Honestly, at this point in my evaluation of my ideas (my approach), I don't think constructing a monoid globally on \( \mathbb{R} \) is hopeful. I definitely got ahead of myself in thinking I could get a monoid \( \mathcal{B} \subset \mathcal{C}^1(\mathbb{R}, \mathbb{R}) \). I think, doing this in a local setting is easier--and far more probable. Even if we focus on the trivial case. That would be when we can just multiply our Schroder functions. By this I mean, if \( f(0)=0=g(0) \) and \( f'(0) = \lambda,\, g'(0) = \mu \) and \( f,g \) are holomorphic in a neighborhood of \( 0 \) and \( |\mu|,|\lambda| \neq 0,1 \), then,
\(
\phi(z) = \Psi^{-1}(\lambda \mu^{-1}\Phi(z))\\
\)
Is holomorphic in a neighborhood of zero. Here,
\(
\Phi(g(z)) = \mu\Phi(z)\\
\Psi^{-1}(\lambda z)= f(\Psi^{-1}(z))\\
\)
And this is of course a group under composition; in which \( \phi \) belongs if \( \lambda \neq \mu \)... I think this may be a more tractable approach to constructing a general categorical theory. This set of sheaves \( \mathcal{B} \) do satisfy the conjugate property. But it's a little useless globally (it just means there's a taylor series in a neighborhood of zero). I do think its doable in the global sense, but probably in a local setting, is the correct way to approach the larger theory of conjugation. I definitely can't show it on the real line, but I may be able to do it locally in the complex plane (not necessarily about a fixed point). I'll have to entirely alter my approach though.
Nonetheless the group, under composition, of sheaves \( \mathcal{B} \),
\(
f(0) = 0\\
f\,\,\text{is holomorphic at}\,0\\
|f'(0)| \neq 0,1\\
\)
Is a very good place to start. Of which the conjugate property is almost satisfied here. And furthermore, this is a GROUP which almost satisfies the conjugate property. We'd just have to allow for \( |f'(0)| =1 \) and somehow massage this case to allow for the conjugate property--while still staying in the group.
I forgot how useless I am at deep questions in real-analysis, so I'll stick to holomorphy. I'm so angry I can't get it on the real-line ):<
Regards, James
\(
\phi(z) = \Psi^{-1}(\lambda \mu^{-1}\Phi(z))\\
\)
Is holomorphic in a neighborhood of zero. Here,
\(
\Phi(g(z)) = \mu\Phi(z)\\
\Psi^{-1}(\lambda z)= f(\Psi^{-1}(z))\\
\)
And this is of course a group under composition; in which \( \phi \) belongs if \( \lambda \neq \mu \)... I think this may be a more tractable approach to constructing a general categorical theory. This set of sheaves \( \mathcal{B} \) do satisfy the conjugate property. But it's a little useless globally (it just means there's a taylor series in a neighborhood of zero). I do think its doable in the global sense, but probably in a local setting, is the correct way to approach the larger theory of conjugation. I definitely can't show it on the real line, but I may be able to do it locally in the complex plane (not necessarily about a fixed point). I'll have to entirely alter my approach though.
Nonetheless the group, under composition, of sheaves \( \mathcal{B} \),
\(
f(0) = 0\\
f\,\,\text{is holomorphic at}\,0\\
|f'(0)| \neq 0,1\\
\)
Is a very good place to start. Of which the conjugate property is almost satisfied here. And furthermore, this is a GROUP which almost satisfies the conjugate property. We'd just have to allow for \( |f'(0)| =1 \) and somehow massage this case to allow for the conjugate property--while still staying in the group.
I forgot how useless I am at deep questions in real-analysis, so I'll stick to holomorphy. I'm so angry I can't get it on the real-line ):<
Regards, James