Dynamical intricacy and average sample complexity

TL;DR

This paper introduces new measures called intricacy and average sample complexity to quantify the balance between freedom and coherence in dynamical systems, extending entropy concepts with applications to shifts of finite type.

Contribution

It defines and analyzes intricacy and average sample complexity, establishing their properties and connections to entropy in dynamical systems, with computations for specific cases.

Findings

Suprema of intricacy and average sample complexity equal entropy.

Computed these measures for shifts of finite type.

Outlined directions for future research.

Abstract

We propose a new way to measure the balance between freedom and coherence in a dynamical system and a new measure of its internal variability. Based on the concept of entropy and ideas from neuroscience and information theory, we define \emph{intricacy} and \emph{average sample complexity} for topological and measure-preserving dynamical systems. We establish basic properties of these quantities, show that their suprema over covers or partitions equal the ordinary entropies, compute them for many shifts of finite type, and indicate natural directions for further research.