Matching for random systems with an application to minimal weight expansions

TL;DR

This paper extends the concept of matching to random dynamical systems on an interval, showing that for many such systems, invariant densities are piecewise constant and analyzing properties of signed binary expansions.

Contribution

It introduces the notion of random matching for random systems and applies it to analyze digit frequencies in signed binary expansions.

Findings

Random matching implies invariant densities are piecewise constant.

For a family of systems, the digit 0 frequency in expansions is at most 1/2.

Almost every parameter in the family exhibits random matching.

Abstract

We extend the notion of matching for one-dimensional dynamical systems to random matching for random dynamical systems on an interval. We prove that for a large family of piecewise affine random systems of the interval the property of random matching implies that any invariant density is piecewise constant. We further introduce a one-parameter family of random dynamical systems that produce signed binary expansions of numbers in the interval [-1,1]. This family has random matching for Lebesgue almost every parameter. We use this to prove that the frequency of the digit 0 in the associated signed binary expansions never exceeds 1/2.