Synchronization–intermittent synchronization transition and the uniqueness of absolutely continuous invariant measure in coupled tent map lattice

We develop a new geometric-combinatorial method to study synchronization, intermittent synchronization, and the absolutely continuous invariant measure of a two-node coupled map lattice (CML) for a class of tent maps whose slope $k\le 2$. We prove that with a coupling $c\in [0,c^{\ast })\cup (1-c^{\ast },1]$, where $c^{\ast }=\frac{1}{2}(1-\frac{1}{k})$, there exists a unique absolutely continuous invariant measure and intermittent synchronization occurs, that is, almost every point enters and exits an arbitrarily small neighborhood of the diagonal infinitely many times. In contrast, for $c\in (c^{\ast },1-c^{\ast })$, synchronization occurs for every point. This shows that $c^{\ast }$ is the transition point of the coupling strength.