Irreversible and dissipative systems

We study some new dynamical systems where the corresponding piecewise linear flow is neither time reversible nor measure preserving. We create a dissipative system by starting with a finite polysquare translation surface, and then modifying it by including a one-sided barrier on a common vertical edge of two adjacent atomic squares, in the form of a union of finitely many intervals. The line flow in this system partitions the system into a transient set and a recurrent set. We are interested in the geometry of these two sets.