Attractors of Piecewise Translation Maps

TL;DR

This paper studies the properties of attractors in Piecewise Translation maps, including their geometry, ergodic behavior, and measure, with extensions to stochastic versions and numerical evidence.

Contribution

It provides new theoretical results on the dynamics, geometry, and measure of attractors in Piecewise Translation systems, including stochastic cases.

Findings

Attractors exhibit specific geometric and ergodic properties.

Stochastic Piecewise Translations almost surely have attractors of zero Lebesgue measure.

Numerical experiments support the conjectures about attractor behavior.

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.