Canonical order spectra in topological dynamical systems

TL;DR

This paper introduces the emergent order spectrum as a new invariant in topological dynamical systems, capturing recurrence phenomena beyond traditional methods, and proves its invariance and independence from metrics.

Contribution

It defines the emergent order spectrum for pairs in topological dynamical systems and demonstrates its invariance and discriminative power over existing recurrence invariants.

Findings

Order spectrum is independent of metric and vanishing sequence.

Order spectrum discriminates recurrence phenomena beyond Conley's decomposition.

Nested acyclic chains can be constructed for chain-related points.

Abstract

In a compact topological dynamical system $(X,f)$, we associate to every pair $(x,y)$ a canonical order-theoretic invariant, its emergent order spectrum $\Omega(x,y)$. We first prove that, if $x$ and $y$ are chain-related, one can always build families of nested and acyclic $\varepsilon_n$-chains ($\varepsilon_n \to 0$). The order spectrum $\Omega(x,y)$ is then defined as the set of countable linear order-types obtained as direct limits of (order-compatible) nested and acyclic $\varepsilon_n$-chains. The order spectrum is independent of the compatible metric and of the vanishing sequence, and invariant under topological conjugacy. Moreover, it discriminates recurrence phenomena that are indiscernible via Conley's decomposition or Auslander's prolongational hierarchy.