  In this paper we introduce a new criteria for classification of dynamical
orbits in order to study the complexity theory of dynamical systems. And then
we find that there exist twelve different orbit structures in hyperbolic
systems and they display strong complexity in the sense of topological entropy,
though most of them is trivial from probabilistic perspective. In this process
we obtain two powerful ergodic properties on entropy-dense property and
saturated property. The criteria is to use upper and lower natural density,
upper and lower Banach density to differ statistical future of dynamical orbits
and we also get some combined observations with many classical concepts
including recurrence, ergodic average together.
