  We study generalized indicators of sensitivity to initial conditions and
orbit complexity in topological dynamical systems. The orbit complexity is a
measure of the asymptotic behavior of the 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 indicators consider the asymptotic behavior
of the speed of divergence of nearby staring orbits.
  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 generalized orbit
complexity, initial condition sensitivity and the dimension of the underlying
space or invariant measure
