Thread Rating:
  • 1 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Universal uniqueness criterion?
#51
(06/22/2009, 09:46 PM)bo198214 Wrote:
(06/22/2009, 07:19 PM)Kouznetsov Wrote:
(06/22/2009, 02:24 PM)bo198214 Wrote: .. Well, but *has* more fixed points. In every strip there is a fixed point of .
Henryk, how about to build up "another" holomorphic tetration that goes to other fixed points at ?
I dont think that there is an initial region connecting other conjugated fixed point pairs than the one closest to the real axis. (Plot the straight line connecting two such fixed points and plot its image under exp. Both lines intersect.)

On the other hand I asked you som time ago to apply your algorithm to other fixed points, but you somehow did not follow that path.

What about using Kneser's approach to produce the alternate tetration functions?
Reply
#52
(07/05/2009, 08:57 PM)BenStandeven Wrote: What about using Kneser's approach to produce the alternate tetration functions?

There is no initial region connecting an alternative fixed point pair.
The image of the straight line connecting a secondary fixed point pair overlaps with itself.
Secondary fixed points lie in a range with imaginary part greater or less than pi.
The vertical line connecting a pair is longer than 2*pi.
This means the image revolves more than once around 0 with constant radius, hence overlapping itself.

I tried to construct different connecting lines of a secondary fixed point pair and failed. I believe there is no initial region connecting two secondary fixed points.
Reply
#53
(07/05/2009, 09:13 PM)bo198214 Wrote:
(07/05/2009, 08:57 PM)BenStandeven Wrote: What about using Kneser's approach to produce the alternate tetration functions?

There is no initial region connecting an alternative fixed point pair.
The image of the straight line connecting a secondary fixed point pair overlaps with itself.
Secondary fixed points lie in a range with imaginary part greater or less than pi.
The vertical line connecting a pair is longer than 2*pi.
This means the image revolves more than once around 0 with constant radius, hence overlapping itself.

I tried to construct different connecting lines of a secondary fixed point pair and failed. I believe there is no initial region connecting two secondary fixed points.

Yeah, that's right; the path would have to pass through a point with imaginary value 2 pi, and also through its conjugate. Then the other side of the region would intersect itself at the exponential of that point.
Reply
#54
(07/06/2009, 12:53 AM)BenStandeven Wrote: the path would have to pass through a point with imaginary value 2 pi, and also through its conjugate. Then the other side of the region would intersect itself at the exponential of that point.

Say the curve is injective and connects two points and with equal real part and with . One needs to show that then there is always a pair of points and with equal real part and with .

This sounds very plausible but I couldnt prove it except for certain simple shapes of , e.g. convex.
Reply
#55
(07/06/2009, 08:56 AM)bo198214 Wrote: One needs to show that then there is always a pair of points and with equal real part and with .

This is equivalent to that and intersect.
If only extend to the right, i.e. , then this is a consequence of the Jordan curve theorem. We have . The closed Jordan curve has the point in its interior and the point in its exterior. Hence there must be an intersection of and (as does not pass [a,b] for .)
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Semi-exp and the geometric derivative. A criterion. tommy1729 0 1,024 09/19/2017, 09:45 PM
Last Post: tommy1729
  A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 9,143 02/17/2017, 05:21 AM
Last Post: JmsNxn
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 12,417 01/09/2017, 02:41 AM
Last Post: Gottfried
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,271 03/19/2016, 10:44 AM
Last Post: fivexthethird
  [2014] Uniqueness of periodic superfunction tommy1729 0 1,658 11/09/2014, 10:20 PM
Last Post: tommy1729
  Real-analytic tetration uniqueness criterion? mike3 25 18,302 06/15/2014, 10:17 PM
Last Post: tommy1729
  exp^[1/2](x) uniqueness from 2sinh ? tommy1729 1 1,919 06/03/2014, 09:58 PM
Last Post: tommy1729
  Uniqueness Criterion for Tetration jaydfox 9 10,029 05/01/2014, 10:21 PM
Last Post: tommy1729
  Uniqueness of Ansus' extended sum superfunction bo198214 4 5,976 10/25/2013, 11:27 PM
Last Post: tommy1729
  A question concerning uniqueness JmsNxn 3 4,975 10/06/2011, 04:32 AM
Last Post: sheldonison



Users browsing this thread: 1 Guest(s)