Attractors of Piecewise Translation Maps

Abstract

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.