Effective symbolic dynamics, random points, statistical behavior, complexity and entropy

We consider the dynamical behavior of Martin-Lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to de. ne 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 entropy of the system, (ii) that the supremum of the complexity of orbits equals the topological entropy. (C) 2009 Elsevier Inc. All rights reserved.