• 1 Vote(s) - 5 Average
• 1
• 2
• 3
• 4
• 5
 Cyclic dynamics f(-x) = T (f(x)) tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 08/24/2015, 11:56 PM Let analytic a(z) = a^[r](z) for all z and some real r <> 1. Cyclic dynamics fundamental equation. F( a(z) ) = T ( F(z) ) With T entire. Less general By using g(q x) = a( g(x) ) , q = exp(2 pi i / r). We get W(q x) = T( W(x) ). --- Example F(-x) = F(x) + pi i Solution ln(x). Can you find F(-x) = g * F(x) + h Note 1) we consider only nonconstant solutions Note 2) we prefer a(x) to have exactly one fixpoint , so that its iterations are Unique ! Regards Tommy1729 tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 08/25/2015, 08:13 AM The solution seems to be constructible by using again a superfunction. Example F(-x) = e F(x) Ln(-x) = ln(x) + pi i Exp(x+1) = e exp(x) So F = exp(ln(x)/(pi i)) = x^(- 1/pi i) Regards Tommy1729 tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 08/25/2015, 08:23 AM Assuming the assumption in the previous post was correct , we need to show that the equation keeps on being valid on the other branches too ... Or when that is not the case. Analytic continuation probably is key here. Its easy when the range and domain are on the same branch ( such as the example x^(- 1/pi i). Then we get A --> B --> A I think this is not so hard. Regards Tommy1729 « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post THE problem with dynamics tommy1729 1 2,047 04/04/2017, 10:52 PM Last Post: tommy1729 Dynamics as alternating waves tommy1729 2 1,964 02/13/2017, 01:01 AM Last Post: tommy1729 Polygon cyclic fixpoint conjecture tommy1729 1 1,968 05/18/2016, 12:26 PM Last Post: tommy1729 Natural cyclic superfunction tommy1729 3 2,689 12/08/2015, 12:09 AM Last Post: tommy1729 [2014] " statistical dynamics " tommy1729 0 1,563 08/31/2014, 11:53 PM Last Post: tommy1729 new results from complex dynamics bo198214 25 31,426 02/25/2013, 01:42 AM Last Post: sheldonison cyclic points tommy1729 3 4,145 04/07/2011, 07:57 PM Last Post: JmsNxn online-book: Milnor, dynamics in one complex variable bo198214 0 5,061 05/23/2009, 04:21 PM Last Post: bo198214 Cyclic complex functions and uniqueness jaydfox 12 11,889 04/24/2008, 08:50 PM Last Post: bo198214

Users browsing this thread: 1 Guest(s)