|Date:||Mon, November 18, 2013|
|Place:||Research II Lecture Hall|
Abstract: Newton's method for polynomials or entire maps can be regarded as a dynamical system on the Riemann sphere or, respectively, on the complex plane. Understanding the topology of its Julia set gives results which are interesting both dynamically and numerically. We present here a recent result which states that the Julia set of Newton's methods is always connected or, equivalently, that its stable regions are simply connected. In the talk however, we shall concentrate mostly on the main tool used to prove this theorem, namely the existence of absorbing regions in Baker domains (components on which all iterates tend to infinity). Absorbing regions (domains which eventually attract all orbits) are known to exist for each type of Fatou component except, until now, for Baker domains. This result takes a much more general form and it is based on work of Cowen on holomorphic maps from the right half plane to itself with no fixed points.