Topological and metric emergence of continuous maps

TL;DR

This work characterizes how the space of invariant measures generated by a dynamical system reflects dimensional and metric structure, through topological and metric emergence. It shows zero topological emergence for order-preserving homeomorphisms on 1D manifolds, while $C^0$-generic conservative maps on higher-dimensional manifolds exhibit maximal topological emergence equal to $\dim X$, with a parallel result in the non-conservative setting. The authors develop pseudo-horseshoe techniques and leverage specification to produce large, well-separated families of ergodic measures, proving both upper and lower bounds that match. They also establish an intermediate-value property for metric emergence and show that, for any $f$, the metric-emergence map over invariant measures attains every value in $[0,\mathcal{E}_{\mathrm{top}}(f)]$, connecting ergodic decompositions to continuum-like emergence behavior. Together, these findings link geometric dimensionality, generic dynamical behavior, and statistical complexity in a robust, measurable framework.

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.

Theorems & Definitions (23)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Remark 2.1
• Definition 2.2