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

Authors:
Stefano Galatolo;Mathieu Hoyrup;Cristóbal Rojas
Affiliations:
Dipartimento di Matematica Applicata, Universita di Pisa, Paris, France;Département d'Informatique -- Ecole Normale Supérieure, Paris, France;Département d'Informatique -- Ecole Normale Supérieure, Paris, France and CREA, Ecole Polytechnique, Paris, France
Venue:
Information and Computation
Year:
2010

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Abstract

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. 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-Sinai entropy of the system, (ii) that the supremum of the complexity of orbits equals the topological entropy.