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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

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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: 1,358

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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: 1,358

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Joined: Feb 2009

ok , i think i feel a proof coming.

why 2 distinct finite fixpoints never match ...