12/06/2022, 01:14 PM
ok some important nuances !
For starters
The riemann mapping in general does not preserve the semi-group iso.
So very likely the Kneser method does not satisfy the semi-group iso.
the 2sinh method uses a fixpoints and suggests having the semi-group iso but it might not have a nonzero radius and be nowhere analytic.
This brings us to the essence :
the 1-periodic theta function is unique ... but maybe there are different paths and multiple paths have each a semi-group iso ??
This would imply to having uniqueness ( and the semi-group iso ) by 2 things : the path and the function we iterate.
But the real line or the real line intervals only have one path.
So there cannot be 2 solutions from R to a subset of R.
Hence the 2sinh must be unique in having the semi-group iso IF IT EVEN HAS IT.
And even if it has it might not be analytic.
And carleman matrices dont make me optimistic.
So the focus lies on the paths taken.
this might simply explain what happens when we use different fixpoints for kneser like ideas before the riemann mapping ( as to using branch logic as tool for explaining things )
So the idea is to change a path A to a path B while preserving the semi-group iso.
lets say a function t does that
t(A) = B
Do we also have a path C with the iso that is then simply :
t(t(A)) = t(B) = C
??
Thereby perhaps linking functional iterations ( of t ) to changing paths ??
For a given function f(z) how does one find t or one of its iterates ??
This might seem like a riemann mapping idea but it is slightly different and not necc analytic !!!
( not analytic for instance mapping the path from kneser ( pre riemann mapping ) to 2sinh )
IT seems the path mapping is more like a conjugate thing ;
g( c^k g_F(inverse(z)) )
where F comes from the fixpoint expansion with derivative c at fixpoint.
By using the conjugate thing , we clearly see the preserving of the semi-group iso.
( yes this resembles 2sinh method alot , but those where similar paths and a real fixpoint with a real value c , here fix and c are non-real )
But again - for the same reasons - g might not even be analytic !!
( we are working with schroder equation satisfying context here , as opposed to abel ... i mix em up without mentioning sometimes sorry )
So how we know that there are multiple paths with semi-group iso ?
Im not sure and need more understanding but
the conjugate fixpoint of exp gives the conjugate path !!
so that are at least 2 paths !
...
what makes me wonder if they use the same path functions to the real line ...
( similar to the riemann mapping what does so )
Short story short : there is still hope for R to R semi-group iso tetration.
we need more research and understanding though.
2sinh method is a candidate.
But it is unclear if we have hope for ALSO analytic.
***
As for the ideas with derivatives ; the most general case is the uniqueness criterion
semi-log is analytic and a bernstein function with semi-exp(- oo ) = given constant.
although that given constant might not be free to choose !
And I only assume existance.
***
end nuances ( short version )
regards
tommy1729
For starters
The riemann mapping in general does not preserve the semi-group iso.
So very likely the Kneser method does not satisfy the semi-group iso.
the 2sinh method uses a fixpoints and suggests having the semi-group iso but it might not have a nonzero radius and be nowhere analytic.
This brings us to the essence :
the 1-periodic theta function is unique ... but maybe there are different paths and multiple paths have each a semi-group iso ??
This would imply to having uniqueness ( and the semi-group iso ) by 2 things : the path and the function we iterate.
But the real line or the real line intervals only have one path.
So there cannot be 2 solutions from R to a subset of R.
Hence the 2sinh must be unique in having the semi-group iso IF IT EVEN HAS IT.
And even if it has it might not be analytic.
And carleman matrices dont make me optimistic.
So the focus lies on the paths taken.
this might simply explain what happens when we use different fixpoints for kneser like ideas before the riemann mapping ( as to using branch logic as tool for explaining things )
So the idea is to change a path A to a path B while preserving the semi-group iso.
lets say a function t does that
t(A) = B
Do we also have a path C with the iso that is then simply :
t(t(A)) = t(B) = C
??
Thereby perhaps linking functional iterations ( of t ) to changing paths ??
For a given function f(z) how does one find t or one of its iterates ??
This might seem like a riemann mapping idea but it is slightly different and not necc analytic !!!
( not analytic for instance mapping the path from kneser ( pre riemann mapping ) to 2sinh )
IT seems the path mapping is more like a conjugate thing ;
g( c^k g_F(inverse(z)) )
where F comes from the fixpoint expansion with derivative c at fixpoint.
By using the conjugate thing , we clearly see the preserving of the semi-group iso.
( yes this resembles 2sinh method alot , but those where similar paths and a real fixpoint with a real value c , here fix and c are non-real )
But again - for the same reasons - g might not even be analytic !!
( we are working with schroder equation satisfying context here , as opposed to abel ... i mix em up without mentioning sometimes sorry )
So how we know that there are multiple paths with semi-group iso ?
Im not sure and need more understanding but
the conjugate fixpoint of exp gives the conjugate path !!
so that are at least 2 paths !
...
what makes me wonder if they use the same path functions to the real line ...
( similar to the riemann mapping what does so )
Short story short : there is still hope for R to R semi-group iso tetration.
we need more research and understanding though.
2sinh method is a candidate.
But it is unclear if we have hope for ALSO analytic.
***
As for the ideas with derivatives ; the most general case is the uniqueness criterion
semi-log is analytic and a bernstein function with semi-exp(- oo ) = given constant.
although that given constant might not be free to choose !
And I only assume existance.
***
end nuances ( short version )
regards
tommy1729