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arXiv · 2511.19777

Order spectra in topological dynamical systems

F. Ciavattini·A. Della Corte·C. Lucamarini
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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 spectrum is independent of the vanishing sequence and invariant under topological conjugacy, and it refines Conley's partial order.

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F. Ciavattini, A. Della Corte, C. Lucamarini. 2025-11-24. Order spectra in topological dynamical systems. https://arxiv.org/abs/2511.19777

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