Emergence in dynamical systems

Abstract

In [Be17], we introduced the notion of emergence to quantify the statistical complexity of a dynamical system. Roughly speaking, emergence measures the number of probability measures required to describe, up to a given precision $\epsilon$, the statistical behavior of most orbits. A system is said to have high emergence when this number grows super-polynomially as $\epsilon\to0$. We survey recent developments in a program aimed at understanding the prevalence of high emergence in differentiable dynamics. We review several notions of emergence and their connections with quantization, ergodic decompositions, and entropy, and discuss examples exhibiting high or maximal emergence in conservative, symplectic, analytic, and dissipative dynamics, as well as in constrained families such as unimodal, H\'enon, and rational maps. We also present new results concerning variants of metric and topological emergence, convexity properties, local emergence, and variational principles, together with a collection of open problems on the typicality of high emergence.