Emergent order spectrum for transitive homeomorphisms

The Emergent Order Spectrum $\Omega(x,y)$ 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 $\Omega_f(X^2)$ is universal at the countable scattered level: every countable scattered order-type together with the order-type of the rationals appears in $\Omega_f(X^2)$. More precisely, there exists a comeagre subset $M\subseteq X^2$ such that, for every $(x,y)\in M$, the individual spectrum $\Omega_f(x,y)$ already realizes all countably infinite scattered order-types; moreover, the order-type of the rationals belongs to $\Omega_f(x,y)$ for every pair $(x,y)\in X^2$.