Matthieu ASTORG


Wandering domains in higher dimension
PhD Defense (in french)

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Julia sets

In holomorphic dynamics, we split the phase space in two disjoint parts: the Julia set and the Fatou set. The chaotic part of the dynamics takes place inside the Julia set, while the dynamics on the Fatou set is stable. Julia sets are almost always complicated fractals; here's a few examples.

Dendrite Cantor Feigenbaum

Those three Julia sets correspond respectively to parameters c=0.4+0.8 i, c=i et c=-1.41... (Feigenbaum constant) in the quadratic family z^2+c. The first one is a dendrite, and the second one is a Cantor set. Now here are some more examples, which now are Julia sets of rational maps that are not polynomials:

un tapis de Sierpinski Cantor de courbes Julia

They are the respective Julia sets of z^2 - 0.01/z^2, z^5+0.01/z^2 and z^5+0.1/z^2. The first one is a Sierpinski carpet, and the second one is a Cantor set of concentric simple curves.

One can also consider Julia sets of holomorphic maps that are not rational, such as exp, sin, tan, etc. Here's an example of Julia set of a holomorphic map on the punctured plane, with a Herman ring. The Julia set is the yellow and pink part, the Fatou set the grey part (the Herman ring is visible in the center). The Julia set is a Cantor bouquet, and is a disjoint union of uncountably many curves that has no interior (this is not obvious from the picture due to numerical artifacts).


Here's another example in the same family, with different colors.


The Mandelbrot set

The Mandelbrot set

The Mandelbrot set represents the different dynamical behaviours of parameters in the quadratic family z^2+c. Each point the image above corresponds to a parameter c with its own Julia set. In shades of blue are the parameters for which the Julia set is not connected (i.e. "is not in one piece"). The shades of pink correspond to the distance to the union of the boundary and of the centers of hyperbolic components (according to the hyperbolic metric of the punctured hyperbolic components).


A zoom near the cusp at c=0.25. The Mandelbrot set is also a fractal, and it contains an infinite number of copies of itself, that are dense in itself.

A tour of the main cardioid: on the right of the picture, the green points turns around a loop (specifically, a closed geodesic). On the left, we can see the corresponding Julia sets. Note that they are all homeomorphic to a Jordan curve, and that they vary continuously (in fact, holomorphically) with respect to the parameter c.

Same thing, except that this time we are moving around in a period 2 hyperbolic component. Again, note that all Julia sets are homeomorphic and move continuously.

Slices of a wandering domain in dimension 2

A wandering domain is a sequence of connected components of the Fatou set that are mapped to each other without ever landing in a cycle of components. A deep theorem due to Dennis Sullivan states that rational maps cannot have wandering domains in dimension one. However, this is not the case in higher dimension.

Wandering domain Wandering domain Wandering domain Wandering domain Wandering domain Wandering domain

These images are slices by complex lines w=constant of the dynamical space of the polynomial skew-product P(z,w)=(z+z^2++0.95z^3+pi^2/4 w, w-w^2). This map has a wandering domain (see this paper). The blue and red part is the filled-in Julia set, meaning the set points whose orbit remains bounded. Each slice is the image under P of the one before. Some components of the wandering domain have been marked in red (the others are blue).
Here is a video illustrating the same phenomenon (same thing, but with more frames). Each slice is shown for the same amount of time; note that marked components (in red) spend a lot of time close to the origin as they pass through the eggbeater.

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