Fiber entropy and algorithmic complexity of random orbits

TL;DR

This paper establishes that in a random dynamical system, the conditional algorithmic complexity of a typical orbit almost surely equals the fiber entropy, extending classical deterministic results to stochastic settings.

Contribution

It generalizes Brudno's theorem by linking algorithmic complexity and fiber entropy in ergodic random dynamical systems.

Findings

Conditional algorithmic complexity equals fiber entropy almost surely.

Extends classical deterministic complexity-entropy relationship to stochastic systems.

Provides a rigorous connection between complexity and entropy in RDS.

Abstract

Let $\Theta$ be a finite alphabet. We consider a bundle of measure preserving transformations $(T_{\theta})_{\theta \in \Theta}$ acting on a probability space $(X,\mu)$, which are chosen randomly according to an ergodic stochastic process $(\Xi,\nu,\sigma)$ with state space $\Theta$. 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 $x, T_{\alpha_{0}}(x),T_{\alpha_{1}}\circ T_{\alpha_{0}}(x),...$ in $X$ along a sequence $\alpha = \alpha_{0}\alpha_{1}\alpha_{2}...$ in $\Xi$ 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.