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

Stefano Galatolo (1) , Mathieu Hoyrup (2) , Cristobal Rojas (3)

1 DMA - Dipartimento di Matematica Applicata [Pisa]
2 CARTE - Theoretical adverse computations, and safety
3 Department of Mathematics [University of Toronto]

We consider the dynamical behavior of Martin-Löf 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-Sinaï entropy of the system, (ii) that the supremum of the complexity of orbits equals the topological entropy.

Mots clés: algorithmic randomness; Kolmogorov-Chaitin complexity; computable partition; effective symbolic dynamics; entropy; orbit complexity

Domaines: Théorie de l'information [cs.IT]; Théorie de l'information et codage [math.IT]; Systèmes dynamiques [math.DS]

Stefano Galatolo, Mathieu Hoyrup, Cristobal Rojas. Effective symbolic dynamics, random points, statistical behavior, complexity and entropy. Information and Computation, 2010, 208 (1), pp.23-41. ⟨10.1016/j.ic.2009.05.001⟩. ⟨inria-00517382⟩