Entropy and complexity in dynamical systems and PDEs

In this chapter we aim at presenting applications of notions from Information Theory to the study of the statistical properties of dynamical systems. In particular we review the notion of Algorithmic Information Content, or Kolmogorov Complexity, and recall the definition of complexity of an orbit of a dynamical system. The main result is that for ergodic dynamical systems, the complexity of an orbit is almost everywhere constant and coincides with the Kolmogorov-Sinai entropy of the system. We remark that the interest in these results has at least two motivations: applications of this approach to time series of e.g. physical or biomedical origin; investigations on statistical properties of dynamical systems which present ``critical'' behaviours with respect to other classical indicators or for which it may be difficult to compute them