conjecture 2 fixpoints - 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: conjecture 2 fixpoints ( /showthread.php?tid=491) |

conjecture 2 fixpoints - tommy1729 - 08/11/2010
a conjecture about 2 fixpoints. it was once asked when 2 fixpoint based real iterates coincide. perhaps an example. conjecture 2 fixpoints : let f(z) be a laurent series meromorphic everywhere apart in circle D with center at origin and radius 1/a. f(z) is not periodic. f(-z) = -f(z) f(z) has only 2 fixpoints. those fixpoints are -1,+1. f ' (-1) = -1/a <=> f ' (1) = 1/a and a > 1. if lim n-> oo a^n (f^[n](z) - f[2n](z)) exists and is meromorphic on C\D then this is the superfunction matching both fixpoints. regards tommy1729 RE: conjecture 2 fixpoints - tommy1729 - 08/12/2010
similarly : let f(z) be a non-periodic entire function. let f(z) = f(-z) f(z) has only 2 fixpoints ; real x and -x. f ' (x) = 1/a and a > 1. if lim n-> oo a^n (f^[n](z) - f^[2n](z)) exists and is entire then this is the superfunction matching both fixpoints. regards tommy1729 RE: conjecture 2 fixpoints - tommy1729 - 08/12/2010
im beginning to doubt ... the symmetry seems very wrong. if f(z) = f(-z) or -f(-z) then how are the fixpoints symmetric ??? i think only the following makes sense : (up to a linear transform ) for a nonperiodic entire function with only attractive fixpoints : superfunction = lim n-> oo (f^(n)[z] - f^(2n)[z])/D f^(n)[z] and using l'hospital rule if necc. the 'D' stands for derivate and perhaps this might not work and will need to be replaced by f ' [f^(3n)[z]] ^n. one of the assumptions is lim n -> oo D f^(n)[z] = O ( f ' [f^(3n)[z]] ^n ) but im not sure about that. if the superfunctions as defined above maps C to C/(const) that would be intresting ; they are candidates for being entire superfunctions consistant with all fixpoints. we might be able to plug in a periodic theta function to get rid of our problems ... but the problem with that is that a small variation might lead f^(n)[z + theta(z)] to go to another fixpoint and thus causing havoc. however not necc for all f(z) and all theta(z) , there is still hope for a theta. i might remove the other posts in this thread later ... tommy1729 RE: conjecture 2 fixpoints - tommy1729 - 08/15/2010
ok , i think i feel a proof coming. why 2 distinct finite fixpoints never match ... |