Information flow and chaotic dynamics

We introduce an information-theory-based concept for the characterization of the information flow in chaotic systems in the framework of symbolic dynamics. The information flow characterizes the loss of information about the initial conditions, i.e. the decay of statistical correlations between the entire past and a point p steps ahead into the future as a function of p. In the case where the partition generating the symbolic dynamics is finite the information loss is measured by an entropy. This entropy which we call "source-p-step-mutual information" is an extension of the source-entropy. More relevant is the study of the evolution of information loss for the cases of infinitesimal partitions which characterize the intrinsic behaviour of the dynamics on an extremely fine scale. The intrinsic information flow is characterized by the evolution of a conditional entropy which generalizes the Kolmogorov-Sinai entropy for the case of observing the uncertainty more than one step ahead into the future. In order to characterize the dynamics on fine scale and to describe the true and intrinsic information flow of chaotic systems an infinitesimal partition were used. The zeta function formalism is applied.