Coupled map lattices: Some topological and ergodic properties

Lattice dynamical systems (LDSs) form the class of extended systems that is the intermediate one between partial differential equations (PDEs) and cellular automata. The most popular class of LDSs is formed by coupled map lattices (CMLs). While being introduced rather recently LDSs allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio-temporal dynamics and to obtain some new, and even rigorous results on space-time chaos, intermittency and pattern formation. We discuss the rigorous results that were obtained in this area, but eventually give also some applications.