That was very interesting. I really like the rephrasing. Got a little lost towards the end with the categories; but other than that seems like a very good standardization of what we think of when we say recursive functions. It'd be interesting to see if you can derive the same analysis if \( f \) and \( g \) are defined with multiple variables. I.e; producing a function
\(
\chi(s,z): \mathbb{C} \times \mathbb{C} \to \mathbb{C}\\
f(s,\chi(s,z)) = \chi(s,g(s,z))\\
\)
...I had an incorrect idea here...
I've only ever worked with \( \chi(s+1,z) = f(s,\chi(s,z)) \) and I was quick to jump the gun
Now I'm actually scratching my head at how in the hell you could solve these equations...
Very interesting though. This reads as a good foundation to all the super-function tricks.
Regards, James
EDIT:
So the best I could think of is solving for the successor case and then through conjugation solving the general case (which is not ideal).
If,
\( \phi \bullet s = f \bullet \phi \)
Then, start with a function,
\(
A(s) = \Omega_{j=1}^\infty e^{s-j} f(z) \bullet z\\
A(s+1) = e^s f(A(s))\\
\)
It may be more helpful to use a different convergent factor other than \( e^{s-j} \); but let's just say it works. Then taking,
\(
F(s) = \lim_{n\to\infty} f^{-n} \bullet A(s+n)\\
\)
This could solve the successor problem. This is for instance, definitely doable if we stick to a real monotonic function, say taking \( \mathbb{R} \to \mathbb{R} \); with a sufficiently well behaved inverse (think like \( \log \)...).
Then, taking \( f,g \)--which are similar functions; then we can construct a \( G \) similarly. Then,
\(
F\bullet G^{-1} = \phi\\
\)
Should be a suitable function on \( \mathbb{R} \) satisfying the required,
\(
\phi g = f \phi\\
\)
...........
I'm actually kind of curious now if I can make this work more generally. Do you think this might give you concrete ground to stand on? If we can make your "convergence criteria" a little more absolute with an example of a space where this works? I.e: holding a space \( f,g \in \mathcal{B} \) such we can always find functions \( \phi \in \mathcal{B} \) such that \( f \phi = \phi g \). Maybe not so perfectly, but I imagine something like this. Makes for good normal subgroup; functor; etc nonsense I imagine. Lmao.
EDIT 2:
One example I'm running through my head, because it's simple, is,
\(
f(z) = 1+z^3\\
\)
which is \( \mathbb{R} \to \mathbb{R} \), it's well behaved, and its inverse is too. Then,
\(
A(s) = e^{s-1} (1+(e^{s-2}(1+...)^3)^3)\\
\)
This certainly converges geometrically. And the limit \( \phi = \lim_{n\to\infty} \sqrt[3]{\sqrt[3]{A(s+n)-1}...-1} \) probably converges (at least from what I'm running it through). Which would solve the equation,
\(
\phi(s+1) = 1+ \phi(s)^3
\)
If we can find a common family of functions like this. I'm sure the exponential convergents will suffice. I'm thinking surjective/injective \( \mathbb{R}\to\mathbb{R} \); at least continuously differentiable. The problem I see is that \( \phi \) will not be in the same family of functions (which is a bummer, but probably to be expected).
\(
\chi(s,z): \mathbb{C} \times \mathbb{C} \to \mathbb{C}\\
f(s,\chi(s,z)) = \chi(s,g(s,z))\\
\)
...I had an incorrect idea here...
I've only ever worked with \( \chi(s+1,z) = f(s,\chi(s,z)) \) and I was quick to jump the gun
Now I'm actually scratching my head at how in the hell you could solve these equations...
Very interesting though. This reads as a good foundation to all the super-function tricks.
Regards, James
EDIT:
So the best I could think of is solving for the successor case and then through conjugation solving the general case (which is not ideal).
If,
\( \phi \bullet s = f \bullet \phi \)
Then, start with a function,
\(
A(s) = \Omega_{j=1}^\infty e^{s-j} f(z) \bullet z\\
A(s+1) = e^s f(A(s))\\
\)
It may be more helpful to use a different convergent factor other than \( e^{s-j} \); but let's just say it works. Then taking,
\(
F(s) = \lim_{n\to\infty} f^{-n} \bullet A(s+n)\\
\)
This could solve the successor problem. This is for instance, definitely doable if we stick to a real monotonic function, say taking \( \mathbb{R} \to \mathbb{R} \); with a sufficiently well behaved inverse (think like \( \log \)...).
Then, taking \( f,g \)--which are similar functions; then we can construct a \( G \) similarly. Then,
\(
F\bullet G^{-1} = \phi\\
\)
Should be a suitable function on \( \mathbb{R} \) satisfying the required,
\(
\phi g = f \phi\\
\)
...........
I'm actually kind of curious now if I can make this work more generally. Do you think this might give you concrete ground to stand on? If we can make your "convergence criteria" a little more absolute with an example of a space where this works? I.e: holding a space \( f,g \in \mathcal{B} \) such we can always find functions \( \phi \in \mathcal{B} \) such that \( f \phi = \phi g \). Maybe not so perfectly, but I imagine something like this. Makes for good normal subgroup; functor; etc nonsense I imagine. Lmao.
EDIT 2:
One example I'm running through my head, because it's simple, is,
\(
f(z) = 1+z^3\\
\)
which is \( \mathbb{R} \to \mathbb{R} \), it's well behaved, and its inverse is too. Then,
\(
A(s) = e^{s-1} (1+(e^{s-2}(1+...)^3)^3)\\
\)
This certainly converges geometrically. And the limit \( \phi = \lim_{n\to\infty} \sqrt[3]{\sqrt[3]{A(s+n)-1}...-1} \) probably converges (at least from what I'm running it through). Which would solve the equation,
\(
\phi(s+1) = 1+ \phi(s)^3
\)
If we can find a common family of functions like this. I'm sure the exponential convergents will suffice. I'm thinking surjective/injective \( \mathbb{R}\to\mathbb{R} \); at least continuously differentiable. The problem I see is that \( \phi \) will not be in the same family of functions (which is a bummer, but probably to be expected).