Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Infinite tetration of the imaginary unit
(06/23/2011, 01:13 PM)sheldonison Wrote: I'm going to read up on what's out there on Fatou sets. Anybody have a suggested reference? Specifically, we're interested in a Periodic Fatou set generated from a complex parabolic fixed point. I'm also trying to formulate a question for math overflow, to verify the existence of real periodic Fatou sets, generated with a superfunction of a function with a periodic parabolic fixed point, similar to what is seen on the Shell-Thron boundary.
- Sheldon
Looks like my intuition is correct. They're called Siegel discs, which are part of the Classification of Fatou components. In particular, Siegel discs occur around "occur around irrationally indifferent periodic points", of exactly the type that we have been describing in this thread. Siegel proved their existence.

I'll do some more reading -- there are requirements on the Brjuno Number of the convergents of the continued fraction expansion of α, which would be 1/period. The requirements don't seem that stringent, which implies that on the Shell-Thron boundary, for many/most cases if the period is irrational, , then there is a real periodic superfunction. Then the real periodic superfunction can be mapped to a unit disc, where the boundary of the unit disc would be generating the superfunction from an initial starting point of 0, where the fractal behavior caused by , or . The interior of the Siegel disc would correspond to where the superfunction is analytic, with a real period, converging to the fixed point as , which would correspond to the center of the disc.
- Sheldon

Messages In This Thread
Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 12:09 AM
solved -- they're called Siegel discs - by sheldonison - 06/23/2011, 06:10 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 1,301 03/20/2018, 12:16 AM
Last Post: tommy1729
  [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 12,724 12/28/2014, 10:22 PM
Last Post: MphLee
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 5,419 05/06/2014, 09:47 PM
Last Post: tommy1729
  Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 7,245 07/17/2013, 09:46 AM
Last Post: Gottfried
  Wonderful new form of infinite series; easy solve tetration JmsNxn 1 4,535 09/06/2012, 02:01 AM
Last Post: JmsNxn
  The imaginary tetration unit? ssroot of -1 JmsNxn 2 5,628 07/15/2011, 05:12 PM
Last Post: JmsNxn
  the infinite operator, is there any research into this? JmsNxn 2 5,682 07/15/2011, 02:23 AM
Last Post: JmsNxn
  Tetration and imaginary numbers. robo37 2 5,607 07/13/2011, 03:25 PM
Last Post: robo37
  Infinite Pentation (and x-srt-x) andydude 20 27,328 05/31/2011, 10:29 PM
Last Post: bo198214
  Imaginary zeros of f(z)= z^(1/z) (real valued solutions f(z)>e^(1/e)) Gottfried 91 97,728 03/03/2011, 03:16 PM
Last Post: Gottfried

Users browsing this thread: 1 Guest(s)