Effective symbolic dynamics, random points, statistical behavior, complexity and entropy
Dynamical Systems
2008-04-29 v2 Information Theory
math.IT
Probability
Abstract
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.
Cite
@article{arxiv.0801.0209,
title = {Effective symbolic dynamics, random points, statistical behavior, complexity and entropy},
author = {Stefano Galatolo and Mathieu Hoyrup and Cristobal Rojas},
journal= {arXiv preprint arXiv:0801.0209},
year = {2008}
}