01/25/2010, 07:30 PM

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.

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.