Allow me to use \( \beta_\lambda(s) \).
Well, since,
\(
\exp(\beta_\lambda(s)) = \beta_\lambda(h(s))\\
\)
We're solving a conjugate equation. This is difficult to solve with infinite compositions. This is precisely the thing Mphlee and I are talking about. If I have \( \beta_\lambda \); we can get \( h \). But if we have \( h \); how do we get \( \beta_\lambda \)?
This is precisely the trouble we're having, solving conjugate equations arbitrarily. The closest I can think of is,
\(
F(s) = \lim_{n\to\infty} \log^{\circ n} h^{\circ n}(s)\\
\)
Which if it converges, satisfies,
\(
F(h(s)) = \exp F(s)\\
\)
But then, we're entering in, where the hell does this converge? Additionally, what if we stick a function \( g \) in between,
\(
G(s) = \lim_{n\to\infty} \log^{\circ n} g(h^{\circ n}(s))\\
\)
How do we derive convergence of this? How is it related to \( F \)?
Using infinite compositions for conjugate equations is out of the question. You can't solve these equations; as they equate to super function equations. You have to use a limiting process that is separate from an infinite composition. To explain this better,
IfÂ
\(
f(s,z)\,\,\text{is holomorphic}\\
\sum_j |f(h^{\circ -j}(s),z)| < \infty\\
F(s) = \Omega_{j=1}^\infty f(h^{\circ -j}(s),z)\\
F(h(s)) = f(s,F(s))\\
\)
The trouble is; you are trying to solve an equation where \( \frac{d}{ds} f(s,z) = 0 \); that is constant in the first argument. Infinite compositions are nearly useless here. Unless; you use the above limiting process; and guess a solution \( g \) using infinite compositions.
It's important to remember, that infinite compositions are perfect for
\(
f(s,z)\,\,\text{is holomorphic with some kind of summation condition; and is not constant in}\,s\\
\)
But if it's constant; there's no such luck. It's the same reason there is no infinite composition that produces tetration (without introducing a limit of some kind after the infinite composition). It's very much the equivalent of trying to make sense of \( \sum_{j=1}^\infty 1 \); it just diverges.
I am totally lost how to solve equivariant maps arbitrarily. Unless of course we open up a fixed point discussion. As the main purpose of my constructed tetration is to avoid relying on a fixed point (like with kneser); we only want to focus on behaviour at infinity. Also, I'm fairly confident that \( \text{tet}_\beta(s) \to \infty \) as \( \Im(s) \to \pm\infty \). We have no normality like with kneser's, which tends to a fixed point or its conjugate. Which, is the feature I want from my tetration; no normality conditions as \( \Im(s) = \infty \). I have a somewhat shoddy proof at the moment; I haven't had the eureka moment yet; but the numbers are implying a lack of normality. And there's good heuristic arguments that it is not normal at \( \Im(s) = \infty \).
Regards, James
Well, since,
\(
\exp(\beta_\lambda(s)) = \beta_\lambda(h(s))\\
\)
We're solving a conjugate equation. This is difficult to solve with infinite compositions. This is precisely the thing Mphlee and I are talking about. If I have \( \beta_\lambda \); we can get \( h \). But if we have \( h \); how do we get \( \beta_\lambda \)?
This is precisely the trouble we're having, solving conjugate equations arbitrarily. The closest I can think of is,
\(
F(s) = \lim_{n\to\infty} \log^{\circ n} h^{\circ n}(s)\\
\)
Which if it converges, satisfies,
\(
F(h(s)) = \exp F(s)\\
\)
But then, we're entering in, where the hell does this converge? Additionally, what if we stick a function \( g \) in between,
\(
G(s) = \lim_{n\to\infty} \log^{\circ n} g(h^{\circ n}(s))\\
\)
How do we derive convergence of this? How is it related to \( F \)?
Using infinite compositions for conjugate equations is out of the question. You can't solve these equations; as they equate to super function equations. You have to use a limiting process that is separate from an infinite composition. To explain this better,
IfÂ
\(
f(s,z)\,\,\text{is holomorphic}\\
\sum_j |f(h^{\circ -j}(s),z)| < \infty\\
F(s) = \Omega_{j=1}^\infty f(h^{\circ -j}(s),z)\\
F(h(s)) = f(s,F(s))\\
\)
The trouble is; you are trying to solve an equation where \( \frac{d}{ds} f(s,z) = 0 \); that is constant in the first argument. Infinite compositions are nearly useless here. Unless; you use the above limiting process; and guess a solution \( g \) using infinite compositions.
It's important to remember, that infinite compositions are perfect for
\(
f(s,z)\,\,\text{is holomorphic with some kind of summation condition; and is not constant in}\,s\\
\)
But if it's constant; there's no such luck. It's the same reason there is no infinite composition that produces tetration (without introducing a limit of some kind after the infinite composition). It's very much the equivalent of trying to make sense of \( \sum_{j=1}^\infty 1 \); it just diverges.
I am totally lost how to solve equivariant maps arbitrarily. Unless of course we open up a fixed point discussion. As the main purpose of my constructed tetration is to avoid relying on a fixed point (like with kneser); we only want to focus on behaviour at infinity. Also, I'm fairly confident that \( \text{tet}_\beta(s) \to \infty \) as \( \Im(s) \to \pm\infty \). We have no normality like with kneser's, which tends to a fixed point or its conjugate. Which, is the feature I want from my tetration; no normality conditions as \( \Im(s) = \infty \). I have a somewhat shoddy proof at the moment; I haven't had the eureka moment yet; but the numbers are implying a lack of normality. And there's good heuristic arguments that it is not normal at \( \Im(s) = \infty \).
Regards, James