# Stability of iterated function systems on the circle

## TL;DR

This paper proves that iterated function systems of circle homeomorphisms with at least one dense orbit are asymptotically stable, with the Perron-Frobenius operator satisfying the e-property and establishing a strong law of large numbers.

## Contribution

It introduces new stability results for circle homeomorphism systems and demonstrates the e-property and strong law of large numbers for these systems.

## Key findings

- Systems with dense orbit are asymptotically stable
- Perron-Frobenius operator satisfies the e-property
- Strong law of large numbers holds for trajectories

## Abstract

We prove that any Iterated Function System of circle homeomorphisms with at least one of them having dense orbit, is asymptotically stable. The corresponding Perron-Frobenius operator is shown to satisfy the e-property, that is, for any continuous function its iterates are equicontinuous. The Strong Law of Large Numbers for trajectories starting from an arbitrary point for such function systems is also proved.