Invariant measures for some piecewise continuous maps.

study some special classes of piecewise continuous maps on a finite smooth partition of the compact phase space and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps), this measure exists for any map. Moreover, they can always be selected non-atomic. use this result to demonstrate that any piecewise translation map is metrically equivalent to an interval exchange map. In higher dimensions, we demonstrate that the invariant measure exists if the number of domains in the partition does not exceed the dimension of the space plus one. Finally, we provide a general conditions on approximation of invariant measures of a piecewise continuous map by invariant measures of piecewise translation maps.