Bounded Analytic Hyper operators

[MphLee]

We do not have to create a normed space that the super function operator acts on, but we do have to talk about values converging to a fixed point under the super function operator. This will be equivalent to the values "contracting" under the point wise norm. Namely the family of functions we use are those such that \( \uparrow^{n} f \to Constant \) as \( n\to\infty \), with some additional conditions attached to \( f \). The \( \uparrow \) notation is the notation for the super function operator that I am using. This also implies under the pointwise norm it is a type of contraction mapping.

Yeah! That sounds awesome! Reading about dynamics I was just guessing about the basin of attraction on the space of functions while studyng the dynamics of the superfunction/subfunction operators.
For example studing the dynamics of the antirecursion(subfunction) we see that the set \( {\rm FR}:=\{f:{\rm rank}(f)<\infty\} \) of finite-rank functions is a proper subset of the set of functions that tends to the successor (wich is a fixed point).
\( {\rm FR}\subseteq \{\phi:\lim_{n\to \infty}{\mathcal S}^{-n}[\phi]=S\} \)
Imho there is alot of interesting stuff about the classifications of functions... I know you are involved in an analytic theory of "iterated iteration" but I'm still working on some synthetic framework (to be precise to an abstract-"algebraic" translation of the problems) that maybe could give us some existence/non-existence proofs inside some simpler models (some well behaved toy models) and then go for the generalization ... maybe not very soon but I hope for it.


BTW I'm sure that you have noticed that the sequence of hyperexponentiations seems to converge to the successor function in the interval [0,1] as the rank grows ... and probably its on "omegation" too (I rememeber Romerio/Rubtsov and also an old post of you here on the TF )

. This is going to be more complicated than my last two papers, and at a few parts I feel like I'm out of my depth, but I'll manage. It will probably take a month or two to get this out well enough that I'm willing to post it.

Ok, good luck!