Dynamical systems and computable information

Abstract

We present some new results which relate information to chaotic dynamics. In our approach the quantity of information is measured by the Algorithmic Information Content (Kolmogorov complexity) or by a sort of computable version of it (Computable Information Content) in which the information is measured by the use of a suitable universal data compression algorithm. We apply these notions to the study of dynamical systems by considering the asymptotic behavior of the quantity of information necessary to describe their orbits. When a system is ergodic, this method provides an indicator which equals the Kolmogorov-Sinai entropy almost everywhere. Moreover, if the entropy is 0, our method gives new indicators which measure the unpredictability of the system and allows to classify various kind of weak chaos. Actually this is the main motivation of this work. The behaviour of a zero entropy dynamical system is far to be completely predictable exept that in particular cases. In fact there are 0 entropy systems which exibit a sort of {\it weak chaos} where the information necessary to describe the orbit behavior increases with time more than logarithmically (periodic case) even if less than linearly (positive entropy case). Also, we believe that the above method is useful for the classification of zero entropy time series. To support this point of view, we show some theoretical and experimenthal results in specific cases.