Invariant measures of torus piecewise isometries

We study measure-theoretical aspects of torus piecewise isometries. Not much is known about this type of dynamical systems, except for the special case of one-dimensional interval exchange mappings. The last case is fundamentally different from the general situation in the presence of an invariant measure (Lebesgue measure), which helps a lot in the analysis. Due to the absence of good methods of analysis of general systems with discontinuities, even the existence of invariant measures of the torus piecewise isometries was an open question. We establish sufficient conditions for the existence/absence of invariant measures for this class of systems. Technically, our results are based on the approximation of the maps under study by weakly periodic ones.