In this paper, we present some results on information, complexity and entropy as defined below and we discuss their relations with the Kolmogorov-Sinai entropy which is the most important invariant of a dynamical system. These results have the following features and motivations:
-we give a new computable definition of information and complexity which allows to give a computable characterization of the K-S entropy;
-these definitions make sense even for a single orbit and can be measured by suitable data compression algorithms; hence they can be used in simulations and in the analysis of experimental data;
-the asymptotic behavior of these quantities allows to compute not only the Kolmogorov-Sinai entropy but also other quantities which give a measure of the chaotic behavior of a dynamical system even in the case of null entropy.
The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. The goal of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a complexity function bounded by a given function $f$ with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the {\it word entropy} $E_W(f)$ associated to a given f…
Let $Θ$ be a finite alphabet. We consider a bundle of measure preserving transformations $(T_θ)_{θ\in Θ}$ acting on a probability space $(X,μ)$, which are chosen randomly according to an ergodic stochastic process $(Ξ,ν,σ)$ with state space $Θ$. This describes a paradigmatic case of a random dynamical system (RDS). Considering a finite partition $\mathcal{P}$ of $X$ we show that the conditional algorithmic complexity of a random orbit $x, T_{α_{0}}(x),T_{α_{1}}\circ T_{α_{0}}(x),...$ in $X$ along a sequence $α= α_{0}α_{1}α_{2}...$ in $Ξ$ equals almost surely the fiber entropy of the RDS with r…
A subshift with linear block complexity has at most countably many ergodic measures, and we continue of the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity is arbitrarily close to linear but has uncountably many ergodic measures, we show that this behavior fails as soon as the block complexity is superlinear. With a different construction, we show that there exists a minimal subshift with an ergodic measure whose slow entropy grows slower than any given rate tending to infinitely but faster than any other rate m…
Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.
It was proved by Brudno that entropy and Kolmogorov complexity for dynamical systems are tightly related. We generalize his results to the case of arbitrary computable amenable group actions. Namely, for an ergodic shift-action, the asymptotic Kolmogorov complexity of a typical point is equal to the Kolmogorov-Sinai entropy of the action. For topological shift actions, the asymptotic Komogorov complexity of every point is bounded from above by the topological entropy, and there is a point attaining this bound.
Categories of measurement functors. Entropy of discrete amenable group representations on abstract categories. Entropy as a bifunctor into $[0,\infty]$
The equivalence of precompactness, zero maximal pattern entropy and bounded mean complexity for finite partitions
Institute for Mathematical Sciences and Artificial Intelligence & Department of Mathematics, Shantou University（汕头大学数学与人工智能研究所及数学系）；Department of Mathematics, Shantou University（汕头大学数学系）
For any non-erasing free monoid morphism $σ: \cal A^* \to \cal B^*$, and for any subshift $X \subset \cal A^\Z$ and its image subshift $Y = σ(X) \subset \cal B^\Z$, the associated complexity functions $p_X$ and $p_Y$ are shown to satisfy: there exist constants $c, d, C > 0$ such that $$c \cdot p_X(d \cdot n) \,\, \leq \,\, p_Y(n) \,\, \leq \,\, C \cdot p_X(n)$$ holds for all sufficiently large integers $n \in \N$, provided that $σ$ is recognizable in $X$. If $σ$ is in addition letter-to-letter, then $p_Y$ belongs to $Θ(p_X)$ (and conversely). Otherwise, however, there are examples where $p_X$ …
