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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
Posts: 1,924
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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
Posts: 1,924
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Joined: Feb 2009
08/12/2010, 07:12 PM
(This post was last modified: 08/12/2010, 07:32 PM by tommy1729.)
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
Posts: 1,924
Threads: 415
Joined: Feb 2009
ok , i think i feel a proof coming.
why 2 distinct finite fixpoints never match ...
