Complexity, initial condition sensitivity, dimension and weak chaos in dynamical systems

We prove quantitative relations between complexity, initial condition sensitivity and dimension which are suitable for 0-entropy dynamical systems. The orbit complexity indicator is a measure of the asymptotic behaviour of the information that is necessary to describe the orbit of a given point. The indicator generalizes Brudno's orbit complexity (which is strongly related to the entropy of the system). The initial condition sensitivity indicators consider the asymptotic behaviour of the speed of divergence of nearby starting orbits. The indicators have non-trivial values also in weakly chaotic (0-entropy) dynamical systems, characterizing various cases of weakly chaotic dynamics. Using constructive mathematics, local relations are proved between generalized orbit complexity, initial condition sensitivity and the dimension of the underlying space or measure. The relations are quantitative and have non-trivial meaning also in the 0-entropy case. The generalized relations are then applied to the study of an important example of weakly chaotic dynamics: the Manneville map.