Stochastic perturbations of iterations of a simple, non-expanding, nonperiodic, piecewise linear, interval-map

TL;DR

This paper investigates the long-term behavior of stochastic perturbations of a specific interval map, demonstrating convergence to a unique invariant measure and refuting previous claims of asymptotic periodicity.

Contribution

It proves the existence and uniqueness of an invariant measure for the perturbed map, contradicting earlier assertions of three-periodic asymptotics.

Findings

Convergence of distributions to a unique invariant measure

Refutation of the 1987 claim of three-periodicity

Demonstration of stability under stochastic perturbations

Abstract

Let g(x)=x/2 + 17/30 (mod 1), let \xi_i, i= 1,2,... be a sequence of independent, identically distributed random variables with uniform distribution on the interval [0,1/15], define g_i(x)=g(x)+ \xi_i (mod 1) and, for n=1,2,..., define g^n(x)=g_n(g_{n-1}(...(g_1(x))...)). For x \in [0,1) let \mu_{n,x} denote the distribution of g^n(x). The purpose of this note is to show that there exists a unique probability measure \mu, such that, for all x \in [0,1), \mu_{n,x} tends to \mu, as n tends to infinity. This contradicts a claim by Lasota and Mackey from 1987 stating that the process has an asymptotic three-periodicity.