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?

  • $\begingroup$ The (set-theoretic) recursion theorem is an obvious place to start. It is discussed in Kunen and Jech's books, for instance. But perhaps you are interested in statements in analysis or other contexts. There is this that may not be ideal, as the outcome "trivializes" as stage $\omega$. There are many natural "derivative" operations on sets, in the context of set-theoretic topology and classical descriptive set theory. Kechris's book is a good starting point. The most famous example here is the Cantor-Bendixson derivative and its variants. $\endgroup$ – Andrés E. Caicedo Jun 2 '14 at 5:43
  • $\begingroup$ Convergence of the iterates of f may fail to have a meaningful limit whether the space is Hausdorff or not. There are many kinds of spaces where the only convergent sequences are eventually constant, for example the Cech-Stone compactification of N. $\endgroup$ – DanielWainfleet Aug 13 '15 at 0:16
  • $\begingroup$ Related: math.stackexchange.com/questions/905712 $\endgroup$ – Watson Sep 16 '16 at 9:14

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