claudeNov 16, 2019

https://arxiv.org/abs/1911.03796 #amreading #maths #MandelbrotSet #ExternalRays

A formula to transform rays from hyperbolic components to the real axis.

New to me terminology:
- vein (not exactly sure what this is, paper needs more diagrams and examples in binary instead of decimal)
- pseudocentre (tip of ray pair or vein)
- complexity (length of binary expansion of dyadic rational / pseudocentre / vein)
- kokopelli component (upper period 4 island)
- maximally diverse sequence

Generalizations of Douady's magic formula

We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components $H$ of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on $H$ to the set of angles of external rays landing on the real axis.

Show threadclaudeNov 16, 2019

I must be missing something as this seems trivial?

Rays landing on H
--inverse tuning-->
Rays landing on P1 cardioid
--Douady magic formula-->
Rays landing on R

But indeed the example has the root of P4 island mapping near the tip of the antenna instead of the cusp.

So maybe all the Hubbard tree gymnastics are necessary and it's more to do with veins than islands.

Show threadclaudeNov 16, 2019
I still don't really know what a Hubbard tree is.
Show threadclaudeNov 17, 2019

The key fact used by the proofs seems to be that if an angle $t$ is nearer to 1/2 than any of its images under doubling (mod 1) then its external ray lands on the real axis. I think this is proven in one of the referenced papers.

The magic formula in section 3 is much more comprehensible than the one based on veins: simply add enough 01111... or 10000... such that two periods worth of angle doubling are taken care of, then the remaining digits have both 0 and 1 in each block of p bits, so must be further from 1/2. And the rays are not renormalizable so they land by Yoccoz' theorem (apparently).

Show threadclaudeNov 17, 2019

Diagram of the rays mentioned in definition 6, plus the rays of the principal 1/3 hub (dotted). Pseudocenters are dashed, ray pairs are solid.

I used my https://mathr.co.uk/mandelbrot/web/ to convert external angles from decimal to binary, and made the diagram with m-perturbator-gtk.

Show threadclaude

If I read this right, different angles landing on the same point can have different Hubbard trees? This seems counterintuitive.

37/224 lands with 39/224, 43/224 but is in (1/7, 39/224)

@OCRbot #HubbardTree #ExternalRays

Nov 17, 2019 at 6:43pmWeb
Show threadOCR BotNov 17, 2019

@mathr
Ray pairs are partially ordered: in fact, we say (0;,0/) ~ (03,03) if the leaf
(0, , 07) separates (63,03) from 0. Let us denote as N(@​) the number of ends of the
Hubbard tree associated to the angle 0. Recall that if 0, ~ 2, then N(01,) < N(62).

A dyadic number 6 defines a combinatorial vein in the Mandelbrot set as follows.

Definition 6. Given a dyadic rational number 09, we define the combinatorial vein
of 09 as the set of ray pairs (01,02) such that:
(1) 09 is the pseudocenter of (61, 02);
(2) N() = Nay
For instance, if 4) = }, then the vein extends all the way to (4, 2), since N(}) = 3,

and one can check easily that N(3) = 3 (the “rabbit”). On the other hand, if 6) = 4,

then the vein extends up to (saa dis, Indeed, N() = 5, and N(@​) = 3if 0 € (2, 33).

Note that condition (2) in the above definition is needed. For instance, if one

considers the ray pair (01,02) = (£5, 8), then its pseudocenter is 0) = 39/128.

However, one checks that N(<4) = 6 < 8 = N(38) (see Figure 1.3).

Definition 7. The complexity of a vein V with pseudocenter 0) is given by
dy = II@​olI — 1.