Fractals

Table of Contents

Notation

Mandelbrot set

The Mandelbrot set is the set complex numbers for which the function

fractals_763ecf1e27734aa242bcc1dde78b299043a8abb5.png

does not diverge when iterated from fractals_212dafde8d2b8be8c0a2822fd38dc60f3bd918bb.png, i.e. remains bounded in absolute value.

Formally, the Mandelbrot set is the set of values of fractals_32e55c6e155c100dc50ef35a36a56df33f4c0687.png in the complex plane for which the orbit of fractals_d83ba0b34682ac87f8d84c8310f6bbd28d1fe65b.png under iteration of the quadratic map

fractals_d0d4b3ece57af2c05643f84d78db96941908247e.png

remains bounded.

The Mandelbrot set fractals_f75d7681f9d0acecef24eb637ee201aa5b39199a.png is defined by a family of complex quadratic polynomials

fractals_8cbb7057fb773be7f988a6517fd436718172b3ba.png

given by

fractals_ce9d690074f6b55c55f91074bd40dc8282776015.png

where fractals_32e55c6e155c100dc50ef35a36a56df33f4c0687.png is a complex parameter. For each fractals_32e55c6e155c100dc50ef35a36a56df33f4c0687.png, one considers the behavior of the sequence

fractals_d0a355bc9dad48948186d328686e65121ab0525d.png

obtained by iteration fractals_733370457c92d628e35bf9c8cc336aab9c4956b5.png starting at critical point fractals_212dafde8d2b8be8c0a2822fd38dc60f3bd918bb.png. This either escapes to infinity or stays within a disk of finite radius; the Mandelbrot set is then the set of all points fractals_32e55c6e155c100dc50ef35a36a56df33f4c0687.png such that the sequence stays within a disk of finite radius.

More formally, if fractals_388d9e692835ca76b8cce4192569859ad2dd633d.png denotes the n-th iterate of fractals_733370457c92d628e35bf9c8cc336aab9c4956b5.png (i.e. fractals_733370457c92d628e35bf9c8cc336aab9c4956b5.png composed with itself fractals_f07b932b12c91bca424fd428d383495413b5b6a9.png times), the Mandelbrot set is the subset of the complex plane given by

fractals_69a35e3d84015e6a6da9697418a22512e8261ee6.png

As explained below, it is in fact possible to simplify this definition by taking fractals_ba1f957a44f8888a02979458b2b7c8a202f2e0f7.png.

Visualizations of Mandelbrot sets

  • Color all points fractals_32e55c6e155c100dc50ef35a36a56df33f4c0687.png that belong to the Mandelbrot set fractals_f75d7681f9d0acecef24eb637ee201aa5b39199a.png black and all other points white

Connection to Julia sets

An equivalent definition of the Mandelbrot set is as the subset of the complex plane consisting of those parameters fractals_32e55c6e155c100dc50ef35a36a56df33f4c0687.png for which the Julia set of fractals_66bff4600551a5a99559343145341faf3c7844c6.png is connected.

Julia set

Let fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png be a holomorphic function from the Riemann sphere to itself.

Such fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png are precisely the complex rational functions, that is

fractals_3631dd1478de035d19edd80cbe576859333ca7c8.png

where fractals_1a0992ed24a704eed765cdac778e5c7db57bbe13.png and fractals_03c1842b33391ce8ee4fa8cdfe857329ca754d6c.png are complex polynomials.

Assume that fractals_fefe9e556d399665a26a37824ec578cbffb0cabe.png and fractals_ab437e1f9b3376761b155efe111c9860607c4b86.png are non-constant, and that they have no common roots (i.e. fractals_ab437e1f9b3376761b155efe111c9860607c4b86.png does not divide fractals_fefe9e556d399665a26a37824ec578cbffb0cabe.png). Then there is a finite number of open sets fractals_ee1859473fda5119e8b6859c8772d468de9112ae.png, that are left invariant by fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png and are such that:

  1. fractals_f89aa7d9d27ba9a47a74c3ad28ed38ace624477d.png is dense in the plane
  2. fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png behaves in a regular and equal way on each of the sets fractals_e3c2acd8fda16fe4a737b744feab5cbfe8ecaf48.png

The last statement means that the termini of the sequences of iterations generated by the points of fractals_e3c2acd8fda16fe4a737b744feab5cbfe8ecaf48.png are either

  • the same set => finite cycle
  • fractals_e3c2acd8fda16fe4a737b744feab5cbfe8ecaf48.png are finite cycles of circular or annular shaped sets centered at the same point

These sets fractals_e3c2acd8fda16fe4a737b744feab5cbfe8ecaf48.png are the Fatou domains of fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png, and their union is the Fatou set fractals_067a2d62f4b0762436abaab6aac2b1cadb7e76c9.png of fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png.

Further, each of the Fatou domains contains at least on critical point of fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png, i.e. a (finite) point fractals_9c15196dd07b1add486b8b54592e74bfe946ed95.png such that

fractals_b4dbe5c4502e829cb3036f9002624fd74af0021e.png

if either of the following is the case:

  • degree of the numerator fractals_1a0992ed24a704eed765cdac778e5c7db57bbe13.png is at least two larger than the degree of the denominator fractals_03c1842b33391ce8ee4fa8cdfe857329ca754d6c.png
  • fractals_a3f7c875663ad44f814560984d470d6d8e5415dc.png for some fractals_32e55c6e155c100dc50ef35a36a56df33f4c0687.png and a rational function fractals_02967cc46651f7c0404443e782c964edfa5393a1.png satisfying this condition

The complement of the Fatou set fractals_067a2d62f4b0762436abaab6aac2b1cadb7e76c9.png is the Julia set fractals_c19f7cfe57dcce78e7e77aef4cee2318c7a3532a.png of fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png.

fractals_c19f7cfe57dcce78e7e77aef4cee2318c7a3532a.png is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers).

Like fractals_067a2d62f4b0762436abaab6aac2b1cadb7e76c9.png, fractals_c19f7cfe57dcce78e7e77aef4cee2318c7a3532a.png is left invariant by fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png, and on this set the iteration is repelling, meaning that

fractals_4e57bab76a28188989c907b333b526de11133b4a.png

for all fractals_fe480f547555026d210b55b5d4ef758235f32832.png in a neighborhood of fractals_9c15196dd07b1add486b8b54592e74bfe946ed95.png (within fractals_c19f7cfe57dcce78e7e77aef4cee2318c7a3532a.png).

This means that fractals_3cde5172d50d013ad7e2a092d995a909e9739757.png behaves chaotically on the Julia set.

There are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and thus they only make up a infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

Equivalent definitions of Julia sets are:

  • fractals_c19f7cfe57dcce78e7e77aef4cee2318c7a3532a.png is the smallest closed set containing at least three points which is completely invariant under fractals_cdd1cc131da6040eca078917132a377727053c44.png (Möbius transformation?)
  • fractals_c19f7cfe57dcce78e7e77aef4cee2318c7a3532a.png is the closure of the set of repelling periodic points
  • For all but at most two points fractals_834a15888a5e3a341491f36d35f3911d21c6c2c2.png, the Julia set is the set of limit points of the full backwards orbit

    fractals_a253e6197106105c2339c30559bbbcb7f0c0acb1.png

    i.e. limit points of the union of pre-images of all the iterations.

  • If fractals_cdd1cc131da6040eca078917132a377727053c44.png is an entire function, then fractals_c19f7cfe57dcce78e7e77aef4cee2318c7a3532a.png is the boundary of the set of points which converge to infinity under iteration.
  • If fractals_cdd1cc131da6040eca078917132a377727053c44.png is a polynomial, then fractals_c19f7cfe57dcce78e7e77aef4cee2318c7a3532a.png is the boundary of the filled Julia set: i.e. those points whose orbits under iterations of fractals_cdd1cc131da6040eca078917132a377727053c44.png remain bounded.

Julia set of a Mandelbrot set

Mandelbrot sets are given by

fractals_d81d1bd7d431bac3014dae0caf113f173f2e43d2.png

where fractals_32e55c6e155c100dc50ef35a36a56df33f4c0687.png is a complex parameter.

The Julia set of for this system is the subset of complex plane given by:

fractals_165c472e90acd0b5533bc49f77da73b01c111247.png

where fractals_048d05a7ce05dd016372bc17fee802d4c400451d.png is the n-th iterator of fractals_87c56f6f5222af6d795b9b92bb8fd090db96ecb6.png, i.e. fractals_311047619570cb30dc4947d2db50ee7749a67fc9.png, n times.