Emergent order spectrum for transitive homeomorphisms

The Emergent Order Spectrum $Ω(x,y)$ is a topological invariant of dynamical systems providing order-types induced by the limit order of order-compatible nested $\varepsilon_n$-chains (with $\varepsilon_n\to 0$) from $x$ to $y$. In this paper, we investigate how rich these spectra can be under natural dynamical hypotheses. For a transitive homeomorphism $f$ of a compact metric space $X$ without isolated points and of cardinality $\mathfrak{c}$, we show that the global spectrum $Ω_f(X^2)$ is universal at the countable scattered level: every countable scattered order-type together with the order-type of the rationals appears in $Ω_f(X^2)$. More precisely, there exists a comeagre subset $M\subseteq X^2$ such that, for every $(x,y)\in M$, the individual spectrum $Ω_f(x,y)$ already realizes all countably infinite scattered order-types; moreover, the order-type of the rationals belongs to $Ω_f(x,y)$ for every pair $(x,y)\in X^2$.

💡 Research Summary

The paper introduces the Emergent Order Spectrum (EOS), denoted Ω(x, y), as a new topological invariant for dynamical systems. Ω(x, y) captures the order‑type of points that appear in a nested sequence of ε‑chains from x to y, where the ε‑values tend to zero and the chains are required to be acyclic and order‑compatible (the relative order of points in one chain is preserved in the next). The authors first recall the classical notions of ε‑chains, the chain relation C, and the refined relations C⊆ and C⪯, showing that for compact systems these three relations coincide (Theorem 1.8). This equivalence allows them to work with nested, order‑compatible families of chains without worrying about the particular choice of the vanishing sequence {εₙ}.

The main focus is on transitive homeomorphisms f on a compact metric space X that has no isolated points and cardinality continuum (𝔠). Under these hypotheses, the set I of points whose forward and backward orbits are both dense is comeagre. Lemma 2.7 proves that for any two points x, y ∈ I with disjoint orbits, the basic scattered order‑types ω + k, h + ω* and h + ζ + k (for arbitrary finite k, h) belong to Ω(x, y). The construction uses carefully chosen ε‑chains that follow a long forward segment of the orbit of x, then a short backward segment of the orbit of y, ensuring the chains are nested and acyclic. This provides the building blocks needed for any countable scattered order.

To reach the full universality claim, the authors invoke Hausdorff’s classification of scattered linear orders. Every countable scattered order can be built from singletons using concatenations indexed by the finite order K, the natural order ω, or its reverse ω*. The authors formalize this combinatorial hierarchy through the notion of an α‑structure (Definition 2.3), which is a recursively defined family of countably infinite subsets of a large set M, organized to depth α (α < ω₁). Lemma 2.6 shows that for any set M of size at least ℵ₁ one can construct a family {A_α | α < ω₁} of pairwise disjoint α‑structures.

The core result, Theorem 2.8, combines these ingredients. First, a comeagre set M ⊂ X² is selected so that for each (x, y) ∈ M the forward and backward orbits are dense and disjoint. Then, for each countable ordinal α, the authors assign to the α‑structure A_α a collection of orbit pieces whose EOS will realize all scattered orders of V D‑rank ≤ α. Using Lemma 2.7 they embed the elementary blocks (ω + k, h + ω*, h + ζ + k) into the chains associated with each orbit piece. By concatenating these blocks according to the index orders K, ω, ω* or ζ, they obtain nested, order‑compatible ε‑chain families whose limit order reproduces any desired scattered order of rank ≤ α. An induction on α then yields that for every countable scattered order L there exists a nested chain family from x to y whose limit order is order‑isomorphic to L. Since η (the dense order type of the rationals) is already present for any transitive system (Theorem 1.17), the global spectrum Ω_f(X²) contains every countable scattered order together with η.

Consequently, the paper demonstrates that the combination of chain recurrence and topological transitivity is sufficient to encode the entire Hausdorff hierarchy of countable scattered linear orders into a single conjugacy invariant. Moreover, the proof is constructive: it not only shows the existence of the required order‑compatible nested ε‑chains but actually builds them, thereby providing an independent existence theorem for the EOS in the transitive setting. This work opens the door to using EOS as a powerful tool for classifying and distinguishing topological dynamical systems beyond traditional invariants.