Mandelbrot set

The Mandelbrot set is a two-dimensional set that is defined in the complex plane as the complex numbers for which the function does not diverge to infinity when iterated starting at, i.e., for which the sequence,, etc., remains bounded in absolute value.
This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.
Images of the Mandelbrot set exhibit an infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a fractal curve. The "style" of this recursive detail depends on the region of the set boundary being examined. Images of the Mandelbrot set are created by determining whether the sequence goes to infinity for each sampled complex number c. The real and imaginary parts of are mapped as image coordinates on the complex plane and coloured based on the point at which the sequence crosses an arbitrary threshold. If is held constant and the initial value of is varied instead, the corresponding Julia set for the point is obtained.
The Mandelbrot set is well-known, even outside mathematics, for how it exhibits complex fractal structures when visualized and magnified, despite having a relatively simple definition, and is commonly cited as an example of mathematical beauty.

History

The Mandelbrot set has its origin in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. The fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups. On 1 March 1980, at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, Benoit Mandelbrot first visualized the set.
Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980. The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard, who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in fractal geometry.
The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books, and an internationally touring exhibit of the German Goethe-Institut.
The cover article of the August 1985 Scientific American introduced the algorithm for computing the Mandelbrot set. The cover was created by Peitgen, Richter and Saupe at the University of Bremen. The Mandelbrot set became prominent in the mid-1980s as a computer-graphics demo, when personal computers became powerful enough to plot and display the set in high resolution.
The work of Douady and Hubbard occurred during an increase in interest in complex dynamics and abstract mathematics, and the topological and geometric study of the Mandelbrot set remains a key topic in the field of complex dynamics.

Formal definition

The Mandelbrot set is the uncountable set of values of c in the complex plane for which the orbit of the critical point under iteration of the quadratic map
remains bounded. Thus, a complex number c is a member of the Mandelbrot set if, when starting with and applying the iteration repeatedly, the absolute value of remains bounded for all.
For example, for c = 1, the sequence is 0, 1, 2, 5, 26,..., which tends to infinity, so 1 is not an element of the Mandelbrot set. On the other hand, for, the sequence is 0, −1, 0, −1, 0,..., which is bounded, so −1 does belong to the set.
The Mandelbrot set can also be defined as the connectedness locus of the family of quadratic polynomials, the subset of the space of parameters for which the Julia set of the corresponding polynomial forms a connected set. In the same way, the boundary of the Mandelbrot set can be defined as the bifurcation locus of this quadratic family, the subset of parameters near which the dynamic behavior of the polynomial changes drastically.

Basic properties

The Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 centred on zero. A point belongs to the Mandelbrot set if and only if for all. In other words, the absolute value of must remain at or below 2 for to be in the Mandelbrot set,, and if that absolute value exceeds 2, the sequence will escape to infinity. Since, it follows that, establishing that will always be in the closed disk of radius 2 around the origin.
File:Verhulst-Mandelbrot-Bifurcation.jpg|thumb|Correspondence between the Mandelbrot set and the bifurcation diagram of the quadratic map
The intersection of with the real axis is the interval. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,
The correspondence is given by
This gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.
Douady and Hubbard showed that the Mandelbrot set is connected. 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. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of. Upon further experiments, he revised his conjecture, deciding that should be connected. A topological proof of the connectedness was discovered in 2001 by Jeremy Kahn.
The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of, 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 the bifurcation locus of the family of quadratic polynomials. In other words, the boundary of the Mandelbrot set is the set of all parameters for which the dynamics of the quadratic map exhibits sensitive dependence on i.e. changes abruptly under arbitrarily small changes of 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, and then interpreting the set of points in the complex plane as a curve in the real Cartesian plane of degree in x and y. Each curve is the mapping of an initial circle of radius 2 under. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.

Other properties

Main cardioid and period bulbs

The main cardioid is the period 1 continent. It is the region of parameters for which the map has an attracting fixed point. The set comprises all parameters of the form where lies within the open unit disk.
Attached to the left of the main cardioid at the point, the period-2 bulb is visible. This region consists of values of for which has an attracting cycle of period 2. It is the filled circle of radius 1/4 centered around −1.
More generally, for every positive integer, there are circular bulbs tangent to the main cardioid called period-q bulbs, which consist of parameters for which has an attracting cycle of period. More specifically, for each primitive th root of unity , there is one period-q bulb called the bulb, which is tangent to the main cardioid at the parameter and which contains parameters with -cycles having combinatorial rotation number. The periodic Fatou components containing the attracting cycle meet at the -fixed point. If these are labelled counterclockwise as, then component is mapped by to the component.

Hyperbolic components

Bulbs that are interior components of the Mandelbrot set in which the maps have an attracting periodic cycle are called hyperbolic components.
It is conjectured that these are the only interior regions of and that they are dense in. This problem, known as density of hyperbolicity, is one of the most important open problems in complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components. For real quadratic polynomials, this question was proved in the 1990s independently by Lyubich and by Graczyk and Świątek.
Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy.
Each of the hyperbolic components has a center, which is a point c such that the inner Fatou domain for has a super-attracting cycle—that is, that the attraction is infinite. This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. Therefore, for some n. If we call this polynomial , we have that and that the degree of is. Therefore, constructing the centers of the hyperbolic components is possible by successively solving the equations. The number of new centers produced in each step is given by Sloane's.

Local connectivity

It is conjectured that the Mandelbrot set is locally connected. This conjecture is known as MLC. By the work of Adrien Douady and John H. 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 work of Jean-Christophe Yoccoz established local connectivity of the Mandelbrot set at all finitely renormalizable parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies. Since then, local connectivity has been proved at many other points of, but the full conjecture is still open.