Proximality, stability, and central limit theorem for random maps on an interval

TL;DR

This paper investigates stochastic dynamical systems on an interval, establishing conditions under which they exhibit proximality, stability, and a central limit theorem, thereby advancing understanding of their long-term behavior.

Contribution

It introduces the concept of μ-injectivity for non-invertible maps and proves that under this condition, key dynamical properties and a CLT hold.

Findings

Proximality is established under μ-injectivity.

Asymptotic stability is proved for systems satisfying the conditions.

A central limit theorem is derived for the systems.

Abstract

Stochastic dynamical systems consisting of non-invertible continuous maps on an interval are studied. It is proved that if they satisfy the recently introduced so-called $\mu$-injectivity and some mild assumptions, then proximality, asymptotic stability and a central limit theorem hold.