  We consider the dynamical behavior of Martin-L\"of random points in dynamical
systems over metric spaces with a computable dynamics and a computable
invariant measure. We use computable partitions to define a sort of effective
symbolic model for the dynamics. Through this construction we prove that such
points have typical statistical behavior (the behavior which is typical in the
Birkhoff ergodic theorem) and are recurrent. We introduce and compare some
notions of complexity for orbits in dynamical systems and prove: (i) that the
complexity of the orbits of random points equals the Kolmogorov-Sina\"i entropy
of the system, (ii) that the supremum of the complexity of orbits equals the
topological entropy.
