  Piecewise Translations is a class of dynamical systems which arises from some
applications in computer science, machine learning, and electrical engineering.
In dimension 1 it can also be viewed as a non-invertible generalization of
Interval Exchange Transformations. These dynamical systems still possess some
features of Interval Exchanges but the total volume is no longer preserved and
allowed to decay.
  Every Piecewise Translation has a well-defined attracting subset which is the
locus of our interest. We prove some results about how fast the dynamics lock
onto the attractor, geometry of the attractor, and its ergodic properties. Then
we consider stochastic Piecewise Translations and prove that almost surely its
attractor has zero Lebesgue measure. Finally we present some conjectures and
supporting numerics.
