Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Does tetration take the right half plane to itself?
This question is rather straightforward. Let . We know that there exists a unique bounded tetration function that is holomorphic for . But is ? Does the bounded tetration function take the right half plane to itself?

If this is true, this manages to prove a lot of things. First of all, it follows that tetration has only one fixed point such that , that is geometrically attracting , which follows by the Schwarz lemma. Secondly for all .  All of the orbits of tend to this fixed point.

This implies we can construct a complex iteration . Therefore giving us pentation that ALSO takes the right half plane into itself. Therefore it ALSO has a unique fixed point, this fixed point is ALSO geometrically attracting, all of the orbits of pentation tend to this fixed point, and now we can rinse and repeat to construct hexation, so on and so forth. 

The great part about this is how it qualifies the sequence of bounded analytic hyper-operators. It gives a lot of great properties.  We get the following things for free.

for is holomorphic in and analytic for  (I still haven't really managed to show the hyper-operators are analytic in the base argument, only continuous; but with this lemma, it follows trivially). 
(before I simply wrote they send to the complex plane and focused on their behaviour in ).
has a purely imaginary period (something that gives a lot of cool things, like a Fourier series representation for example).
if is the fixed point of or the limit at infinity of , then for for some . (Exponential decay is always nice.)
...(I could go on)...
And most importantly, these functions satisfy the holy grail of functional equations

The functional equation is something I could never truly get perfect, because I only managed to show it on the real positive line, without sending the right half plane to itself, the composition in the complex plane is technically ill defined.

So all in all, I've boiled a whollllllllleeeee swash of questions into one question.

Does ?

If anyone's curious, I can explain how I've approached the question. It's a little convoluted, so I'll leave it out till someone asks me.

Messages In This Thread
Does tetration take the right half plane to itself? - by JmsNxn - 05/10/2017, 07:46 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Why the beta-method is non-zero in the upper half plane JmsNxn 0 343 09/01/2021, 01:57 AM
Last Post: JmsNxn
  Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 2,122 09/09/2019, 10:55 PM
Last Post: tommy1729
  Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) Gottfried 1 3,087 09/09/2019, 10:50 PM
Last Post: tommy1729
  Half-iteration of x^(n^2) + 1 tommy1729 3 7,878 03/09/2017, 10:02 PM
Last Post: Xorter
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 28,208 01/09/2017, 02:41 AM
Last Post: Gottfried
  [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 65,827 03/27/2015, 11:28 PM
Last Post: tommy1729
  [entire exp^0.5] The half logaritm. tommy1729 1 4,417 05/11/2014, 06:10 PM
Last Post: tommy1729
  Does the Mellin transform have a half-iterate ? tommy1729 4 7,363 05/07/2014, 11:52 PM
Last Post: tommy1729
  Simple method for half iterate NOT based on a fixpoint. tommy1729 2 6,275 04/30/2013, 09:33 PM
Last Post: tommy1729
  half-iterates of x^2-x+1 Balarka Sen 2 6,796 04/30/2013, 01:14 AM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)