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
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