A note on the topological synchronisation of unimodal maps

TL;DR

This paper analyzes the topological synchronization of coupled unimodal maps, focusing on invariant measures and convergence properties as coupling varies, thereby completing previous work and clarifying measure uniqueness and convergence rates.

Contribution

It computes the invariant measure limits at extreme coupling values and proves geometric convergence of the Markov chain to its stationary measure, extending prior analysis.

Findings

Invariant measure limits at coupling extremes

Uniqueness of the invariant measure for the random system

Geometric convergence of the Markov chain

Abstract

In this note we complete the analysis carried on in \cite{CGSV} about the topological synchronisation of unimodal maps of the interval coupled in a master-slave configuration, by answering to the questions raised in that paper. Namely, we compute the weak limits of the invariant measure of the coupled system as the coupling strength $k\in\left( 0,1\right) $ tends to $0$ and to $1$ and discuss the uniqueness of the invariant measure of its random dynamical system counterpart, proving that the convergence of the associated Markov chain to its unique stationary measure is geometric. [CGSV] Caby Th., Gianfelice M., Saussol B., Vaienti S. "Topological synchronisation or a simple attractor?" Nonlinearity Vol. 36, no. 7, pp. 3603-3621 (2023).