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what is known about Coo half-iterates that do not converge in the neighbourhood of the fixpoints ( even not with a mittag-leffler expansion ) ?
(01/08/2010, 12:36 AM)tommy1729 Wrote: [ -> ]what is known about Coo half-iterates that do not converge in the neighbourhood of the fixpoints ( even not with a mittag-leffler expansion ) ?

nothing ?
(01/10/2010, 11:44 PM)tommy1729 Wrote: [ -> ]what is known about Coo half-iterates that do not converge in the neighbourhood of the fixpoints ( even not with a mittag-leffler expansion ) ?

I'm not sure what you mean by "based on the fixed point". An example of the half iterate equation would be
$\text{half-iterate}_e(x)=\text{sexp}_e(\text{slog}_e(x)+0.5)$

I haven't seen any graphs of the half iterate function, in the real domain or the complex domain. Using Dmitrii Kouznetsov's sexp could be an example, since it is based on sexp(x>-2)=real number. A true fixed point version of sexp/slog would exponentially diverge away from the fixed point at approximately 0.318 + 1.337i, for small values, and would not be real valued at the real axis. If the slog/sexp is based on the real number line, and the base is greater than $\eta$ than it would give different results than sexp/slog based on the fixed point. (Kouznetsov's sexp is also based on the fixed point in that it converges to the fixed point at +/- i*$\infty$).

The base change version of sexp/slog is not even defined in the complex plane, so the complex fixed point has no meaning, but the base change version of sexp/slog is Coo, so I'm guessing that would be an example as well.

I'm not sure what Tommy is looking for. Do you want to compare the behavior of different super-function algorithms, which would lead to different half-iterates? Or are you looking for a particular kind of super-function, which has a particular kind of non-converging half-iterate?
- Shel
(01/11/2010, 04:39 AM)sheldonison Wrote: [ -> ]
(01/10/2010, 11:44 PM)tommy1729 Wrote: [ -> ]what is known about Coo half-iterates that do not converge in the neighbourhood of the fixpoints ( even not with a mittag-leffler expansion ) ?

I'm not sure what you mean by "based on the fixed point". An example of the half iterate equation would be
$\text{half-iterate}_e(x)=\text{sexp}_e(\text{slog}_e(x)+0.5)$

- Shel

i didnt say " based on the fixed point " and if i would have said it , i would have meant regular iteration around the fixpoint.
( by using taylor series )

im talking about half-iterates that diverge in a neighbourhood of the fixed points.

regards

tommy1729
(01/13/2010, 12:02 AM)tommy1729 Wrote: [ -> ]im talking about half-iterates that diverge in a neighbourhood of the fixed points.

...and converge somewhere else.

regards

tommy1729
The half-iterate sheldon mentiones
(01/11/2010, 04:39 AM)sheldonison Wrote: [ -> ]$\text{sexp}_e(\text{slog}_e(x)+0.5)$

does not converge, i.e. has a branch-point at the primary fixed points.

Generally any half-iterate that is not the regular at a fixed point, does not converge there (in the sense of not of not being holomorphic); given that the fixed point is also a fixed point of the half-iterate.
(06/24/2010, 07:43 AM)bo198214 Wrote: [ -> ]The half-iterate sheldon mentiones
(01/11/2010, 04:39 AM)sheldonison Wrote: [ -> ]$\text{sexp}_e(\text{slog}_e(x)+0.5)$

does not converge, i.e. has a branch-point at the primary fixed points.

Generally any half-iterate that is not the regular at a fixed point, does not converge there (in the sense of not of not being holomorphic); given that the fixed point is also a fixed point of the half-iterate.

but ? which sexp and which slog ? we have various tetration functions here on the forum ?

and what do you mean by " not regular at a fixed point " ?
does that mean branch point at the fixed point ?

are you guys saying that around the fixpoints of exp , $\text{sexp}_e(\text{slog}_e(x)+0.5)$ always has a branch point , nomatter what kind of tetration we use ?

regards

tommy1729
(06/24/2010, 12:15 PM)tommy1729 Wrote: [ -> ]and what do you mean by " not regular at a fixed point " ?

I wrote "not the regular": Every half iterate that is not the regular half iterate has a singularity (e.g. branch point or whatever) at that fixed point.
(06/26/2010, 03:37 AM)bo198214 Wrote: [ -> ]
(06/24/2010, 12:15 PM)tommy1729 Wrote: [ -> ]and what do you mean by " not regular at a fixed point " ?

I wrote "not the regular": Every half iterate that is not the regular half iterate has a singularity (e.g. branch point or whatever) at that fixed point.

ah ok. got it. thanks.