Ergodic and geometric properties of piecewise isometries - two-dimensional generalizations of interval exchange transformations

A piecewise isometry is a finite or countably infinite collection of isometries acting on disjoint polygonal domains. One dimensional piecewise isometries, known as interval exchanges, have been extensively studied and their typical ergodic properties are now understood. In two dimensions the landscape of phenomena appears to be in contrast to one dimensional counterparts. In this talk we will illustrate several piecewise isometric systems with different dynamics, one of which (a joint work with Peter Ashwin) features a coexistence of an infinite number of periodic components as well as uniquely ergodic components on which the map acts as an interval exchange transformation. These periodic components form an irregular polygonal set. Since piecewise isometric systems are nonhyperbolic and they are discontinuous, many standard tools are not useful in the analysis of the orbit behavior. One of the available techniques in case of rational piecewise isometries is to study the return maps using cyclotomic fields. We show that the return actions preserve some properties of cyclotomic integers defining the maps. The talk will be augmented by multimedia.