04/16/2012, 04:21 AM

(04/15/2012, 04:30 PM)sheldonison Wrote: This post is copied from: Problems from the Problem Sessions at the Snowbird Conference on the 25th Birthday of the Mandelbrot Set

6. (Douady) Persistence of the Fatou Coordinate:

First consider the polynomial for a real small . This polynomial has two repelling fixed points z1,z2 such that . Let be the vertical line connecting them. is a curve connecting the fixed points to the right of .

We define a Fatou coordinate on a neighborhood that contains , the region between them, but NOT the endpoints z1; z2 as the map satisfying

.

We can vary into the complex plane. As its argument increases we can follow . The deformed curves and spiral into the fixed points. As 1/4 + crosses into the main cardioid of the Mandelbrot set, one of the fixed points, say z1, becomes attracting. is still defined. The curves spiral in at the attracting fixed point; is attracted "inside" . When we reach the boundary of the cardioid again (with argument of ), the spiral breaks. That is, because the fixed point becomes indifferent, the spiral is no longer defined; cannot be attracted "inside" . Thus the continuation of the Fatou coordinate breaks down at the second crossing (below the real axis) of the cardioid.

The construction can be made exactly the same way by making the argument of negative. Thus there are two overlapping regions of the parameter plane in which a Fatou coordinate is defined. Each can only be defined for a single crossing of the cardioid.

The Fatou coordinate can be defined near a parabolic point for an arbitrary rational or entire map. The question is to understand the obstruction to extending the coordinate in this setting. Is the obstruction local or global?

I don't know the status of Douady's problem. In his question, is the Abel function or Fatou coordinate near a parabolic fixed point, analogous to the Fatou coordinate, for the bipolar tetration solution for where . One can compare Douady's problem above, with my post a few weeks ago, on the analogous problem for tetration. Both problems behave similarly, and both may have analogous Riemann surfaces. Also, I have generated some solutions for the superfunction, for , and there is some pretty bizarre behavior if corresponds to an indifferent fixed point with a rational period, corresponding somewhat to pictures I posted in this post, for the analogous tetration bipolar solution where the fixed point has a period=5.

If anyone else has any related information, or publications, I'd be very interested.

- Sheldon Levenstein

(03/07/2012, 12:08 AM)sheldonison Wrote: ....

But first, I briefly want to describe the merged tetration solution and the Shell Thron boundary, and briefly describe reasons why one might not expect there to be solutions at the Shell Thron boundary boundary. Briefly, start with real tetration at the real axis, for bases>eta=e^(1/e), and rotate either clockwise, or counter clockwise. Initially, both fixed points are repelling. Consider the case where we rotate counterclockwise, through increasing imag(z). Initially, the period for the upper fixed point will have positive real and imaginary components. At the Shell Thron boundary, the upper fixed point has a real period; the lower fixed point is still repelling. I call this the first Shell Thron boundary crossing. Before continuing, I would briefly state that as we continue rotating around eta counterclockwise, we will reach the real axis for points<eta, and then continuing further, we will reach the Shell Thron boundary a second time. I believe there is a singularity at the second Shell Thron boundary crossing, but there is not a singularity at the first Shell Thron boundary crossing.

....

Very interesting. I noticed here:

http://math.eretrandre.org/tetrationforu...e=threaded

that as the STR boundary is approached for the second crossing, the lines of singularities in the upper half plane get denser. It seems that when we hit the boundary, they should become "infinitely dense", killing the holomorphizm and apparently signalling the reaching of a natural boundary of analyticity at the STR boundary. This behavior would be much akin to that of the single-fixpoint regular iterations, though in this case it does not prevent the existence of a branch (e.g. the principal one) that is defined over the whole plane minus some singularities (and a cut, if it must be continuous over its whole domain).