Canonical order spectra in topological dynamical systems
F. Ciavattini, A. Della Corte, C. Lucamarini
TL;DR
The paper introduces the Emergent Order Spectrum (EOS), an order-theoretic invariant that captures fine-grained recurrence between chain-related points in compact topological dynamical systems. By constructing nested, order-compatible sequences of $\varepsilon_n$-chains, EOS assigns to each pair $(x,y)$ a set of countable linear order-types $\Omega_f(x,y)$ that survive under taking direct limits of the chain orders. It proves that, in the compact case, $\mathcal{C}=\mathcal{C}_{\subseteq}=\mathcal{C}_{\preceq}$, using Hausdorff-limit projections and a transfinite pruning procedure to ensure acyclicity, and shows EOS is invariant under topological conjugacy and independent of the metric and vanishing sequence. The EOS refines Conley’s decomposition and Auslander's prolongational hierarchy, encoding a richer recurrence structure that can distinguish metastable patterns and perturbation-induced dynamics not detected by prior invariants. In transitive systems, EOS can realize a wide array of order-types, including all countable scattered and dense orders, highlighting its potential to compare dynamical models at high resolution.
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 $Ω(x,y)$. We first prove that one can always build families of nested $\varepsilon_n$-chains ($\varepsilon_n \to 0$) whose linear orders are eventually compatible under inclusion. $Ω(x,y)$ is then defined as the set of countable linear order-types obtained as direct limits of order-compatible nested $\varepsilon_n$-chains (empty if and only if $x$ is not in chain relation with $y$). The order spectrum is independent of the compatible metric and of the vanishing sequence, and invariant under topological conjugacy. Moreover, it reveals tight connections with the underlying dynamics and discriminates recurrence phenomena that are indiscernible via Conley's decomposition or Auslander's prolongational hierarchy.