# On Emergence and Complexity of Ergodic Decompositions

## TL;DR

This paper develops the concept of emergence in dynamical systems, quantifying the complexity of their statistical behaviors through topological and metric emergence, and provides examples of systems with maximal emergence properties.

## Contribution

It introduces the notions of topological and metric emergence, relates them to classical entropy concepts, and constructs examples of systems with maximal emergence, including generic smooth diffeomorphisms.

## Key findings

- Topological emergence measures the size of the set of ergodic measures.
- Metric emergence quantifies non-ergodicity of a reference measure.
- Examples show existence and genericity of systems with maximal emergence.

## Abstract

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms.