Information, Initial Condition Sensitivity and Dimension in Weakly Chaotic Dynamical Systems

We study generalized indicators of sensitivity to initial conditions and orbit complexity in topological dynamical systems. The orbit complexity is a measure of the asymptotical behavior of information that is necessary to describe the orbit of a given point. The indicator generalizes, in a certain sense, the Brudno's orbit complexity (which is strongly related to the entropy of the system). The initial condition sensitivity indicator generalizes in some sense the Brin-Katok local entropy. The indicators have non trivial values also in weakly chaotic dynamical systems, characterizing various cases of weakly chaotic dynamics. Then, using constructivity, local relations are proved between complexity, initial condition sensitivity and the dimension of the underlying space or invariant measure