Department of Numerical Mathematics Faculty of Mathematics and Physics Charles University

What we (don’t) know about iteration - a brief introduction to holomorphic dynamics and chaos

Abstract: Newton's method is an example of the simple process of iterating a nonlinear mapping on a given space. In the numerical community results on the convergence of Newton's and similar methods are usually local (convergence on a neighborhood of the exact solution). However, in the past hundred years there have been extensive endeavors to answer the question what happens globally. The field of holomorphic dynamics deals with these questions in the complex plane and leads to well known objects in fractal and chaos theory. We give an overview of known results along with some open questions and their relation to Newton's method.