# Emergence via non-existence of averages

## TL;DR

This paper introduces pointwise emergence as a new way to quantify the non-existence of averages in dynamical systems, showing that many systems exhibit super-polynomial emergence on large sets.

## Contribution

It defines pointwise emergence and demonstrates its prevalence in systems with the specification property and within Newhouse open sets.

## Key findings

- High pointwise emergence occurs on residual sets in systems with the specification property.
- Super-polynomial emergence is dense in Newhouse open sets on positive measure subsets.
- The concept offers a new quantitative perspective on non-ergodic behavior.

## Abstract

Inspired by a recent work by Berger, we introduce the concept of pointwise emergence. This concept provides with a new quantitative perspective into the study of non-existence of averages for dynamical systems. We show that high pointwise emergence on a large set appears for abundant dynamical systems: Any continuous maps on a compact metric space with the specification property have super-polynomial pointwise emergence on a residual subset of the state space. Furthermore, there is a dense subset of any Newhouse open set each element of which has super-polynomial pointwise emergence on a positive Lebesgue measure subset of the state space.