Iterated function systems with a given continuous stationary distribution

TL;DR

This paper presents a method to construct an iterated function system (IFS) with probabilities that guarantees a specified continuous probability measure on the real line as its unique stationary distribution.

Contribution

The authors develop a novel construction technique for IFSs that ensures any given continuous probability measure is the unique stationary distribution.

Findings

Successfully constructs IFS with a prescribed continuous stationary measure

Proves uniqueness of the stationary measure for the constructed IFS

Provides a general framework applicable to any continuous probability measure

Abstract

For any continuous probability measure $\mu$ on ${\mathbb R}$ we construct an IFS with probabilities having $\mu$ as its unique measure-attractor.