A note on the topological synchronization of unimodal maps

In this note we complete the analysis carried on in Caby et al (2023 Nonlinearity36 3603-21) 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 is an element of ( 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.