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

TL;DR

This paper investigates the behavior of Martin-Löf random points in computable dynamical systems, demonstrating they exhibit typical statistical properties, recurrence, and their orbit complexity aligns with system entropy measures.

Contribution

It introduces an effective symbolic model for computable dynamical systems and establishes the equivalence of orbit complexity of random points with system entropy.

Findings

Random points have typical statistical behavior as per Birkhoff's theorem.

Orbit complexity of random points equals the system's Kolmogorov-Sina"i entropy.

Supremum of orbit complexity matches the topological entropy.

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.