Mandelbrot set

The Mandelbrot set is a particularly interesting fractal that has become popular far outside of mathematics both for its aesthetic appeal and its complicated structure, arising from a simple definition. This is largely due to the efforts of Benoit Mandelbrot and others, who worked hard to communicate this area of mathematics to the general public.

Mathematically, the Mandelbrot set $M\,$ is defined as the connectedness locus of the family

f_c:\mathbb{C}\to\mathbb{C}; z\mapsto z^2 +c.\,

of complex quadratic polynomials. That is, the Mandelbrot set is the subset of the complex plane consisting of those parameters

c\,

for which the Julia set of

f_c\,

is connected.

An equivalent way of defining $M\,$ is as the set of parameters $c\,$ for which the critical point $0\,$ does not tend to infinity. That is,

f_c^n(0)\not\to\infty,\,

where

f_c^n\,

is the

n\,

-fold composition of

f_c

with itself. This definition lends itself immediately to the production of computer generated renderings, (see below).

History

The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. The first pictures of it were drawn in 1978 by Brooks and Matelski as part of a study of Kleinian Groups.Robert Brooks and Peter Matelski, The dynamics of 2-generator subgroups of PSL(2,C), in "Riemann Surfaces and Related Topics", ed. Kra and Maskit, Ann. Math. Stud. 97, 65–71, ISBN 0691082642

Mandelbrot studied the parameter space of quadratic polynomials in an article which appeared in 1980.Benoit Mandelbrot, Fractal aspects of the iteration of $z\mapsto\lambda z(1-z)\,$ for complex $\lambda,z\,$ , Annals NY Acad. Sci. 357, 249/259 The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard,Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985) who established many fundamental properties of $M\,$ , and named the set in honor of Mandelbrot.

The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. It would be futile to attempt to make a list of all the mathematicians who have contributed to our understanding of this set since then, but such a list would certainly include Mikhail Lyubich, Curt McMullen, John Milnor, Mitsuhiro Shishikura and Jean-Christophe Yoccoz.

Basic properties

The Mandelbrot set is a compact set, contained in the closed disk of radius 2 around the origin. In fact, if

c\,

belongs to the Mandelbrot set, then

|f^n(c)|\leq 2\,

for all

n\geq 0\,

The intersection of $M\,$ with the real axis is precisely the interval ^*\,. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,

z\mapsto \lambda z(z-1),\quad \lambda\in

^*.\,

Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, they constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk.

(Mandelbrot had originally conjectured that the Mandelbrot set is disconnected, but it turns out that this conjecture was based on computer pictures generated by programs which are unable to detect the thin filaments connecting different parts of $M\,$ . Upon further experiments, he revised his conjecture, deciding that $M\,$ should be connected.)

The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of $M\,$ , gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms, and form the backbone of the Yoccoz parapuzzle.

The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters $c\,$ for which the dynamics changes abruptly under small changes of $c.$ It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting p₀=z, p_n=p_n-1²+z, and then interpreting the set of points |p_n(z)|=1 in the complex plane as a curve in the real Cartesian plane of degree 2ⁿ⁺¹ in x and y.

The main cardioid and period bulbs

Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This main cardioid is the region of parameters $c\,$ for which $f_c\,$ has an attracting fixed point. It consists of all parameters of the form

c = \frac{1-(\mu-1)^2}{4}

for some

\mu\,

in the open unit disk.

To the left of the main cardioid, attached to it at the point $c=-3/4\,$ , a circular-shaped bulb is visible. This bulb consists of those parameters $c\,$ for which $f_c\,$ has an attracting cycle of period 2. This set of parameters is an actual circle, namely that of radius $1/4\,$ around -1.

There are many other bulbs attached to the main cardioid: for every rational number $\frac{p}{q}$ , with p and q coprime, there is such a bulb attached at the parameter : $c_{\frac{p}{q}} = \frac{1 - \left(e^{2\pi i \frac{p}{q}}-1\right)^2}{4}.$

This bulb is called the $\frac{p}{q}$ -bulb of the Mandelbrot set. It consists of parameters which have an attracting cycle of period $q\,$ and combinatorial rotation number $\frac{p}{q}$ . More precisely, the $q\,$ periodic Fatou components containing the attracting cycle all touch at a common point (commonly called the $\alpha\,$ -fixed point). If we label these components $U_0,\dots,U_{q-1}\,$ in counterclockwise orientation, then $f_c\,$ maps the component $U_j\,$ to the component $U_{j+p\,(\operatorname{mod} q)}$ .

The change of behavior occurring at $c_{\frac{p}{q}}$ is known as a bifurcation: the attracting fixed point "collides" with a repelling period $q\,$ -cycle. As we pass through the bifurcation parameter into the $\frac{p}{q}$ -bulb, the attracting fixed point turns into a repelling fixed point (the $\alpha\,$ -fixed point), and the period $q\,$ -cycle becomes attracting.

Hyperbolic components

All the bulbs we encountered in the previous section were interior components of the Mandelbrot set in which the maps

f_c\,

have an attracting periodic cycle. Such components are called hyperbolic components.

It is conjectured that these are the only interior regions of $M\,$ . This problem, known as density of hyperbolicity, may be the most important open problem in the field of complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" components.

For real quadratic polynomials, this question was answered positively in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the Feigenbaum diagram. So this result states that such windows exist near every parameter in the diagram.)

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. However, such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).

Little Mandelbrot copies

The Mandelbrot set is self-similar in the sense that small copies of itself can be found at arbitrarily small scales near any point of the boundary of the Mandelbrot set. This phenomenon is explained by Douady and Hubbard's theory of renormalization.

Local connectivity of the Mandelbrot set

It is conjectured that the Mandelbrot set is locally connected. This famous conjecture is known as MLC (for Mandelbrot Locally Connected). By the work of Douady and Hubbard, this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important hyperbolicity conjecture mentioned above.

The celebrated work of Yoccoz established local connectivity of the Mandelbrot set at all finitely-renormalizable parameters; that is, roughly speaking those which are contained only in finitely many small Mandelbrot copies. Since then, local connectivity has been proved at many other points of $M\,$ , but the full conjecture is still open.

Further results

The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 by a result of Mitsuhiro Shishikura.Mitsuhiro Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. 147 (1998) p. 225-267. (First appeared in 1991 as a Stony Brook IMS Preprint, available as arXiv:math.DS/9201282.) It is not known whether the boundary of the Mandelbrot set has positive planar Lebesgue measure.

In the Blum-Shub-Smale model of real computation, the Mandelbrot set is not computable, but its complement is computably enumerable. However, many simple objects (e.g., the graph of exponentiation) are also not computable in the BBS model. At present it is unknown whether the Mandelbrot set is computable in models of real computation based on computable analysis, which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.

Relationship with Julia sets

As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set.

This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proves that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane. Similarly, Yoccoz first proves the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters. Adrien Douady phrases this principle as

Plough in the dynamical plane, and harvest in parameter space.

Geometry of the Mandelbrot set

Recall that, for every rational number

\frac{p}{q}

, where

p

and

q

are relatively prime, there is a hyperbolic component of period

q

bifurcating from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the $\frac{p}{q}$ -limb. Computer experiments suggest that the diameter of the limb tends to zero like

\frac{1}{q^2}

. The best current estimate known is the famous Yoccoz-inequality, which states that the size tends to zero like

\frac{1}{q}

A period $q$ -limb will have $q-1$ "antennas" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.

Generalizations

Sometimes the connectedness loci of families other than the quadratic family are also referred to as the Mandelbrot sets of these families.

The connectedness loci of the unicritical polynomial families $f_c = z^d + c\,$ for $d>2$ are often called Multibrot sets.

For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation locus, which is a natural object to study even when the connectedness locus is not useful.

It is also possible to consider similar constructions in the study of non-analytic mappings. Of particular interest is the tricorn, the connectedness locus of the anti-holomorphic family

z \mapsto \bar{z}^2 + c.\,

The tricorn (also sometimes called the Mandelbar set) was encountered by Milnor in his study of parameter slices of real cubic polynomials. It is not locally connected. This property is inherited by the connectedness locus of real cubic polynomials.

Computer drawings of the Mandelbrot set

A simple algorithm

The definition of the Mandelbrot set, together with its basic properties, suggests a simple algorithm for drawing a picture of the Mandelbrot set. The region of the complex plane we are considering is subdivided into a certain number of pixels. To color any such pixel, let

c\,

be the midpoint of that pixel. We now iterate the critical value

c\,

under

f_c\,

, checking at each step whether the orbit point has modulus larger than 2.

If this is the case, we know that the midpoint does not belong to the Mandelbrot set, and we color our pixel white. Otherwise, we keep iterating for a certain (large, but fixed) number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black.

In pseudocode, this algorithm would look as follows.

For each pixel on the screen do:
{
  x = x0 = x co-ordinate of pixel
  y = y0 = y co-ordinate of pixel
  x2 = x*x
  y2 = y*y
 
  iteration = 0
  maxiteration = 1000
 
  while ( x2 + y2 < (2*2)  AND  iteration < maxiteration ) {
 
    y = 2*x*y + y0
    x = x2 - y2 + x0
    x2 = x*x  	 
    y2 = y*y
    iteration = iteration + 1
  }
 
  if ( iteration == maxiteration )
    colour = black
  else
    colour = iteration
}

Problems

The Mandelbrot set has some very thin filaments, so even if a given pixel does intersect the Mandelbrot set, it is quite likely that its midpoint will nevertheless escape.

As a result, the algorithm does not behave stably under small perturbations, and will generally not detect small features of the Mandelbrot set. It is precisely this property which led Mandelbrot to conjecture that $M\,$ is disconnected.

A common way around this problem, which also results in more aesthetically pleasing pictures, is to color any pixel whose midpoint is determined to be an escaping parameter according to the number of iterations it requires to escape. Since parameters closer to the Mandelbrot set will take longer to escape, this method will make the connections between different parts of the Mandelbrot set visible.

Distance estimates

The proof of the connectedness of the Mandelbrot set in fact gives a formula for the uniformizing map of the complement of

M\,

(and the derivative of this map). By the Koebe 1/4-theorem, one can then estimate the distance between the mid-point of our pixel and the Mandelbrot set up to a factor of 4.

In other words, provided that the maximal number of iterations is sufficiently high, one obtains a picture of the Mandelbrot set with the following properties:

Every pixel which contains a point of the Mandelbrot set is colored black.
Every pixel which is colored black is close to the Mandelbrot set.

Optimizations

One way to improve calculations is to find out beforehand whether the given point lies within the cardioid or in the period 2 bulb.

To prevent having to do huge numbers of iterations for other points in the set, one can do "periodicity checking"—which means check if a point reached in iterating a pixel has been reached before. If so, the pixel cannot diverge, and must be in the set.

Art and the Mandelbrot set

Some people have a hobby of searching the Mandelbrot set for interesting pictures and transform them into still artistic images. Some examples are shown in the fractal art article.

Mandelbrot Animation1.gif is an animation showing increasing levels of detail (warning, large file). Although it does not start from a view of the whole set, towards the end it clearly shows the recursive nature of the fractal at all scales when a shape similar to the whole comes into view.

St. Lois filmmaker Bill Boll has produced a DVD called 'The Amazing Mandelbrot Set' containing several high-quality deep zooms of the Mandelbrot set, along with a tutorial.

The Australian band GangGajang has a song Time (and the Mandelbrot set) where the term Mandelbrot set is used liberally in the lyrics.

The American singer Jonathan Coulton has a song titled Mandelbrot Set on his EP Where Tradition Meets Tomorrow about the history of the Mandelbrot set, and of Benoit Mandelbrot himself.

Blue Man Group's first album Audio features tracks titled "Mandelgroove", "Opening Mandelbrot", and "Klein Mandelbrot". The album was nominated for a Grammy in 2000. Also, a hidden track on their second album, The Complex, is entitled Mandelbrot No. 4.

References

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993, ISBN 0387979425
John W. Milnor, Dynamics in One Complex Variable (Third Edition), Annals of Mathematics Studies 160, Princeton University Press 2006, ISBN 0-691-12488-4 (First appeared in 1990 as a Stony Brook IMS Preprint, available as arXiV:math.DS/9201272.)

External links

Fractal Explorer explains in simple terms how the Mandelbrot and Julia sets are generated. It also features an interactive Mandelbrot set explorer and a nice image gallery.
The Fractal Microscope allows one to explore the Mandelbrot set (and quadratic Julia sets), using a Java interface.
Mandelbrot Explorer. A java applet by Robin Hu that can draw Multibrot sets.
The Mandelbrot and Julia sets Anatomy
FAQ on the Mandelbrot set
Xaos, an open source fractal explorer with Mandelbrot/Julia exploration
FFFF an opensource Mandelbrot generator for Windows, Mac OS X, Linux, IRIX (can use Graphics Processing Unit for fast calculation)
Mandelbrot Set Fractal Gallery Mandelbrot Set images created with Fractint.
Another Mandelbrot Set Applet A more complex applet with color cycling for special effects.
Mandelbrot Set Applet A simple applet for exploring the Mandelbrot Set.

Fractals

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