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Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main cardioid involving exp(2 pi i theta) but I could not find the formula.

I'm particularly interested in finding the exact location of the attachment points of the bulbs 3 (the largest bulbs on the side of the main cardioid).

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I believe the point you are interested is the root of the basilica component and is at c=-3/4. – DJBruce May 6 '14 at 19:53
Also the equation for the main cardioid is given by $.5e^{i\theta}-(.5e^{i\theta})^2$. – DJBruce May 6 '14 at 19:58
Thank you, Bruce. Yes, this is the equation I was looking for. Actually the points I was looking for are the ones where the "1/3"(?) bulbs connect to the main cardioid at x=1/8 on the top and bottom. My interest was to prove or disprove that these two points together with the point at x=-1.25 (the left most side of the 0.5 radii circle located at x=-1) form an equilateral triangle, which appear that they actually do. – DolphinDream May 7 '14 at 21:09
up vote 5 down vote accepted

To elaborate a bit more on my comments:

Given a hyperbolic component $U\subset M$ of the Mandelbrot set -- i.e. a connected component of the interior of $M$ such that for all $c\in U$ the polynomial $p_c(z)=z^2+c$ has an attracting cycle -- the length of attracting cycle of $p_c(z)$ is constant for all $c\in U$. Now if $U$ is an hyperbolic component of period $n$ one can show that the multiplier map $\mu:U\rightarrow \mathbb{D}$ given by $c\mapsto (p^n)'(Z(c))$ where $Z( c)$ is a point in the attracting $n$-cycle of $p_c(z)$ is an isomorphism. In fact this map can be extend continuously to an injective map over the boundary of $U$.

We then define the root of a hyperbolic component $U$ to be $\mu^{-1}(1)$. Note that this is necessarily a point in the boundary of $U$, and that if $\mu^{-1}(1)=c_1$ then the polynomial $p_{c_1}(z)=z^2+c_1$ has a parabolic cycle (of length possibly less then n) of multiplier one.

There are two different shapes of hyperbolic components primitive components - ones that have a cusp likes the main cardioid - and satellite components ones that don't have a cusp like the basilica component. Similarly, we call the root of primitive component a primitive root and likewise for the root of a satellite component. These satellite roots are the points of attachment, which are what you appear to be interested in.

For example, if you are interested in the basilica component -- the hyperbolic component of period two (i.e. the large one to the left of the main cardioid) -- then the root is $-3/4$. If solve for the two cycles of $f(z)=z^2-3/4$ you'll notice that the solutions are degenerate; in that they are fixed points with the solutions being $z=-1/2$ and $z=3/2$. Calculating $(f^2)'(z)$ of each of these points we see that $(f^2)'(-1/2)=1$ and so as claimed $-3/4$ is ththe root of this component. If you are to look at a picture of the Mandelbrot set you'll see that this appears to be where the main cardioid meets the basilica component. Note this is an example where the root parameter has smaller cycle than its hyperbolic component.

Finally, this can be used to find the equation for the boundary of the main cardioid. The main cardioid is the hyperbolic component $U\subset M$ such that for all $c\in U$ the polynomial $p_c(z)=z^2+c$ has an attracting fixed point. Now the boundary of $U$ is $\mu^{-1}(\partial\mathbb{D})$. Using this one can then solve for the equation giving the boundary of $U$, which is $.5e^{i\theta}-(.5e^{i\theta})^2$.

I think this might be discussed in Complex Dynamics by Carleson and Gamelin, but I do not currently have a copy to double check.

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in the parameterization where the main cardioid is a circle, the bulbs are attached at rational angles $\phi=2\pi m/n$: see R.L. Devaney, The Mandelbrot bulbs.

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I was referring to the main cardioid (the heart looking shape part of the Mandelbrot set). – DolphinDream May 7 '14 at 21:00

Actually the answer to my question seem to have been there also on Wikipedia already under the Mandelbrot Set! in the section "Main cardioid and period bulbs".

Also, R.L. Devaney's kind response to my question confirms this:

"Here is the parameterization. On the boundary of the main cardioid, the map $z^2 + c$ has a neutral fixed point, i.e., a fixed point for which the derivative is $e^{2 \pi i \theta}$. So the boundary is given by solving the two equations:

$$z^2 + c = z $$ (that give a fixed point) $$2z = e^{2 \pi i \theta}$$

Solving these equations yields:

$$c = z - z^2 = \frac{1}{2}e^{2 \pi i \theta} - \frac{1}{4}e^{4 \pi i \theta}$$

That is the parameterization of the boundary of the main cardioid. When $\theta$ is rational, you are at a point where a bulb just meets this boundary. "

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