# Markovian Random Iterations of Maps

## TL;DR

This paper extends classical results on i.i.d. random iterations to Markovian settings, establishing a correspondence between stationary and invariant measures, and demonstrates local synchronization for circle homeomorphisms.

## Contribution

It introduces a novel correspondence between stationary and invariant measures for Markovian random iterations, generalizing classical i.i.d. results.

## Key findings

- Established a one-to-one correspondence between stationary and invariant measures.
- Proved local synchronization for Markovian random iterations of circle homeomorphisms.
- Extended classical results to Markovian frameworks.

## Abstract

In this paper, we study Markovian random iterations of maps on standard measurable spaces. We establish a one-to-one correspondence between stationary measures and a certain class of invariant measures of a Markovian random iteration, extending a similar classical result of independent and identically distributed random iterations. As an application, we prove a local synchronization property for Markovian random iterations of homeomorphisms of the circle $S^{1}$.