Emergent transfinite topological dynamics

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 $\geω$. 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 $λ$ a countable limit ordinal, we study orbits, recurrence, limit sets and attractors at level $λ$, 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 $λ=ω$ 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.