A constructive version of Birkhoff's ergodic theorem for Martin-Löf random points

A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties

In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets $A_{i}$ with effectively summable measures, there are computable points which are not contained in infinitely many $A_{i}$. As a consequence of this we obtain the existence of computable points which follow the \emph{typical statistical behavior} of a dynamical system (they satisfy the Birkhoff theorem) for a large class of systems, having computable invariant measure and a certain ``logarithmic'' speed of convergen…

遍历定理与算法随机性

Ergodic theorem and algorithmic randomness

本文证明了Birkhoff遍历定理的构造性版本，基于一个明确的组合学陈述，并提出了相关问题。

On effective Birkhoff's ergodic theorem for computable actions of amenable groups

We introduce computable actions of computable groups and prove the following versions of effective Birkhoff's ergodic theorem. Let $Γ$ be a computable amenable group, then there always exists a canonically computable tempered two-sided Følner sequence $(F_n)_{n \geq 1}$ in $Γ$. For a computable, measure-preserving, ergodic action of $Γ$ on a Cantor space $\{0,1\}^{\mathbb N}$ endowed with a computable probability measure $μ$, it is shown that for every bounded lower semicomputable function $f$ on $\{0,1\}^{\mathbb N}$ and for every Martin-Löf random $ω\in \{0,1\}^{\mathbb N}$ the equality \[ \…

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

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-S…

Randomness and Non-ergodic Systems

We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transforma…

A central limit theorem for time-dependent dynamical systems

The work [8] established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random, i. e. specific examples are also found, which are not small perturbations of a given…