Effects of randomization on asymptotic periodicity of nonsingular transformations

TL;DR

This paper investigates how introducing randomness into piecewise expanding transformations affects the long-term periodic behavior of their density functions, which is crucial for understanding real-world systems with noise.

Contribution

It provides new insights into the impact of randomization on the asymptotic periodicity of densities in nonsingular transformations, extending deterministic results to stochastic settings.

Findings

Randomization can alter the asymptotic periodicity of densities.

The study characterizes conditions under which periodicity persists or changes.

Results are applicable to systems with measurement errors or external noise.

Abstract

It is known that the Perron--Frobenius operators of piecewise expanding $\mathcal{C}^2$ transformations possess an asymptotic periodicity of densities. On the other hand, external noise or measurement errors are unavoidable in practical systems; therefore, all realistic mathematical models should be regarded as random iterations of transformations. This paper aims to discuss the effects of randomization on the asymptotic periodicity of densities.