Emergent order spectrum for transitive homeomorphisms
Filippo Ciavattini, Marco Farotti, Camilla Lucamarini
TL;DR
The paper studies the Emergent Order Spectrum $Ω_f(x,y)$ for transitive homeomorphisms on compact spaces and proves a universality phenomenon: the global spectrum $Ω_f(X^2)$ contains every countable scattered order-type as well as the rationals’ order-type $\eta$; for a comeagre set of pairs $(x,y)$, the individual spectrum $Ω_f(x,y)$ already realizes all countably infinite scattered orders. The results rely on a constructive transfinite framework using $VD$-rank to organize orbit classes into $α$-structures and on the theory of Hausdorff for scattered orders, combining these with order-compatible nested $\varepsilon_n$-chains to realize sums indexed by $K$, $\omega$, $\omega^*$, or $\zeta$. The findings show how chain recurrence and topological transitivity encode the entire Hausdorff hierarchy of countable scattered orders within a single conjugacy invariant, advancing the understanding of dynamical invariants for transitive systems.
Abstract
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 $f$ on a compact metric space $X$ with $\lvert X\rvert=\mathfrak{c}$, 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 appear 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 $(x,y)\in X^2$.