Dynamics of finite-type entire functions

The study of complex dynamics has received much attention over the past two decades. The fascinating phenomena encountered in this area, such as 'chaotic behaviour' and self-similarity, have been widely popularized, and computer graphics of 'fractal' objects like the Mandelbrot set have become famous far outside of mathematics.A central problem in one-dimensional dynamics is the *hyperbolicity conjecture*. Roughly speaking, this question can be described as follows: although general one-dimensional dynamical systems may have very complicated and unstable behavior, is it at least possible to *perturb* any such system to a simpler one for which most points tend to stable states? For the case of polynomials (such as those which give rise to the Mandelbrot set), there has been spectacular progress on this problem over the past 20 years. Following on from major breakthroughs by Lyubich, and independently Graczyk and Swiatek, this culminated in the real case with the proof of the hyperbolicity conjecture for real polynomials by Kozlovski, Shen and van Strien.Transcendental dynamics - that is, the iteration of non-polynomial functions in the complex plane, such as the exponential map - is far less understood than that of polynomials, yet the two are intimately related. In this project, we propose to advance the understanding of this field in two ways:a) Prove the hyperbolicity conjecture for a large class of entire transcendental functions. (This will be joint work with Prof. Sebastian van Strien from Warwick University.)b) Extend some important results and ideas from polynomial dynamics, which are of fundamental importance there, to the - at first glance entirely different - transcendental setting. (This will be the research project of the proposed research student, Ms. Helena Mihaljevic-Brandt.)