Emergent transfinite topological dynamics

TL;DR

This paper extends topological dynamics to transfinite iterations, revealing a hierarchical structure of dynamical phenomena at different ordinal levels and introducing new invariants for classification.

Contribution

It introduces a canonical framework for transfinite topological dynamics, including new invariants and a refined notion of conjugacy, expanding classical results to a richer ordinal-based landscape.

Findings

Finitely convergent sequences induce a poset structure with countable ordinal levels.

Transfinite conjugacy refines classical conjugacy, detecting phenomena at each ordinal level.

Standard topological dynamics results are recovered at the first ordinal level, with richer structures at higher levels.

Abstract

We present a canonical extension of topological dynamics to transfinite iterations, which makes precise the idea of dynamical phenomena stabilizing at different time-scales. Specifically, consider a sequence of self-maps $F=\{f_n\}$ of a compact metric space $X$. If $F$ is finitely convergent, i.e. $f_n(x)=f(x)$ for $n>N(x)$, the $f_n$-orbits exhibit an emergent poset structure. A maximal initial segment of this poset is isomorphic to a countable ordinal $\ge\omega$. The construction is canonical: every finitely convergent sequence induces, at each point, a unique maximal transfinite orbit that is independent of any finite initial segment of the sequence and invariant under step-by-step conjugacy at each $n$. For $\lambda$ a countable limit ordinal, we study orbits, recurrence, limit sets and attractors at level $\lambda$, and the interplay of different ordinal levels. Moreover, we introduce the natural notion of transfinite conjugacy, that sharply refines conjugacy of limit maps alone but is strictly weaker than step-by-step conjugacy. We describe a family of new invariants of transfinite conjugacy that detect recurrence and attraction phenomena at each ordinal level. Particularizing to $\lambda=\omega$ recovers (and in some cases strengthens) classical results of topological dynamics, revealing that the standard theory is the first level of a richer structural landscape.