[UFO] - a contradiction in assuming continuous tetration?

[mike3]

I tried plotting it with "exact" value and the bounds were the same as on my first graph down to 5 digits, so the curves would not be visibly distinguishable. Though my tet-approximation is only accurate to like 14-15 digits or so here. So the regular is somewhat close, but not equal, to the tetrational at this line. That makes sense, though, since the tetrational is asymptotic to the regular iteration (and the corresponding one at the conjugate fixed point) in the direction of .

[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

[Gottfried]

I tried plotting it with "exact" value and the bounds were the same as on my first graph down to 5 digits, so the curves would not be visibly distinguishable. Though my tet-approximation is only accurate to like 14-15 digits or so here. So the regular is somewhat close, but not equal, to the tetrational at this line. That makes sense, though, since the tetrational is asymptotic to the regular iteration (and the corresponding one at the conjugate fixed point) in the direction of .

Ok, thanks! Good to know. Perhaps we could produce an overlay-plot to make the difference visible (for instance for illustration in some FAQ or introductory text)? I could send you my data or vice versa.

I started some different tries as an answer to the other questions - the consequences of the multivaluedness, but the matter is somehow extremely dense; perhaps I can put my further problems in readable words tomorrow.

Good night -

Gottfried

[Gottfried]

Hi Mike,

I think I've the source of the problem/of my lack of understanding now.
I couldn't accespt, that the well known many-to-one-transformation of the exp, which does not include a one-to-many transformation (multivaluedness) for the integer-values of a (iteration-)height-parameter, should be find a generalization that for such fractional positive delta-heights actually also a one-to-many-relation occurs.

To make this visually I tried to understand the key-issue if I compared two assumptions of real height-trajectories:
a) the pathes of the many-to-one come from some height z(h-1), meet at z(h) and all following z(h+k), but have different continuous trajectories for fractional {dh} at z(h+{dh})
b) after the trajectories meet at z(h) all trajectories follow the same path.

Here is a sketch illustrating a): (rightclick to enlarge)




and here another one illustrating b):




Well, intuitively I meant that some fixed function f°dh(x), which implements the real-height-iteration from some point x=z0 , for instance by a certain powerseries, cannot provide discrete differing outputs or even continuously "smeared" directions if I assume only one real parameter changing, so this would prefer version b) over a). On the other hand, if I got the following corrctly: "two powerseries, which produce the same continuous curve over the same finite range x0..x1 must be the same, thus the two powerseries must be identical - and can not have different values elsewhere" (paraphrased from memory) so it would be a strange adventure to try to find some solution for this model...

To improve my intuition I looked at a simpler example, the iteration of the function f(x) = x^3. For this a simple continuous iteration can be formulated F(x,h) = x^(3^h) . Clearly, applied to negative or even complex values z0 noninteger heights h can produce chaotic values due to the multivaluedness of log. (I'll provide some pictures in a second post). What I discussed was the application of continuous iteration starting at some complex value z0 = 1+ 0.5*I .
Because for any value z we have the three complex sources w0<>w1<>w2 where w0^3 (= w0^^1) = w1^3 (=w1^^1) = w2^3 ( = w2^^1) = z and also w0^^2 = w1^^2 = w2^^2 = z^^1 = z^3 I assumed, the trajectories for fractional heights h beginning at w0,w1 and w2 would combine at z0 and proceed from there jointly. But this is not true - and the functions f(x) and F(x,h) are easy enough to trust the correctness of computation.

The (possibly) best answer is illuminating: if I express the complex w1,w2 in height-differences of the same source w0 then they have complex heights with *different imaginary* parts.
Well, it happens (what I wouldn't have believed) that some complex value (w0) has descendants whose rectangular complex representation matches when iterated to some different complex heights h0, h1 and h2.
But in my example I assume only real height-differences, which means that only the real part of the height parameter changes. And thus the continuous trajectories from different w0,w1,w2 along *real* heights is in general distinct (and may sometimes match).

The essential here seems to be, that delta-h is in fact complex and not only real; it has a "length" and a "direction" , thus two, and not only one parameter - contrary to my intuition, that delta-h is only real.

---

Well, that all shall need a certain time for me to be taken in more depth and to get the picture for the tetrational function sharper.


I have two more points yet.

• First is, that I played with the curve for z0^^h with z0=1+0.5*I using the sexp, which we looked at earlier in this thread. When developed one more degree of integer height the curve becomes extremely chaotic. Still I suspect, that this chaos is some two-dimensional Runge-oscillation similar to the known Runge-effect in the one-dimensional case if we interpolate functions by approximation using polynomials of increasing orders. Since interpolations for the tetration are often developed using that same paradigm (stepwise extension of finite polynomials to infinite powerseries, regular tetration as I employ that) I have always some "alarm" in some edge of my brain.
  So I tried what would happen, if I correct the trajectory between, say, h1=1 and h2=2 a bit, manually, and see what would the resulting curve for h2..h3 look like then. It was impressive, that the double winding actually could be calmed down to a non-winded, (even "shorter") path. Well, I could not yet make a system out of this, and due to the unquestionable points at integer heights there will still be some winding and self-crossing, but I am impressed, that an improvement (?) is possible at all!
  
  Here we see a reduced winding compared to our first curve at this height interval:
  
  
  
  and here the detail of the range where I manually adapted the interpolation as found by regular tetration. I used changes at the set of only 8 coordinates, but with a small program I'm confident we could do this at a much finer grid (don't know whether I'll have time next week)
  
  
  
  

  ---

  
  
• The second is only a question:
  
  

  That makes sense, though, since the tetrational is asymptotic to the regular iteration (and the corresponding one at the conjugate fixed point) in the direction of .

  
  In which way do you see there an asymptotic, or better(?): by what formula is such an asymptotic existent?
  

Regards -

Gottfried

[tommy1729]

darn. i wanted to post that today.

f(x) = x^2

f(x) = x^3

f(x) = x^sqrt(2)

are imho the best examples to understand and investigate the seemingly contradiction.

i often argue about branches and (-1)^sqrt(2).
( related to the irrationality of sqrt(2) and thompsons lamp paradox )

sometimes even consider (-1)^(irrational real) to be the unit circle , which is controversial ( branches are considered 'countable' even when 'dense' but binary , ternary or continued fraction expansions may lead to another ' interpretation ' ).

afterall after considering the superfunctions of x^sqrt(2) , x^2 , x^3 and x^sqrt(3) or their inverses , we see a striking resemblence that cannot be ignored !!

( an animation of changing riemann surfaces with a real parameter would be nice )

square roots and cube roots are school examples of functions that are complex abs continu , but not complex differentiable. ( more *nice* examples are welcome , where *nice* probably means analytic almost everywhere , however i suppose al *nice* examples are inverses of polynomials or similar looking .. not ? )

im very intrested in complex abs continu functions and i believe they are important and natural.

but its a long time ago i studied these , is there a terminology for complex abs continu functions defined on all of z ( as analogue of entire function ) ?

i especially like complex abs continu functions that are still continu at oo too.

furthermore

x^a has a superfunction agreeing on all fixpoints !!

( or at least for integer a and especially 0 and 1 )

( i notice that what makes this rare property may be the combination of complex abs continu + = to secondary , ternary , ... fixpoint expansion ?!? )

i would like to see more superfunctions agreeing on their fixpoints !

btw , maybe g^[-1]((g(x))^a) for suitable g(x) can be used to construct all superfunctions that agree upon fixpoints and are complex abs continu ???

( like we did for solving the iterations of exp(x)+x ; replacing the fixpoint at - oo to finite )

however if im not mistaken in my rapid brainstorm , this puts restrictions on the fixpoints ( repelling in direction region Q etc ) so maybe there may be other examples of superfunctions agreeing on fixpoints that match the other possible properies of the fixpoints ...

btw , how does one link the disagreement of half-iterates on multiple fixpoints to the superfunction with the fixpoints at complex oo ?

i assume thats simple : either the superfunction does not go to oo ( not defined beyond abs(z) = X ) , or it is not constant near the fixpoint but periodic , since if at fixpoint 2 the halfiterate gives value B then the half-iterate of B = fixpoint 2 ...

intresting stuff. basic yet intresting.

forgive my brainstorm type of writing ...

regards

tommy1729

[Gottfried]

Well, here are some pictures for the iteration of the cubic.

This is the plot for the first half-iterate from the three cubic roots z0,z1,z2 of Z=1+0.5*I. Just applied four times using z=z^(3^0.5), for the three z0,z1,z2, then z=z^(3^0.5), and so on, just a couple of times. clearly the naive computation of complex z to the squareroot(3) does not provide meaningful graphs - neither for the integer and even less for the spline-interpolated coordinates of a plot. Well, that's true for the half-iterates of z1 and z2, for the half iterates of z0 the plot seems meaningful:

   

The direct source of such a chaos is the behave of the software to provide the imaginary value for such a complex root as positive or negative value in a restricted range for arg(z) between -Pi..Pi because it is just modular to 2*Pi. Correcting at least for the appropriate phi for the quadrant in which z resides gives that far better looking graph, where we even can insert smaller iteration steps.

   

but still with some problem, when the iteration reaches the border of a quadrant, the arg()-function for the iterates oscillates and must be corrected. The detail shows an entertaining oscillation at the negative real axis...

   

Another correction for the imaginary values, when iterated is tried:

   

and is even better, but still not perfect. At least we see the three roots much clearer and how they do not lay on the same trajectory. But the computation is still not optimal: now the crossing of the real axis creates problems. See the detail:

   

Thus I improved the computation using integer iteration to some small roots first, say instead of z1 using z1^^-10 as basis for a trajectory, compute the regular fractional iterates for one unit interval, say y1, and then retransform to y^^10. This gives the acceptable trajectories of the next pictures, shown in the following post (the MyBB-software seems to not allow more than 4 or five attachments per post...)

Gottfried

[tommy1729]

gottfried , i usually like your plots.

what do you use ?

[Gottfried]

Next I show the last two plots with final(?) computation of the regular iteration.
First we see, that as expected, at integer heights the trajectories match, and the integer iterates of the three roots join at the expected powers of Z

   

However, the trajectories do not combine after the first match, what were that, what I expected initially. The trajectories cross each other without joining, and the trajectory of z2 seems does two times more windings than that of z1

   

The two following plots simply show the extreme winding if the height is increased to the next unit interval. I've no more comment there, only that this introduces the question, whether there could be some normalizing of the continuous iteration-process, which allows to trajectories to join. And if that is possible (and meaningful) whether such a "normalization" has some impact for the sexp-iteration.

Gottfried

   

   

[Gottfried]

gottfried , i usually like your plots.

what do you use ?

Thanks! It's just excel, to which I transfer the data computed using Pari/GP

Gottfried