# Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also equipped with the structure of a Hausdorff topological space. Then $f^\omega : X \rightarrow X$ also kind of makes sense; basically, for any $x \in X$, we say that $f^\omega(x)$ is well-defined iff the sequence $i < \omega \mapsto f^{i}(x)$ converges, in which case $f^\omega(x)$ is taken to equal $\mathrm{lim}_{i<\omega}f^{i}(x).$ Proceeding in this way, we should be able to raise $f$ to the power of an arbitrary ordinal. Along a similar vein, suppose that $\alpha$ is an ordinal and that $f : \alpha \rightarrow \mathrm{Par}(X,X)$ is a sequence of partial function $X \rightarrow X$. Then we can define $\bigcirc_{i<\alpha}f_i$ in the obvious way, by taking compositions at successor ordinals and pointwise limits at limit ordinals.

The canonical examples are as follows. Consider the endofunction on the cardinal numbers $S$ given by $\kappa \mapsto \kappa^+$. Then we can define the aleph numbers by writing $\aleph_\alpha = S^\alpha(\aleph_0).$ Similarly, if we let $P$ denote the continuum endofunction on the cardinal numbers, namely $\kappa \mapsto 2^\kappa$, then we can define the beth numbers by writing $\beth_\alpha = P^\alpha(\aleph_0).$

Question. Do any sources consider these notions in a systematic way? If so, where can I learn more?