Tetration Forum

In some talks with dynamics systems experts it appears they solved some problems we are interested in, though they dont know about it. This is about existence of Abel functions defined on a sickle between two fixed points (in the complex plane) and also about uniqueness of such Abel functions.

Complex dynamics is a currently flourishing field and uses a different terminology than we use. For example what we call "Abel function", they call it "Fatou coordinates".

The keyword is "parabolic implosion", you even find online articles when googeling. I will base my explanations on the article of Mitsushiro Shishikura in the book "The mandelbrot set, theme and variations". His article in the book is named "Bifurcation of parabolic fixed points", and is rather advanced.

The basic idea is the following:
We start with a parabolic fixed point \( z_0 \) of a holomorphic function \( f \), i.e. \( f(z_0)=z_0 \) and \( f'(z_0)=1 \).
If we slightly perturb this function by a complex \( \epsilon \), \( g(z)=f(z)+\epsilon \), then the one fixed point splits up into several fixed points (at least 2). But the perturbed function still behaves quite similar to the original function.

From the classic theory about Abel functions at a parabolic fixed point (basically developed by Ecalle), we know that there is a Leau-Fatou-flower (see the online book of Milnor: "dynamics in one complex variable" for detailed explanation), i.e. alternating attracting and repulsing petals, of a flower with center in \( z_0 \). The number of petals is \( 2(n+1) \) where \( n \) is the number of vanishing derivatives in \( z_0 \) after the first derivative (which is 1).

For example for the function \( f(x)=b^x \) with \( b=e^{1/e} \). Its second derivative is non-zero, so \( n=0 \) and hence there are two petals. And indeed one petal covers the real axis \( x>e \), where \( f \) is repelling and the other petal covers the real axis \( x<e \) where \( f \) is attracting.

If we slightly perturb this function by adding \( \epsilon>0 \) or by just increasing the base a little then the one parabolic fixed point splits up into two conjugate repelling fixed points. If we slightly perturb with \( \epsilon<0 \) then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling).

The classic unperturbed theory says now that we can develop an injective Abel function/Fatou coordinate \( \alpha \) on each petal. And this Abel function is uniquely determined by the demand that the resulting fractional/complex iterates \( f^{[t]}(z)=\alpha^{[-1]}(t+\alpha(z)) \) have an asymptotic power series development in \( z_0 \). I always call it here the regular Abel function and the regular iterates.

The current complex dynamics theory says that for small enough epsilon, we find a sickle between the perturbed fixed point pair, and an injective Fatou coordinate on that sickle, which has also a not so explicit formulated uniqueness. Moreover this Fatou coordinate depends holomorphically on \( \epsilon \)!

In some other post on this forum I mentioned that the regular Abel function/regular iterate of \( f(x)=b^x \) at the lower fixed point for \( b<e^{1/e} \) is probably not holomorphically continuable in \( b \) to \( b=e^{1/e} \). However if one uses the perturbed Fatou coordinates for \( b<e^{1/e} \) and not the regular ones at the lower fixed point, then it (and the resulting fractional iterates) depends holomorphically on \( b \)!
I have to revalidate this statement, but it appears very promising.

As to the uniqueness argument, it indeed appears to be similar to the one I developed in my article.

(01/25/2010, 07:30 PM)bo198214 Wrote: [ -> ]In some other post on this forum I mentioned that the regular Abel function/regular iterate of \( f(x)=b^x \) at the lower fixed point for \( b<e^{1/e} \) is probably not holomorphically continuable in \( b \) to \( b=e^{1/e} \). However if one uses the perturbed Fatou coordinates for \( b<e^{1/e} \) and not the regular ones at the lower fixed point, then it (and the resulting fractional iterates) depends holomorphically on \( b \)!
I have to revalidate this statement, but it appears very promising.

Do you have proof that this is the case? As this could mean the regular iteration is the "wrong" way to do tetration, not the "right" one. It would be interesting to compare the graph of tetration at some base, say \( \sqrt{2} \), obtained through the regular iteration, to that obtained through this method, esp. on the complex plane, and also to determine the magnitude of the disagreement between the two at the real axis.

(01/26/2010, 06:21 AM)mike3 Wrote: [ -> ]It would be interesting to compare the graph of tetration at some base, say \( \sqrt{2} \), obtained through the regular iteration, to that obtained through this method, esp. on the complex plane, and also to determine the magnitude of the disagreement between the two at the real axis.

Yes, yes, yes. I am still not that familiar with the theory and technique to give definitive answers. So far it seems these pertubed Fatou coordinates only give real analytic functions for conjugated complex fixed point pairs. To give it a distinguished name (and as "perturbured Fatou coordinates" does not really hit the point) I will call it from here bipolar iteration in mnemonic that the Abel function is defined on a sickel between *two* fixed points, in opposite to the regular iteration which is defined in a neighborhood of *one* fixed point.

First, I dont know yet effective methods to compute these bipolar Fatou coordinates, though I think that Dmitriis algorithm should yield the bipolar superfunction.

Second, the bipolar Fatou coordinates may exist between two real fixed points (though even that is not completely clear to me and not backed by the theory) but it may not be real analytic.

So the question is whether there is at all a real tetration on (1,oo) which is particularly analytic in the base at e^(1/e) ...

(03/08/2010, 11:59 AM)bo198214 Wrote: [ -> ]Yes, yes, yes. I am still not that familiar with the theory and technique to give definitive answers. So far it seems these pertubed Fatou coordinates only give real analytic functions for conjugated complex fixed point pairs. To give it a distinguished name (and as "perturbured Fatou coordinates" does not really hit the point) I will call it from here bipolar iteration in mnemonic that the Abel function is defined on a sickel between *two* fixed points, in opposite to the regular iteration which is defined in a neighborhood of *one* fixed point.

First, I dont know yet effective methods to compute these bipolar Fatou coordinates, though I think that Dmitriis algorithm should yield the bipolar superfunction.

Second, the bipolar Fatou coordinates may exist between two real fixed points (though even that is not completely clear to me and not backed by the theory) but it may not be real analytic.

So the question is whether there is at all a real tetration on (1,oo) which is particularly analytic in the base at e^(1/e) ...

Hmm. This suggests the possibility there is an alternative solution instead of regular iteration for the bases in \( (1, e^{1/e}] \) (after all, if it's analytic at \( e^{1/e} \) and the regular isn't, the two can't be equal if they're both analytic in that interval.).

(03/10/2010, 03:10 AM)mike3 Wrote: [ -> ]Hmm. This suggests the possibility there is an alternative solution instead of regular iteration for the bases in \( (1, e^{1/e}] \) (after all, if it's analytic at \( e^{1/e} \) and the regular isn't, the two can't be equal if they're both analytic in that interval.).

Oh, my explanation was misunderstandable, the bipolar Abel function is always holomorphic and injective on a sickel between the two fixed points. But it may not be real on the real axis, or perhaps not even defined on the real axis, in the case of two real fixed points. You know the sickel would be above or below the real axis or possibly even wind around the fixed points, so being defined on pieces of the real axis; and it is not clear whether it can be extended to the real axis between the fixed points.

On the other hand to obtain alternative solutions you can always build up linear combinations of the two regular Abel functions \( \alpha \), \( \beta \):
\( \gamma(z)=c\alpha(z)+(1-c)\beta(z) \) which is again an Abel function:
\( \gamma(f(z))=c(\alpha(z)+1)+(1-c)(\beta(z)+1)=\gamma(z)+1 \)

(03/10/2010, 11:13 AM)bo198214 Wrote: [ -> ]

(03/10/2010, 03:10 AM)mike3 Wrote: [ -> ]Hmm. This suggests the possibility there is an alternative solution instead of regular iteration for the bases in \( (1, e^{1/e}] \) (after all, if it's analytic at \( e^{1/e} \) and the regular isn't, the two can't be equal if they're both analytic in that interval.).

Oh, my explanation was misunderstandable, the bipolar Abel function is always holomorphic and injective on a sickel between the two fixed points. But it may not be real on the real axis, or perhaps not even defined on the real axis, in the case of two real fixed points. You know the sickel would be above or below the real axis or possibly even wind around the fixed points, so being defined on pieces of the real axis; and it is not clear whether it can be extended to the real axis between the fixed points.

On the other hand to obtain alternative solutions you can always build up linear combinations of the two regular Abel functions \( \alpha \), \( \beta \):
\( \gamma(z)=c\alpha(z)+(1-c)\beta(z) \) which is again an Abel function:
\( \gamma(f(z))=c(\alpha(z)+1)+(1-c)(\beta(z)+1)=\gamma(z)+1 \)

However, for it to be not real valued in \( (1, e^{1/e}] \) and real valued in \( (e^{1/e}, \infty) \) would imply there is a singularity/branchpoint at \( b = e^{1/e} \), hence not holomorphic there, eh?

(03/20/2010, 08:12 AM)mike3 Wrote: [ -> ]However, for it to be not real valued in \( (1, e^{1/e}] \) and real valued in \( (e^{1/e}, \infty) \) would imply there is a singularity/branchpoint at \( b = e^{1/e} \), hence not holomorphic there, eh?

ya I would say so, that each non-integer iterate has a singularity as a function in \( b \) at \( e^{1/e} \).

(03/21/2010, 01:04 PM)bo198214 Wrote: [ -> ]

(03/20/2010, 08:12 AM)mike3 Wrote: [ -> ]However, for it to be not real valued in \( (1, e^{1/e}] \) and real valued in \( (e^{1/e}, \infty) \) would imply there is a singularity/branchpoint at \( b = e^{1/e} \), hence not holomorphic there, eh?

ya I would say so, that each non-integer iterate has a singularity as a function in \( b \) at \( e^{1/e} \).

So does this mean the original hypothesis that it "depends holomorphically on \( b \)" (apparently across \( e^{1/e} \) since you contrast this behavior with that of the "usual" regular iteration) was wrong?

(03/21/2010, 11:03 PM)mike3 Wrote: [ -> ]So does this mean the original hypothesis that it "depends holomorphically on \( b \)" (apparently across \( e^{1/e} \) since you contrast this behavior with that of the "usual" regular iteration) was wrong?

Quite probably. I am still not sure about numeric computation of the bipolar Abel/super function. But my guess that it is not real valued for \( b<e^{1/e} \) stems also from a statement in Shishikura's article (proposition 3.2.3) which quite resembles Dmitrii Kouznetsov's algorithm to compute the superfunction. It says that the inverse of the bipolar Abel function, i.e. the superfunction \( \phi_f \) satisfies:
\( \lim_{\Im(w)\to +\infty}\phi_f(w) = p \) and \( \lim_{\Im(w)\to -\infty} \phi_f(w) = q \)
where \( p \) and \( q \) are the two fixed points of \( f \).
(which is exactly what Dmitrii uses for his construction of the superfunction.)
This implies that \( \phi_f \) is only real-valued if \( \overline{p}=q \) because real-analytic functions \( \phi \) satisfy \( \overline{\phi(z)}=\phi(\overline{z}) \).

Hmm. This suggests there are two quite distinct approaches to the tetration using fixpoints, each of which covers one of two seemingly vastly different domains. Namely, we have the Shell-Thron region wherein the regular iteration is used, which yields a solution that is real valued at the real axis, but this solution has (may have? Still need more rigorous proof) a natural boundary at the region border, so it cannot (might not?) be extensible outside said region. Outside that region, we have the rest of the plane, for which the extension would be achieved via the bipolar method, which may not be extensible inside the STR, or if it is, it cannot be real-valued for \( 1 < b < e^{1/e} \).

(BTW, I've been playing around with another tetration method based on trying to use the Borel summation on Ansus' continuum-sum formula. If you want, I can post some rough observations from an attempt at numerical approximation. I'm still not sure if it converges, as it seems to take tons of precision and terms to work, so I can't really press past more than a few decimals of accuracy.)