Transfinite Topological Dynamics

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 refines conjugacy of limit maps alone but is strictly weaker than step-by-step conjugacy.

We describe a family of invariants of transfinite conjugacy that detect recurrence and attraction phenomena at each ordinal level.

Particularizing to $\lambda=\omega$ recovers (and in some cases refines) classical results of topological dynamics.