  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.
