[UFO] - a contradiction in assuming continuous tetration?

[tommy1729]

P.s. the self-crossings of the curve are really impressive! Do I really want a function like this...?

to know the truth you have to know exactly what you dont want

well e^z is not injective so most functions look like this.

i think we can - for general functions - put such curves into categories.

( ignoring subdivision for now )

the category then depends only on the initial z_0.

type 0 the curve converges to oo or a fixpoint.

type 1 the curve cycles without self-crossing.

type 2 the curve crosses itself at some places with 1 intersection.

type 3 the curve crosses itself at some places with 1 and 2 intersections. ( there must be places where 2 intersections occur , 2 intersections is meant as 3 curves crossing a common point )

type 4 the curve crosses itself at some places with more than 2 intersections.

( if i recall correct , it was conjectured or proven that in case of type 4 with infinite intersections there exists a point where n intersections exists for all n )

some questions follow naturally :

( in the recently posted thread by me : " period of exp exp exp " i already talked about it but apparantly it wasnt understood )

i called the first intersection of the curve caused by z_0 : the period of exp exp exp for z_0. where period means the amount of iterations (exp) we need to take to get to this point.

so when i ask for the period , i mean the first intersection of the curve.

can we compute or define it a priori ?

can we define it in terms of superfunctions and similar ?

it is known that in any neighbourhood of any nonreal z_0 there is a nonreal z_1 that goes to oo if z_0 doesnt , or doesnt go to oo if z_0 does , where 'goes to' means the limit of oo integer iterations of exp.

despite that it might still be possible that z_0 and z_1 have the same period / first intersection ?

( i assume so , and chaos then occurs at the following intersections comparisements )

( all of this is very related to fractals and chaos )

the pics posted by gottfried show nonreal z_0 curves of type 2 for exp.

probably there exist curves of type 0 and 1 for exp and nonreal z_0 too.

( even if z_0 isnt a fixpoint ? for entire functions with attractive fixpoints there must always be a type 0 curve even if z_0 isnt a fixpoint )

but does there exist a z_0 with 0 < im(z_0) < pi/2 such that the curve for exp is of type 3 ?

if i can follow sheldon , the answer to that last is yes.

but i would like to see it.

regards

tommy1729