A note on the topological synchronisation of unimodal maps

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 $k\in (0,1)$ 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).