Predictability: a way to characterize Complexity

Different aspects of the predictability problem in dynamical systems are
reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai
entropy, Shannon entropy and algorithmic complexity is discussed. In
particular, we emphasize how a characterization of the unpredictability of
a system gives a measure of its complexity. Adopting this point of view, we
review some developments in the characterization of the predictability of
systems showing different kind of complexity: from low-dimensional systems
to high-dimensional ones with spatio-temporal chaos and to fully developed
turbulence. A special attention is devoted to finite-time and
finite-resolution effects on predictability, which can be accounted with
suitable generalization of the standard indicators. The problems involved
in systems with intrinsic randomness is discussed, with emphasis on the
important problems of distinguishing chaos from noise and of modeling the
system. The characterization of irregular behavior in systems with discrete
phase space is also considered.