Fiber entropy and algorithmic complexity of random orbits

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.