In a topological dynamical system the complexity of an orbit is a measure of
the amount of information (algorithmic information content) that is necessary
to describe the orbit. This indicator is invariant up to topological
conjugation. We consider this indicator of local complexity of the dynamics and
provide different examples of its behavior, showing how it can be useful to
characterize various kind of weakly chaotic dynamics. We also provide criteria
to find systems with non trivial orbit complexity (systems where the
description of the whole orbit requires an infinite amount of information). We
consider also a global indicator of the complexity of the system. This global
indicator generalizes the topological entropy, taking into account systems were
the number of essentially different orbits increases less than exponentially.
Then we prove that if the system is constructive (roughly speaking: if the map
can be defined up to any given accuracy using a finite amount of information)
the orbit complexity is everywhere less or equal than the generalized
topological entropy. Conversely there are compact non constructive examples
where the inequality is reversed, suggesting that this notion comes out
naturally in this kind of complexity questions.Comment: 23 page