Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[split] Understanding Kneser Riemann method
#1
For the inspiration for this thread, see this thread

@sheldonison
(01/13/2016, 01:37 PM)sheldonison Wrote:
just as a point of order, I think you meant 1 instead of 0.

I've tried to understand your Pari/GP scripts, but I think my fundamental issue is with the Kneser construction / Riemann mapping thing. So here is my understanding so far. Whatever the Kneser construction is, it seems to produce results consistent with regular iteration. Regular iteration produces power series with complex coefficients for , because the fixed points are complex for these bases. That makes sense to me. Regular iteration produces power series with real coefficients for , because the fixed points are real for these bases, or for all bases in the closed set of the region of convergence of the infinitely iterated exponential. I would also expect that tetration for "period 3" bases (approximately all complex bases with negative real part) to produce power series with complex coefficients, not only because the fixed points are complex, but also because of homotopy considerations, or that the orbits of 3 points that converge to a 3 cycle would require pushing any "lines" into a "round thing" (not sure if that's rigorous, but it makes sense to me).

So in this context, the Riemann mapping step is a method to find a function that somehow maps power series with complex coefficients to power series with real coefficients. The value of such a construction is that it allows us to compare regular iteration and intuitive iteration for . But there are too many unknowns for me: what are the properties of this Riemann mapping? how do we find it? what is the result? is it analytic? wouldn't this just be equivalent to



and if this is the method for calculating the Riemann mapping, then we can't expect to learn anything about the two methods of iteration. Perhaps I should revisit this when I'm less confused.
Reply


Messages In This Thread
[split] Understanding Kneser Riemann method - by andydude - 01/13/2016, 04:01 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  My interpolation method [2020] tommy1729 1 286 02/20/2020, 08:40 PM
Last Post: tommy1729
  Kneser method question tommy1729 9 1,334 02/11/2020, 01:26 AM
Last Post: sheldonison
  Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 602 09/09/2019, 10:55 PM
Last Post: tommy1729
  2 fixpoints , 1 period --> method of iteration series tommy1729 0 1,872 12/21/2016, 01:27 PM
Last Post: tommy1729
  Tommy's matrix method for superlogarithm. tommy1729 0 1,871 05/07/2016, 12:28 PM
Last Post: tommy1729
  Periodic analytic iterations by Riemann mapping tommy1729 1 2,497 03/05/2016, 10:07 PM
Last Post: tommy1729
  Kouznetsov-Tommy-Cauchy method tommy1729 0 2,212 02/18/2015, 07:05 PM
Last Post: tommy1729
  Problem with cauchy method ? tommy1729 0 2,069 02/16/2015, 01:51 AM
Last Post: tommy1729
  [MO] something practical on Riemann-mapping Gottfried 0 2,057 01/05/2015, 02:33 PM
Last Post: Gottfried
  picturing method. tommy1729 4 4,364 07/03/2014, 08:23 AM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)