Mandelbrot Set - Basic Properties

Basic Properties

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

for all .

In other words, if the absolute value of ever becomes larger than 2, the sequence will escape to infinity.

The intersection of 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,

The correspondence is given by

In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.

The area of the Mandelbrot set is estimated to be 1.50659177 ± 0.00000008.

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. This conjecture was based on computer pictures generated by programs which are unable to detect the thin filaments connecting different parts of . Upon further experiments, he revised his conjecture, deciding that 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, 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 for which the dynamics changes abruptly under 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 p₀=z, p_n=p_n-12+z, and then interpreting the set of points |p_n(z)|=2 in the complex plane as a curve in the real Cartesian plane of degree 2n+1 in x and y.