maps without a fixed point that happen to also have a point whose orbit has bounded deviation from the mean rotation. Such maps are seen to be akin to circle maps with irrational rotation number; the irrationally sloped foliation leads to the map being semi-conjugate to an irrational translation of T ².