Dynamics of transcendental functions with escaping singular orbits and infinite-dimensional Teichmüller theory
Final Report Abstract
This project was one of our most ambitious, especially in the technical aspects related to infinite-dimensional Teichmüller theory. It was successful in the intended classification of transcendental functions with escaping singular values in the restricted case of maps that are compositions of polynomials and exponentials, and it was successful in developing the tools that make it possible to extend the results essentially to the generality originally intended: entire functions of finite type and finite order. It even opens up perspectives in greater generality when the escaping set is no longer organized in the form of dynamic rays. Moreover, this work has led to a better understanding of infinite-dimensional Teichmüller theory, where some (until then) apparently natural conjectures were disproved. We developed work-arounds for our purposes (so that it suffces to focus on an invariant compact subset), but a better understanding of these spaces in general remains a desirable goal. In terms of perspectives, it connects to results that were developed in the community since the beginning of the project: the development of dreadlocks (in cases where dynamic rays do not exist) opens up the perspective of work beyond maps of finite order, and recent work on Local Connectivity of the Mandelbrot set seems to require results from transcendental dynamics that are very close in spirit to the ones we developed, but for transcendental maps that (at this time) we have not investigated.
Publications
- Infinite-dimensional Thurston theory and transcendental dynamics with escaping singular orbits. PhD Thesis, Aix-Marseille Université,15 December 2020
Konstantin Bogdanov
