1. By closure in the monoid, I mean specifically closure under Cauchy sequences using uniform convergence on all compact subsets. By this I mean,
\(
f \in \overline{\mathcal{B}}\,\,\text{implies}\,\,\exists f_n \in \mathcal{B}\,\,\text{s.t.} \forall [a,b] \subset \mathbb{R}\,\,||f(x) - f_n(x)||_{x \in [a,b]} \to 0\,\,\text{as}\,\,n\to\infty\\
\)
This should still be a monoid; and will look close enough to \( \mathcal{B} \) for our purposes.
2. I'm not too sure how convexity enters the discussion. I've never used convexity really. The intuitive manner I view this is as an attracting fixed point at infinity. Where \( f'(x) \to \lambda < \infty \) is geometrically attracting, and \( f'(x) \to \infty \) as a super-attracting fixed point.
3. Yes, it is to avoid fixed points, keep the orbits simple, and keep things well adjusted. Which will in turn ensure that our superfunction is surjective. Which makes talking about the conjugate property much simpler.
A1. Yes, that's my thinking exactly.
A2. Yes, I did this construction in hopes this would be a diffeomorphic group (whatever the hell you call it), a subgroup of \( \text{Diff}(\mathbb{R}) \). I avoided this language largely because I'm not familiar enough with the language.
I have attached a revamped version of this PDF. I realized that the exponential convergents will only work when \( f(x) \ge e^{\mu x} \) for some \( \mu > 0 \) and everywhere else become sort of overkill and don't help us. So instead I decided to use convergents \( p(x-j) \) where \( p(x) \sim \frac{1}{\sqrt{f(x)/x}} \) as \( x \to \infty \). This requires us taking our auxiliary function \( \Phi(x) \) to satisfy \( \Phi(x+1) \sim f(\Phi(x))/\sqrt{f(x)/x} \). This means that \( \Phi(x) \) will now grow slower than our superfunction rather than faster. When I was initially designing this I was thinking too much about \( \phi(s) \) which grows faster than tetration; and so I assumed we need our function \( \Phi(x) \) to grow faster than our superfunction. This proves to fail, especially when we take simple functions like \( f(x) \sim x^{1+\delta} \) for \( x \to \infty \); where the exponential convergents don't work, and any convergent which makes \( \Phi(x) \) grow faster than the super function also won't work.
Anyway, here it is revamped a good amount. I'm still fiddling with this, I want it to work more rigorously. I think this has potential for a very interesting paper, so this will probably be the last update until I'm fully satisfied. I'm mostly just posting this here to correct my assumption about exponential convergents being the give all end all. We unfortunately have to use more complicated convergents in the general case.
Regards, James
EDIT: I think I'm going to alter the manner of convergence from what I've posted here, as I think I can choose a more universal form of the convergents; which works for pretty much every function; rather than brute forcing it based on each \( f \). I'll see though. Give me a week or so and I'll try to have a comprehensive first draft.
\(
f \in \overline{\mathcal{B}}\,\,\text{implies}\,\,\exists f_n \in \mathcal{B}\,\,\text{s.t.} \forall [a,b] \subset \mathbb{R}\,\,||f(x) - f_n(x)||_{x \in [a,b]} \to 0\,\,\text{as}\,\,n\to\infty\\
\)
This should still be a monoid; and will look close enough to \( \mathcal{B} \) for our purposes.
2. I'm not too sure how convexity enters the discussion. I've never used convexity really. The intuitive manner I view this is as an attracting fixed point at infinity. Where \( f'(x) \to \lambda < \infty \) is geometrically attracting, and \( f'(x) \to \infty \) as a super-attracting fixed point.
3. Yes, it is to avoid fixed points, keep the orbits simple, and keep things well adjusted. Which will in turn ensure that our superfunction is surjective. Which makes talking about the conjugate property much simpler.
A1. Yes, that's my thinking exactly.
A2. Yes, I did this construction in hopes this would be a diffeomorphic group (whatever the hell you call it), a subgroup of \( \text{Diff}(\mathbb{R}) \). I avoided this language largely because I'm not familiar enough with the language.
I have attached a revamped version of this PDF. I realized that the exponential convergents will only work when \( f(x) \ge e^{\mu x} \) for some \( \mu > 0 \) and everywhere else become sort of overkill and don't help us. So instead I decided to use convergents \( p(x-j) \) where \( p(x) \sim \frac{1}{\sqrt{f(x)/x}} \) as \( x \to \infty \). This requires us taking our auxiliary function \( \Phi(x) \) to satisfy \( \Phi(x+1) \sim f(\Phi(x))/\sqrt{f(x)/x} \). This means that \( \Phi(x) \) will now grow slower than our superfunction rather than faster. When I was initially designing this I was thinking too much about \( \phi(s) \) which grows faster than tetration; and so I assumed we need our function \( \Phi(x) \) to grow faster than our superfunction. This proves to fail, especially when we take simple functions like \( f(x) \sim x^{1+\delta} \) for \( x \to \infty \); where the exponential convergents don't work, and any convergent which makes \( \Phi(x) \) grow faster than the super function also won't work.
Anyway, here it is revamped a good amount. I'm still fiddling with this, I want it to work more rigorously. I think this has potential for a very interesting paper, so this will probably be the last update until I'm fully satisfied. I'm mostly just posting this here to correct my assumption about exponential convergents being the give all end all. We unfortunately have to use more complicated convergents in the general case.
Regards, James
EDIT: I think I'm going to alter the manner of convergence from what I've posted here, as I think I can choose a more universal form of the convergents; which works for pretty much every function; rather than brute forcing it based on each \( f \). I'll see though. Give me a week or so and I'll try to have a comprehensive first draft.