An application of functional equations for generating $\varepsilon$-invariant measures

TL;DR

This paper explores a functional equation approach to generate measures that are approximately invariant under a measurable transformation, providing conditions for existence and uniqueness of solutions related to $ ext{L}^1$ functions.

Contribution

It introduces a novel method using functional equations to construct and analyze $ ext{ε}$-invariant measures for measurable transformations.

Findings

Established existence and uniqueness conditions for solutions in $ ext{L}^1$.

Derived a specific functional equation involving derivatives of functions.

Connected the solutions to measures that are absolutely continuous and approximately invariant.

Abstract

Let $(X,{\mathcal A},\mu)$ be a probability space and let $S\colon X\to X$ be a measurable transformation. Motivated by the paper of K. Nikodem [Czechoslovak Math. J. 41(116) (4) (1991) 565--569], we concentrate on a functional equation generating measures that are absolutely continuous with respect to $\mu$ and $\varepsilon$-invariant under $S$. As a consequence of the investigation, we obtain a result on the existence and uniqueness of solutions $\varphi\in L^1([0,1])$ of the functional equation $$ \varphi(x)=\sum_{n=1}^{N}|f_n'(x)|\varphi(f_n(x))+g(x), $$ where $g\in L^1([0,1])$ and $f_1,\ldots,f_N\colon[0,1]\to[0,1]$ are functions satisfying some extra conditions.