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For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :

Is it possible to find a part of the parameter plane, scanned with a given limited precision (rasterised) such that:

  • every pixel intersects the Mandelbrot set (or even the boundary of the Mandelbrot set), and
  • this part is not inside a hyperbolic component of the Mandelbrot set?
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3 Answers 3

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The Hairiness Conjecture, formulated by Milnor and proved by Lyubich ("Feigenbaum-Coullet-Tresser universality and Milnor’s Hairiness Conjecture", Annals of Mathematics, 1999) states that, near any real Feigenbaum parameter, the rescalings of the Mandelbrot set converge to the whole complex plane in the Hausdorff metric.

This means that, if you fix any resolution that you want, then at a sufficiently deep zoom near such a point the distance-estimate method will show you only black pixels.

EDIT. Let me elaborate also on Alex's answer (since this is going to be too long for a comment), and the difference between the Misiurewicz and Feigenbaum parameters.

As Alex points out, suppose that you fix your resolution, say 5000x5000 pixels. For simplicity, let us assume that our picture is always square, so that the resolution is determined by a single natural number, N, e.g. N=5000.

Then Alex's point is that you can find a Misiurewicz parameters $c_0$ such that, for sufficiently small magnification, the picture drawn around $c_0$ at resolution N by N contains only black pixels. Or, more mathematically: For all sufficiently small $\varepsilon$, if $c$ is such that $|c-c_0|<\varepsilon$, then $$\operatorname{dist}(c,\mathcal{M}) \leq \varepsilon/N \tag{*}$$.

If you know a little bit about the Mandelbrot set, this is quite obvious: take one of the "star" Misiurewicz parameters, which are at the end of a bulb of period $q$ bifurcating from the main cardioid, and which have $q$ "arms" attached to them. These arms are essentially equally spaced out (this follows e.g. from the picture in the dynamical plane, which is asymptotically self-similar, and which is repeated in the parameter plane by a theorem of Tan Lei). If the number $q$ is much bigger than the number $N$, then the claim follows.

On the other hand, if you fix a Misiurewicz parameter $c_0$ first, and then choose $N$ sufficiently large, you will always see something of the complement of the Mandelbrot set, no matter how closely you zoom in. (This is essentially for the same reason.)

The hairiness theorem implies that, for any real Feigenbaum parameter, this is not the case: No matter which resolution $N$ you choose, (*) will hold when $\varepsilon$ is sufficiently small.

For an even easier (though less direct) argument - again using a nontrivial theorem - recall that Shishikura proved that the boundary of the Mandelbrot set has Hausdorff dimension two. In particular, its box-counting dimension is two, which immediately implies the desired property ...

Moreover, the set of parameters at which the dimension is locally two is topologically generic in the boundary Mandelbrot set, so zooms around a generic boundary parameter will result in the desired picture.

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  • $\begingroup$ "The magnifications of M near any fixed-point of renormalization c are self-similar and fill the plane." math.harvard.edu/~ctm/expositions/home/text/talks/norway/… $\endgroup$
    – Adam
    Commented Dec 31, 2017 at 19:38
  • $\begingroup$ @Adam, yes this refers precisely to the hairiness conjecture/theorem. $\endgroup$
    – Lasse Rempe
    Commented Jan 15, 2018 at 0:31
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Yes, these are the neighborhoods of certain Misiuriewicz points, see for example the right picture in the bottom here: http://classes.yale.edu/fractals/MandelSet/MandelBoundary/Mis.html

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  • $\begingroup$ Yes, but some points in the neighborhood of this Misiurewicz point ( red pixel) are white pixels so probably do not contain any point of Mandelbrot set. $\endgroup$
    – Adam
    Commented Nov 23, 2014 at 8:14
  • $\begingroup$ All depends on the size of your pixels. Given a size, you can choose a proper Misiuriewicz point, and a neighborhood of it, so that the M-set will intersect every pixel. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 23, 2014 at 14:19
  • $\begingroup$ I can imagine it but only with "very big" pixel size ( very small number of pixels). In practica one take 1000x1000 pixels image. Choose subset of plane, then pixel size is constant. If I take smaller subset then pixel size is also smaller ( zoom). In such condition s I can't find subset of parameter plane where all pixels contain point of Mandelbrot set. Can you give such example ? $\endgroup$
    – Adam
    Commented Nov 23, 2014 at 15:48
  • $\begingroup$ In my previous comment I told you what is true. Mathematical result. This is what the theory says. If you implement this on a computer, the whole screen will look black of course. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 23, 2014 at 16:17
  • $\begingroup$ I suppose that difference between your and mine point of view is similar to : The probability that any random number : is irrational is almost 1 ( in theory because of cardinality ), is rational is 1 ( in numerical computations because of limited precision ) $\endgroup$
    – Adam
    Commented Nov 24, 2014 at 13:48
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enter image description here

Here are images of principal Misiurewicz points of wakes from 1/2 to 1/1024

Limit of this seequence is a dense image. More info is here

and here Zoom About Feigenbaum Point In The Mandelbrot Set Showing Hairiness ( Images by Claude Heiland-Allen ) enter image description here

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