05/27/2016, 04:03 AM
Hey Mphlee. I corrected some of the mistakes I made on my old arxiv page. They were rather small (the same result holds), the new version is also being edited by my professor, and just some double checking being done by him.
And to your question on the super function operator, I have added a little bit about this.
If \( H \) is a set in which all its elements \( \phi(z) \) are holomorphic for \( \Re(z) > 0 \), take the real positive line to itself, have a fix point \( x_0 \in \mathbb{R}^+ \) such that \( 0<\phi'(x_0) < 1 \) and \( \phi^{\circ n}(x) \to x_0 \) for all \( x\in[0,x_0] \)
then..
There is a semi group of operators \( S \) such that \( S \) acts on \( H \). Where more beneficially, the semi group is isomorphic to \( \{\mathbb{R}^+,+\} \)
This semi group being \( \uparrow^t \) for \( t \ge 0 \) where
\( \uparrow^t \uparrow^{s} = \uparrow^{t+s} \)
and if
\( F_t(x) = \uparrow^t \phi(x) \)
then
\( F_t(F_{t+1}(x)) = F_{t+1}(x+1) \)
I have no idea to be honest. This is where I'm looking. The analysis is extensively simplified using ramanujan's master theorem. and some key points arise. I have a few well thought out points but nothing too expansive.
\( \alpha \uparrow^t z \) is injective in \( t \ge 0 \) and \( 0 < \Im(z) < \ell_t \) (where \( \ell_t \) is the imaginary period in \( z \).
The operator \( \uparrow^t \) is injective as well on the function space \( H \) as defined above.
The only fixed point of \( \uparrow \) I can think of are constant functions. Since
\( \uparrow \phi(z) = \phi^{\circ z}(1) \)
and if \( \phi(z) = \alpha \) then surely \( \phi^{\circ z}(1)= \alpha \)
It always follows that \( \uparrow^n \phi(x) \to \phi(1) \) though. This is exemplified by \( \alpha \uparrow^n x \to \alpha \)
And to your question on the super function operator, I have added a little bit about this.
If \( H \) is a set in which all its elements \( \phi(z) \) are holomorphic for \( \Re(z) > 0 \), take the real positive line to itself, have a fix point \( x_0 \in \mathbb{R}^+ \) such that \( 0<\phi'(x_0) < 1 \) and \( \phi^{\circ n}(x) \to x_0 \) for all \( x\in[0,x_0] \)
then..
There is a semi group of operators \( S \) such that \( S \) acts on \( H \). Where more beneficially, the semi group is isomorphic to \( \{\mathbb{R}^+,+\} \)
This semi group being \( \uparrow^t \) for \( t \ge 0 \) where
\( \uparrow^t \uparrow^{s} = \uparrow^{t+s} \)
and if
\( F_t(x) = \uparrow^t \phi(x) \)
then
\( F_t(F_{t+1}(x)) = F_{t+1}(x+1) \)
(05/23/2016, 07:25 PM)MphLee Wrote: I would like to ask you: have you some ideas/intuition on the behavior of this map \( \uparrow:H\to H \) and its dynamics in general?
I have no idea to be honest. This is where I'm looking. The analysis is extensively simplified using ramanujan's master theorem. and some key points arise. I have a few well thought out points but nothing too expansive.
Quote: Is it injective?
\( \alpha \uparrow^t z \) is injective in \( t \ge 0 \) and \( 0 < \Im(z) < \ell_t \) (where \( \ell_t \) is the imaginary period in \( z \).
The operator \( \uparrow^t \) is injective as well on the function space \( H \) as defined above.
Quote:Has it fixed points?
The only fixed point of \( \uparrow \) I can think of are constant functions. Since
\( \uparrow \phi(z) = \phi^{\circ z}(1) \)
and if \( \phi(z) = \alpha \) then surely \( \phi^{\circ z}(1)= \alpha \)
It always follows that \( \uparrow^n \phi(x) \to \phi(1) \) though. This is exemplified by \( \alpha \uparrow^n x \to \alpha \)