Algorithmically random points in measure preserving systems, statistical behaviour, complexity and entropy
Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas · arXiv (Cornell University) · 2008
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 define a sort of effective symbolic model for the dynamics. Trough 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 notion of complexity for orbits in dynamical systems and we prove that the complexity of the orbits of random points equals the entropy of the system.