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.