Topological and metric emergence of continuous maps

TL;DR

This paper investigates the concept of emergence in continuous maps, showing that in one-dimensional cases it is zero, while in higher dimensions it is maximal, and that metric emergence exhibits an intermediate value property.

Contribution

It establishes the dimension-dependent behavior of topological emergence and proves the intermediate value property for metric emergence in continuous maps.

Findings

Homeomorphisms on 1D manifolds have zero topological emergence.

In higher dimensions, generic conservative homeomorphisms have maximal topological emergence.

Metric emergence of continuous maps has the intermediate value property.

Abstract

We prove that the homeomorphisms of a compact manifold with dimension one have zero topological emergence, whereas in dimension greater than one the topological emergence of a C^0-generic conservative homeomorphism is maximal, equal to the dimension of the manifold. Moreover, we show that the metric emergence of continuous self-maps on compact metric spaces has the intermediate value property.