Order spectra in topological dynamical systems

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 ${\epsilon }_n$-chains (${\epsilon }_n\to 0$). 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 ${\epsilon }_n$-chains. The spectrum is independent of the vanishing sequence and invariant under topological conjugacy, and it refines Conley's partial order.