Canonical order spectra in topological dynamical systems

In a compact topological dynamical system $(X,f)$, we associate to every pair $(x,y)$ a canonical order-theoretic invariant, its emergent order spectrum $\mathrm{\Omega }(x,y)$. We first prove that, if $x$ and $y$ are chain-related, one can always build families of nested and acyclic ${\epsilon }_n$-chains (${\epsilon }_n\to 0$). The order spectrum $\mathrm{\Omega }(x,y)$ is then defined as the set of countable linear order-types obtained as direct limits of (order-compatible) nested and acyclic ${\epsilon }_n$-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.