Dynamics and relaxation properties of complex systems with memory

The author introduces the concept of a map with memory associated to a given map f and to a normalized weight sequence P. The stabilization properties of maps with memory are established. It is shown that fixed points and periodic orbits, in the neighbourhood of which a map f has an oscillating behaviour, can be 'coherentized' by the use of a memory. Maps with exponential decaying fading memory are characterized as displaying a phase transition near the critical value beta c of the relaxation constant beta with universal exponent nu =1.