Matthieu ASTORG


Wandering domains in higher dimension
PhD defense (in french)

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Mandelbrot set

Wandering domain

The Mandelbrot set represents the different dynamical behaviours in the quadratic family f(z)=z^2+c. Every pixel on the image above corresponds to a parameter c. In blue are the parameters for which the Julia set is not connected.

Wandering domain

A zoom near the cusp at c=0.25.

Slices of a wandering domain in dimension 2

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Quick explanation : Those images are slices by complex lines w=constant of the dynamical space of the fibered polynomial P(z,w)=(z+z^2+0.95 z^3 + pi^2/4 w, w-w^2). This map has a wandering domain (see this article). The red and blue part is the wandering domain. Each slice is the image of the preceding. A few components have been marked in red to illustrate the dynamics.

Wandering domain

This is a Herman ring for a self-antipodal transcendental map of the punctured plane. The yellow part corresponds to points escaping to zero, and the purple to infinity. The grey part is the Fatou set, and contains a Herman ring.