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arXiv 2108.13019math.DS
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纤维熵与随机轨道的算法复杂性

Fiber entropy and algorithmic complexity of random orbits

Elias Zimmermann

发表信息：
Discrete Contin. Dyn. Syst., 2022, 42 (11): 5289-5308
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摘要：

设$Θ$为有限字母表。我们考虑一束保测变换$(T_θ)_{θ\in Θ}$作用在概率空间$(X,μ)$上，根据遍历随机过程$(Ξ,ν,σ)$随机选择，状态空间为$Θ$。这描述了随机动力系统（RDS）的典型情况。考虑一个有限的分区$\mathcal{P}$的$X$，我们表明，条件算法复杂性的随机轨道$x, T_{α_{0}}(x),T_{α_{1}}\circ T_{α_{0}}(x),...$在$X$沿序列$α= α_{0}α_{1}α_{2}...$在$Ξ$等于几乎肯定的纤维熵的RDS相对于$\mathcal{P}$，只要后者是遍历的。这推广了A. A. Brudno连接算法复杂性和确定性动力系统中的熵。

英文摘要：

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 respect to $\mathcal{P}$, whenever the latter is ergodic. This extends a classical result of A. A. Brudno connecting algorithmic complexity and entropy in deterministic dynamical systems.

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