Complexities of differentiable dynamical systems

TL;DR

This paper explores the complexity of differentiable dynamical systems by examining properties like periodic point growth, entropy, and emergence, and establishes connections between entropy and emergence theories.

Contribution

It introduces the concept of localizable properties and relates key complexity notions, providing a unified perspective on entropy and emergence in dynamical systems.

Findings

Fast growth of periodic points is typical in differentiable systems

Positive entropy correlates with complex dynamical behavior

High emergence indicates rich and intricate system structures

Abstract

We define the notion of localizable property for a dynamical system. Then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems. These notions are the fast growth of the number of periodic points, the positive entropy and the high emergence. We finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence.