Dynamical intricacy and average sample complexity

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 intricacy and 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.

KEYWORDS:

• Entropy
• complexity
• intricacy
• ergodic theory
• topological dynamics

2010 Mathematics Subject Classification::

• Primary 37B40
• 37A35
• 54H20
• 28D20

Acknowledgments

This paper is based on the UNC-Chapel Hill Ph.D. dissertation of the second author [26], written under the direction of the first author. We thank Mike Boyle, Jérôme Buzzi, Tomasz Downarowicz, Kevin McGoff, Jean-Paul Thouvenot, and Lorenzo Zambotti for illuminating conversations and the referee for helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.