Let $S_n$ be an ordered set of numbers indexed by a countable ordinal $n\in\omega^\omega$ such as:
$\ldots 7,49,343,\ldots5,25,125,\ldots,3,9,27,\ldots,2,4,8,\ldots$
Then let this be a topological space such that every subsequence of the form $p_n,p_n^2,p_n^3,\ldots$ converges to $p_{n-1}$ or to $1$ in the case of $2^\infty$
Let some function $f:\Bbb N\to\Bbb N$ send each number to its successor: $f(s_n)=s_{n+1}$ e.g. $2\mapsto4$.
What topological or sequential term describes the property of this function $f$ that it is directed towards $2^\omega$? Is this what transfinite induction refers to?
I can't say $f$ converges to $\langle2\rangle$ but how do I say it converges to $\langle2\rangle$ on transfinite induction (or whatever the correct term is)? I'm aware I can simply say something like the orbit of $f$ is directed. Maybe that's the only answer.
I'm aware I can say let $\overline f=\lim_{m\to\infty}f^m(x)$ then $\lim_{n\to\infty} \overline f^{\text { }n}(x)=1$
In its most general form: What is the term for a function which converges not only when composed infinitely many times, but when one takes it to a limit and then composes infinitely many times again, possibly through infinitely many limit points, it eventually stabilises.
A useful phrase to google is transfinite superpositions, although I think at the present time “transfinite composition” would be a better phrase to use if you intend to write about this.
Nina Karlovna Bary [Bari] (1901-1961) was probably the first person to introduce the idea of transfinite iterations (compositions) of a function, this being in the following two papers.
[1] Mémoire sur la représentation finie des fonctions continues. Première Partie: Les superpositions de fonctions absolument continues, Mathematische Annalen 103 (1930), 185-248.
[2] Mémoire sur la représentation finie des fonctions continues. Deuxième Partie: Le théorème fondamental sur la représentation finie, Mathematische Annalen 103 (1930), 598-653.
Besides the following 2 papers and 1 conference talk by John Todd (1911-2007), I don’t know to what extent Bary or others followed up with this work:
[3] Superpositions of functions (I): Transfinite superpositions of absolutely continuous functions, Journal of the London Mathematical Society (1) 10 #3 (July 1935), 166-171.
[4] Superpositions of functions (II): Transfinite superpositions of absolutely continuous functions, Proceedings of the London Mathematical Society (2) 41 (1936), 433-439.
[5] Transfinite superpositions of absolutely continuous functions, pp. 110-111 in Comptes Rendus du Congrès International des Mathématiciens (Oslo, 1936), 1937, xvi + 289 pages.
Regarding transfinite sequences of functions in general, there have probably been over 100 papers dealing with this topic, mostly published in the last 4 or 5 decades. See this google search and this google scholar search.