Synchronization in minimal iterated function systems on compact manifolds

TL;DR

This paper investigates synchronization phenomena in iterated function systems on compact manifolds, proving the existence of a unique attracting invariant graph under certain conditions, extending previous circle-based results to general manifolds.

Contribution

It establishes the existence of a unique attracting invariant graph for synchronized systems on arbitrary compact manifolds, generalizing prior circle-specific findings.

Findings

Existence of a unique attracting invariant graph under transitivity and negative fiber Lyapunov exponents.

Synchronization occurs as orbits converge to this invariant graph.

Results extend synchronization theory from circle to general compact manifolds.

Abstract

We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence of synchronization. The result extends previous results for iterated function systems by diffeomorphisms on the circle, to arbitrary compact manifolds.