Walter Bergweiler
Mathematisches Seminar
Christian-Albrechts-Universität zu Kiel

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A gallery of complex dynamics pictures. (The pictures may be enlarged by clicking on them.)

These three pictures show attracting basins and Julia sets of the anti-holomorphic functions \[ \alpha\,\overline{\zeta(z)}+\beta\,\overline{z}, \] with the Weierstraß \(\zeta\)-function and certain parameters \(\alpha\) and \(\beta\). This is related to the number of critical points of Green's function on a torus; see the paper "Green's function and anti-holomorphic dynamics on a torus" with Alexandre Eremenko for details. picture picture picture

These four pictures show the Fatou and Julia sets of certain hyperbolic entire functions. Some Fatou components are bounded while others are unbounded. For a discussion of when which case occurs see the paper "Hyperbolic entire functions with bounded Fatou components" with Núria Fagella Rabionet and Lasse Rempe-Gillen for details. picture
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These two pictures show the attracting basins of the anti-holomorphic function \[ f(z)=\frac{k}{\sin\overline{z}}+w \] for certain parameters \(k\) and \(w\). For the relevance of this to gravitational lensing see the paper "On the number of solutions of a transcendental equation arising in the theory of gravitational lensing" (pdf) with Alexandre Eremenko. picture picture

These three pictures show for certain families of functions with a parabolic fixed whether there is more than one critical point contained in the corresponding parabolic basin or not; see the paper "On the number of critical points in parabolic basins" (ps, dvi, pdf) for details. picture picture picture

The red region on the left side of this picture is a Baker domain whose boundary is a Jordan curve; see the paper "Invariant domains and singularities" (ps, dvi, pdf) for details. picture
The red regions in these two pictures are examples of completely invariant domains of entire functions which do not contain all singularities of the inverse function; see the paper "A question of Eremenko and Lyubich concerning completely invariant domains and indirect singularities" (ps, dvi, pdf) for details. picture picture

These pictures show completely invariant domains of meromorphic functions. The Julia sets are Jordan curves (on the sphere), but they are not quasicircles; see the paper "Meromorphic functions with two completely invariant domains" (ps, dvi, pdf) with Alexandre Eremenko for details.
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This picture shows the Julia set of a non-semihyperbolic entire function which has no asymptotic values, no parabolic points and no recurrent critical point; see the paper "Semihyperbolic entire functions" (ps, dvi, pdf) with S. Morosawa for details. Further pictures related to iteration of entire functions (and in particular to semihyperbolicity of entire functions) can be found on his gallery.


Two further pictures related to semihyperbolicity, showing the Julia sets of \[ f(z)=\frac{3\pi^2}{\pi^2-4z}\cos\sqrt{z} \quad\text{and}\quad g(z)=\pi^2-2\pi^2\frac{\sin\sqrt{z}}{\sqrt{z}}. \]

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These two pictures are connected to Newton's method of finding zeros applied to solutions of Bessel differential equations. For parameters in the colored regions, Newton's method converges for an open dense set of starting values, and for parameters in the black set it diverges in some open set; see the paper "On the zeros of solutions of linear differential equations of the second order" (ps, dvi, pdf) with Norbert Terglane for details.

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These pictures here are essentially the same as the previous ones. They arise from Newton's method for Bessel functions - and differ from the previous ones only by a parameter transformation, and a slightly different coloring scheme.

The right picture is obtained by magnifying a part of the left picture. Here is a zoom (4.2 MB) showing this.

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The function \[f(z)=(a+1)\tan z -az\] has a parabolic point at 0 with two parabolic basins. The pictures show these basins for \(a=0.5\), \(a=0.6\) and \(a=0.7\).

The range shown is \(|\mbox{Re}\; z|\leq 20\), \(|\mbox{Im}\;z|\leq 20\).

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Both \(\Gamma(z)\) and \(1/\Gamma(z)\) have the attracting fixed point 1. The attracting basin is shown in green and blue. In addition, \(1/\Gamma(z)\) has the parabolic fixed point 0 whose basin is shown in orange.

The range shown is \(-2.5\leq \mbox{Re}\; z\leq 5.5\), \(|\mbox{Im}\;z|\leq 2.4\).

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These pictures show the basins of attraction for Newton's method applied to the polynomial \[ p(z)=(z^2+4)(z-9). \] The basin of 2i is colored green, that of -2i is colored red, and that of 9 is colored blue. Newton's method does not converge in the yellow set. The left picture shows the range \(|\mbox{Re}\; z|\leq 12\), \(|\mbox{Im}\;z|\leq 12\) and the right picture shows a close-up of the range \(2\leq \mbox{Re}\; z\leq 4\), \(|\mbox{Im}\;z|\leq 1\).

There is also a movie (0.6 MB) showing how the basins of attraction change if the roots at 2i and -2i are fixed and the third root moves on a circle of radius 9, and another one (0.5 MB) where the third root moves along the real axis.

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The Mandelbrot set and some parts of it.

More pictures, programs, and some mathematics concerning the Mandelbrot set, Julia sets, etc., can be found here.

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Here are (in alphabetical order) some links to other mathematicians' websites containing pictures - and sometimes also programs - related to complex dynamics:

Arnaud Chéritat has various pictures related to complex dynamics.

Bob Devaney has various applets for the generation of Julia sets and related sets, a Mandelbrot set explorer, and more.

Wolf Jung has a program for drawing the Mandelbrot set and Julia sets.

Tomoki Kawahira has a gallery with fractal images as well as some java applets.

Lorelei Koss has a page with Graphics of Julia Sets of Weierstrass Elliptic Functions .

Curt McMullen's gallery contains some complex dynamics - and lots of other interesting stuff; there are also programs.

Shunsuke Morosawa has a gallery with pictures and animations of Julia sets of entire functions.

Lasse Rempe has some pictures for iteration of exponential maps. He also organized a Chaos and fractals exhibition.

Mitsuhiro Shishikura has a gallery with pictures of the Mandelbrot set and Julia sets, as well as programs for Macintosh.

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