07/02/2010, 03:52 PM
(This post was last modified: 07/02/2010, 05:40 PM by sheldonison.)

(06/28/2010, 11:18 PM)tommy1729 Wrote: (question 3)I think all of the other fixed points involve non-primary branch points of the logarithm. Those other fixed points show up via their primary natural logarithm branches. In Knesser's solution, the primary fixed point via its primary branch is at +/- i*infinity, but it does show up via its non primary branches.

what happens to limit cycles and n-ary fixpoints ??

sure we can set the fixpoints exp(L) = L at oo i but how about the fixpoints of exp(exp(.. q)) = q and limit cycles of the exp iterations ...

e.g. let e^q1 = q2 , e^q2 = q1 , if we want half-iterates , 1/3 iterates and sqrt(2) iterates to have the same fixpoints , this is a problem , not ?

perhaps we can 'hide' L at +/- oo i ( like kouznetsov ) and 'hide' the other points at - oo ??

(question 4)

For example, the secondary fixed point of "e" is ~2.062+i7.589, but it's primary logarithm is ~2.062+i1.305. The inverse superfunction of (2.062+i1.305) is well defined.

tommy1729 Wrote:i meant sqrt(e) !Tommy is referring to his proposed solution based on iterates of e^(kx)+e^(-kx). The equation has a fixed point of zero. The fixed point is diverging for k>0.5, and for k>0.5 the superfunction grows superexponentially. K=0.5 corresponds to the sqrt(e). Then you iterate logarithms for base e^k (analogous to the base change). However, the resulting equation may only be well defined for real numbers.

because my method works for bases > sqrt(e) , not for all bases > e^(1/e).

- Sheldon