Conjecture:
For any locally-analytic complex function \(f:\mathbb{C}\to\mathbb{C}\), analytically continued whether multivalued or singlevalued, there always lies a fixed point \(L\), which satisfies either:
\(f'(z)\) 's value at \(L\) exists and not in the form \(q\in\mathbb{Q}\cap(0,1),f'(L)=e^{2\pi qi}\)
or: there's a conjugacy g of f by h: \(g=h^{-1}fh\) which maps \(L\to{L_0}\), where \(f'(L)\) doesn't exists but some directional derivative \(\lambda=f'(L_0)\) exists, or the limit \(\lambda=\lim_{a\to{L}}{f'(a)}\) exists, and \(\lambda\) not in the form \(q\in\mathbb{Q}\cap(0,1),\lambda=e^{2\pi qi}\)
This may not be true.
Sub-conjecture: Only take polynomial functions instead of any such f into consideration, the result is true.
This is proved correct for linear functions and quadratic functions.
Conjecture:
Denote P as the successor operator. For any operator T, which can act on a number or another operator, despite multivalued-ness, there always exist an operator Q or M which fits QT=PQ or TM=MP where multiplication refers to compositions between operator.
A specific example is hyperoperation.
An equivalent expression is for any operator T, there always lies exp(T) or log(T).
For any locally-analytic complex function \(f:\mathbb{C}\to\mathbb{C}\), analytically continued whether multivalued or singlevalued, there always lies a fixed point \(L\), which satisfies either:
\(f'(z)\) 's value at \(L\) exists and not in the form \(q\in\mathbb{Q}\cap(0,1),f'(L)=e^{2\pi qi}\)
or: there's a conjugacy g of f by h: \(g=h^{-1}fh\) which maps \(L\to{L_0}\), where \(f'(L)\) doesn't exists but some directional derivative \(\lambda=f'(L_0)\) exists, or the limit \(\lambda=\lim_{a\to{L}}{f'(a)}\) exists, and \(\lambda\) not in the form \(q\in\mathbb{Q}\cap(0,1),\lambda=e^{2\pi qi}\)
This may not be true.
Sub-conjecture: Only take polynomial functions instead of any such f into consideration, the result is true.
This is proved correct for linear functions and quadratic functions.
Conjecture:
Denote P as the successor operator. For any operator T, which can act on a number or another operator, despite multivalued-ness, there always exist an operator Q or M which fits QT=PQ or TM=MP where multiplication refers to compositions between operator.
A specific example is hyperoperation.
An equivalent expression is for any operator T, there always lies exp(T) or log(T).
Regards, Leo 