Attempt to find a limit point but each step needs doubling the precision...

[Gottfried]

Looking at the pics I can't help wonder :

How different is imaginary height for functions with 2 fixpoints from

functions with 2 parabolic fixpoints whose derivatives (at the fixpoints) are nonreal.

??

regards

tommy1729

Well the iteration of real x with real height (base b=sqrt(2)):

1) take any point x at the half line -oo to 2: forward-iteration y=(b^x) then y is again on the real halfline but nearer to the fixpoint 2; backwards iteration (y = log(x)/log(b)) if x>0 then y is still on the half line but farther away from 2; if x<0 then y becomes complex; if x=0 then y becomes negative infinity . Result for forward iteration: point t=2 is limit point and cannot be overstepped by as many forward iterations as you want.


2) take any point x at the real interval 2 to 4 (4 not included): forward-iteration y=(b^x) then y is again on that interval but nearer to the fixpoint 2; backwards iteration (y = log(x)/log(b)) y is nearer to the upper fixpoint 4. Result for forward iteration: point t=2 is limit point and cannot be overstepped by as many forward iterations as you want. For backward iterations: point t=4 is limit point and cannot be overstepped by as many forward iterations as you want.


3) take any point x at the half line 4 to +oo : forward-iteration y=(b^x) then y is again on the real halfline but farther away from the fixpoint 4; backwards iteration (y = log(x)/log(b)): y is still on the half line but nearer to 4; Result for backward iteration: point t=4 is limit point and cannot be overstepped by as many forward iterations as you want.


Iterations with complex height:
x is a point on the halfline -infinity to 2: y is in general complex and if h=Pi*I/log(log(2)) then y is real and in the interval 2..4. Thus with complex heights you can "overstep" the fixpoint to the other side.
That's what I'm interested in here.

But this gives an interesting relation: we choose some point y in the interval 2..4 and find its relative x in the negative halfline, call it y_0 when x_0 = 1, call it y_(-1) when x_(-1)=0 , call it y_(-2) when x_(-2) = - infinity. But there is also y_(-3) ; now what is x_(-3)? Is it log(-infinity)/log(log(2)) ? Anyway: we have some y_(-3) just by y_(-3) = log(y_(-2))/log(log(2)) and so on. Numerical computation usually shall give a *complex* value near the upper real halfline >4 and the software does not allow to find the expected, exactly real, value x_(-3).

And here is now my problem: if you can approximate the complex height very near to h = \rho = Pi*I/log(log(2)) then you come a bit nearer to the real axis, and when you get even nearer to \rho then cou come also nearer to the real axis - but to what rate? In the Trappmann/Kousnetzov-article at best we find some white area in the graph for F_(2,-1) . And my hope is that by some analytical formula we find that the true limit is something on the real axis, possibly positive infinity ( but I don't know).


Well, I'm not (yet) aware of some analytical expression for tetrations with imaginary or complex heights so at the moment I/we can only try to extend the precision to come nearer to the result and see the tendency.



Gottfried