Intermittent synchronization in non-weakly coupled piecewise-linear expanding map lattice: A geometric-combinatorial approach

Coupled (chaotic) map lattices (CMLs) characterize the collective dynamics of a spatially distributed system whose local units are linked either locally or globally. Previous research on the dynamical behavior of CMLs, based primarily on the Perron-Frobenius operator framework, has focused mainly on the weakly coupled case. In this paper, we develop a novel geometric-combinatorial method to study the dynamical behavior of CMLs beyond the weak-coupling regime, specifically a two-node system with identical piecewise-linear expanding maps. We derive a necessary and sufficient condition for two facts: the uniqueness of the absolutely continuous invariant measures and the occurrence of intermittent synchronization-i.e., almost every orbit enters and leaves an arbitrarily small neighborhood of the diagonal infinitely often.