Predictability: a way to characterize Complexity

Different aspects of the predictability problem in dynamical systems arereviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinaientropy, Shannon entropy and algorithmic complexity is discussed. Inparticular, we emphasize how a characterization of the unpredictability ofa system gives a measure of its complexity. Adopting this point of view, wereview some developments in the characterization of the predictability ofsystems showing different kind of complexity: from low-dimensional systemsto high-dimensional ones with spatio-temporal chaos and to fully developedturbulence. A special attention is devoted to finite-time andfinite-resolution effects on predictability, which can be accounted withsuitable generalization of the standard indicators. The problems involvedin systems with intrinsic randomness is discussed, with emphasis on theimportant problems of distinguishing chaos from noise and of modeling thesystem. The characterization of irregular behavior in systems with discretephase space is also considered.