Algorithmically random points in measure preserving systems, statistical behaviour, 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 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.

[1]  Mathieu Hoyrup,et al.  Computability of probability measures and Martin-Löf randomness over metric spaces , 2007, Inf. Comput..

[2]  Klaus Weihrauch,et al.  Computability on Computable Metric Spaces , 1993, Theor. Comput. Sci..

[3]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[4]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[5]  Mathieu Hoyrup,et al.  An effective Borel-Cantelli Lemma. Constructing orbits with required statistical properties , 2007, ArXiv.

[6]  A. Brudno.  Entropy and the complexity of the trajectories of a dynamical system , 1978 .

[7]  S. Galatolo.  Orbit complexity by computable structures , 2000 .

[8]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[9]  Mathieu Hoyrup,et al.  A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties , 2009, Theor. Comput. Sci..

[10]  Péter Gács,et al.  Uniform test of algorithmic randomness over a general space , 2003, Theor. Comput. Sci..

[11]  C. Caramanis.  What is ergodic theory , 1963 .

[12]  Lai-Sang Young,et al.  What Are SRB Measures, and Which Dynamical Systems Have Them? , 2002 .

[13]  V. V'yugin,et al.  Effective Convergence in Probability and an Ergodic Theorem forIndividual Random Sequences , 1998 .

[14]  Владимир Вячеславович Вьюгин,et al.  Эффективная сходимость по вероятности и эргодическая теорема для индивидуальных случайных последовательностей@@@Effective convergence in probability and an ergodic theorem for individual random sequences , 1997 .