Wandering domains in higher dimension
PhD defense (in french)
Click on pictures for a higher resolution.
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.
A zoom near the cusp at c=0.25.
Slices of a wandering domain in dimension 2
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.
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.