Title: Attracting domains for holomorphic maps tangent to the identity
Speaker: Sara Lapan
Speaker Info: Northwestern University
One of the guiding questions behind the study of holomorphic dynamics is: given a germ of a holomorphic self-map of C^m that fixes a point (say the origin), can it be expressed in a simpler form? If so, then the dynamical behavior of the map can be more easily understood. In general, we want to know how points near the origin behave upon iteration of the map f. More specifically, we want to know when there exists a domain whose points are attracted to the origin under iteration by f and, if such a domain exists, when its points converge tangentially to a given direction v. In dimension one, the Leau-Fatou Flower Theorem tells us, among other things, of the existence of such domains. In higher dimensions, Hakim showed that given some assumptions on f and the direction v, a domain of attraction whose points converge to the origin tangentially to v does exist.Date: Tuesday, November 05, 2013
In this talk, we will begin by discussing the existence of attracting domains in dimension one to better visualize and understand what happens in higher dimensions. Then we will discuss what happens in higher dimensions and introduce Hakim's theorem. We will focus the discussion on a collection of maps in C^2 that does not satisfy some of the assumptions in Hakim's theorem; in particular, maps with a unique (and non-degenerate) characteristic direction. It turns out that such maps will also have a domain of attraction. We will discuss this result as well as some of the techniques used in its proof.