  We prove that transport in the phase space of the "most strongly chaotic"
dynamical systems has three different stages. Consider a finite Markov
partition (coarse graining) $\xi$ of the phase space of such a system. In the
first short times interval there is a hierarchy with respect to the values of
the first passage probabilities for the elements of $\xi$ and therefore finite
time predictions can be made about which element of the Markov partition
trajectories will be most likely to hit first at a given moment. In the third
long times interval, which goes to infinity, there is an opposite hierarchy of
the first passage probabilities for the elements of $\xi$ and therefore again
finite time predictions can be made. In the second intermediate times interval
there is no hierarchy in the set of all elements of the Markov partition. We
also obtain estimates on the length of the short times interval and show that
its length is growing with refinement of the Markov partition which shows that
practically only this interval should be taken into account in many cases.
These results demonstrate that finite time predictions for the evolution of
strongly chaotic dynamical systems are possible. In particular, one can predict
that an orbit is more likely to first enter one subset of phase space than
another at a given moment in time. Moreover, these results suggest an algorithm
which accelerates the process of escape through "holes" in the phase space of
dynamical systems with strongly chaotic behavior.
