Hedgehogs in Holomorphic Dynamic

When

October 11, 2024 | 2:30 p.m. – 3:30 p.m.
Refreshments provided

Where

University Hall 4002

Speaker

Tanya Firsova

Abstract

One of the fundamental questions in holomorphic dynamics is understanding the behavior of a dynamical system in the neighborhood of a fixed point. In dimension one, this question has been extensively studied by Schroder, Koenings, Böttcher, Fatou, Julia, and many others. The most difficult and intriguing cases involve Cremer fixed points, where the dynamics is non-linearizable. Using uniformization theory, Pérez-Marco proved the existence of nontrivial invariant compact sets, called "hedgehogs," in the neighborhood of a Cremer fixed point. Drawing on deep results from the theory of analytic circle diffeomorphisms developed by Yoccoz, Pérez-Marco demonstrated that even when a map in the neighborhood of the origin is not conjugate to an irrational rotation, the points in the hedgehog are recurrent and continue to move under the influence of the rotation.

In a joint work with M. Lyubich, R. Radu, and R. Tănase, we were able to construct 'hedgehogs' for multidimensional semi-Cremer germs and to study their dynamical properties. Our methods are purely topological and also provide an alternative proof for the existence of hedgehogs in dimension one. The talk will be focused on and accessible to graduate students. All are encouraged to attend.