• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Iterating at fixed points of b^x jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 09/11/2007, 04:31 PM bo198214 Wrote:I dont know whether I am the first one who realizes that regularly iterating $\exp$ at a complex fixed point yields real coefficients! Moreover they do not depend on the chosen fixed point! To be more precise: If we have any fixed point $a$ of $\exp$, then the power series $\tau_{-a}\circ \exp\circ \tau_a$ (where $\tau_a(x):=x+a$) is a power series with fixed point 0 and with first coefficient $\exp'(a)=\exp(a)=a$, particularly the series is non-real. By the previous considerations the fixed point is repelling so we have the standard way of hyperbolic iteration (on the main branch) which yields again a non-real power series. If we however afterwards apply the inverse transformation $\exp^{\circ t}:=\tau_a\circ (\tau_a^{-1}\circ \exp\circ \tau_a)^{\circ t}\circ \tau_a^{-1}$ we get back real coefficients and they do not depend on the fixed point $a$. Though I can not prove it yet, this seems to be quite reliable. And of course those coefficients are equal to those obtained by the matrix operator method, which in turn equals Andrew's solution after transformation. Somehow you lost me here. I don't doubt what you say; I'm just not sure what you meant. Perhaps my problem is that I'm misunderstanding how to apply your tau function. And I'm not entirely sure what you mean by "power series". Are you talking about the coefficients of a Taylor series expansion, or are you talking about a series of successive exponentials (i.e., a series of powers). I think I've possibly been sloppy in my own usage of the term "power series", which is leading to my present confusion. ~ Jay Daniel Fox « Next Oldest | Next Newest »

 Messages In This Thread Iterating at fixed points of b^x - by bo198214 - 09/08/2007, 10:02 AM The fixed points of e^x - by bo198214 - 09/08/2007, 10:34 AM The fixed points of b^x - by bo198214 - 09/08/2007, 11:36 AM RE: Iterating at fixed points of b^x - by jaydfox - 09/12/2007, 06:23 AM RE: Iterating at fixed points of b^x - by bo198214 - 09/12/2007, 09:54 AM RE: Iterating at fixed points of b^x - by GFR - 10/03/2007, 11:03 PM RE: Iterating at fixed points of b^x - by Gottfried - 10/04/2007, 06:53 AM RE: Iterating at fixed points of b^x - by bo198214 - 10/04/2007, 01:30 PM RE: Iterating at fixed points of b^x - by bo198214 - 10/04/2007, 05:54 PM RE: Iterating at fixed points of b^x - by Gottfried - 10/04/2007, 11:05 PM RE: Iterating at fixed points of b^x - by GFR - 01/31/2008, 03:07 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Are tetrations fixed points analytic? JmsNxn 2 2,765 12/14/2016, 08:50 PM Last Post: JmsNxn Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,474 03/19/2016, 10:44 AM Last Post: fivexthethird Derivative of exp^[1/2] at the fixed point? sheldonison 10 9,653 01/01/2016, 03:58 PM Last Post: sheldonison [MSE] Fixed point and fractional iteration of a map MphLee 0 2,031 01/08/2015, 03:02 PM Last Post: MphLee iterating x + ln(x) starting from 2 tommy1729 2 2,554 04/29/2013, 11:35 PM Last Post: tommy1729 tetration from alternative fixed point sheldonison 21 27,387 12/06/2011, 02:43 PM Last Post: sheldonison attracting fixed point lemma sheldonison 4 9,287 06/03/2011, 05:22 PM Last Post: bo198214 cyclic points tommy1729 3 4,281 04/07/2011, 07:57 PM Last Post: JmsNxn iterating non-analytic tommy1729 0 1,745 02/08/2011, 01:25 PM Last Post: tommy1729 Branch points of superlog mike3 0 2,235 02/03/2010, 11:00 PM Last Post: mike3

Users browsing this thread: 1 Guest(s)