Hi James

I guess that I tried to bring something similar to the forum's attention long ago.

Try to look here

http://math.eretrandre.org/tetrationforu...hp?tid=945
In my humble opinion this seems like something really near to what you are looking for. I red often of "extension" of recursion theory(aka computability) to real valued functions and, it seems to me, it was always near topics like measure spaces but or things like descriptive set theory (that also deals with computabiliy of subset of real numbers and of their cartesian product - i. e. functions, functionals, relations and so on). Serch about Campagnolo and Moore stuff.

Also this reminds that also bo198214 seemed to pursuit something related to this, at least judging from some MO questions

note: (is interesting that the majorisation order is proven to be a well order on two subclasses defined by Skolem of E_3 (Kalmar's elementary functions ) and E_4, see here http://link.springer.com/chapter/10.1007...4_2#page-1 )
http://mathoverflow.net/questions/60264/...-functions
http://mathoverflow.net/questions/60093/...-functions
Yes, those are only hints, I tried this path some years ago... but my ignorance of basic analytical method and lack of enough time scared me away from this.

This fact is really interesting since in my opinion the real core of all this matter (ranks and friends) lies in the group or monoid structure of the collection of functions investigated and I have strong evidences of this... and to be more precise in the conjugation relation (a relaxed version in the monoid but this still makes sense). Conjugation is strictly connected to...guess what? Hom-functors of special dynamical systems-like-categories (and home functors are something that remind alot exponentiation)...

But aside form these speculations there is a subtle connection going on here between ranks, computation complexity and dimension.(*)

Back to the wild speculations... seems like the dimensions, degrees, continuity classes and alike are to the concept of rank AS their algebraic, graded, topological, differential kind of structured collection of functions are to an unknown recursion-related algebraic structure.

Also something really obvious is that the structure of ranks (linked with orders of infinity and) seems too complex to be "parametrized" by the field of real numbers or complex numbers... maybe it is the case that the structure of the ranks (the quotient of the set of functions by the same-rank-equivalence relation) forms an important new algebraic structure...

Try also these (I did not read these actually, maybe partially bot not totally understood but I hope something can be useful for you):

http://www.sciencedirect.com/science/art...4X04000408
http://math.isa.utl.pt/~mlc/phdthesis.pdf
https://www.reading.ac.uk/web/files/math..._paper.pdf
(*)

http://www.personal.psu.edu/t20/papers/sdedc.pdf (Symbolic dynamics: entropy = dimension = complexity)

I apologize in advance for the mistakes, If I, or you, will spot some errors I'll edit ASAP.