Periodicity and periodicity on average in coupled map lattices. Rigorous results.

. - We prove the existence and genericity in a sense of periodic on average behavior for systems of coupled map lattices, when the behavior of the system is chaotic, and the existence of periodic behavior for such systems, when the uncoupled one is chaotic. The stability of both of these phenomena with respect to small random perturbations is also proved. Key words: coupled map lattice, periodic behavior, Sinai-Bowen-Ruelle (SBR) invariant measure. H.Chate and P.Manneville (1) showed numerically the possibility for a system with a large number of components to have a chaotic behavior at a local scale (for individual subsystems), but on average periodic at a global scale (for space averages). Recently Y.Pomeau (2) proposed a beautiful heuristic explanation of this phenomenon for the case of coupled map lattices. His explanation bases on two following assumptions. At first he considered the uncoupled system (consisted of quadratic maps with the period-three window chaotic regime) and..