2 fixpoint failure - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: 2 fixpoint failure (/showthread.php?tid=534) 2 fixpoint failure - tommy1729 - 11/09/2010 so why isnt there an entire superfunction S(x) of a nontrivial entire function F(x) where F(x) has two fixpoints , and the superfunctions agree on both fixpoints ? i think i have an idea : to find a superfunction we usually solve f ( F(x) ) = a ( f(x) ) where 'a' is a linear function. that works well for 1 fixpoint because 'a' is lineair and has thus only 1 fixpoint itself. so f ( F(x) ) = b ( f(x) ) with 'b' being a function with 2 (distinct) fixpoints is logical for 2 (distinct) fixpoints expansions. BUT b^[r] might cycle different from F(x)^[r] thus b^[r] is entire (in r) IFF F(x) is. hence no entire solution for this one ... b must be a moebius function. and hence f(x) must have poles , thus be at best meromorphic over the complex plane. but we might consider the equation f ( F(x) ) = 'moebius' ( f(x) ) to be unsatifying : we dont know how to solve it !! we can however say that it must have a solution IFF we have the property that there is agreement on both fixpoint expansions. then again , thats very nonconstructive and maybe not so usefull. ( without a method to solve the equation ) ( modified for clarity , i realized it was not so clear ) a penny for your thoughts. regards tommy1729 RE: 2 fixpoint failure - tommy1729 - 11/13/2010 i modified a bit