# Combinatorics in Transcendental Dynamics

## Periodic Reporting for period 1 - CoTraDy (Combinatorics in Transcendental Dynamics)

Reporting period: 2017-09-01 to 2019-08-31

The topic of this project is in Dynamical System, a wide and variegated area of mathematics concerned with the study of the evolution of systems of any kind, from biological models to abstract mathematical systems.

The main purpose of this project was to study a specific type of abstract system given by the iteration of some functions called entire transcendental maps, and to deepen our understanding of such systems by investigating specific subsystems (the 'curves escaping points', or 'rays') which can be correlated with other simple and well understood examples. This type of approach takes the name of 'combinatorial study'.

This is a project in pure mathematics, so its direct impact for society in terms of research is long-term and difficult to predict. Other works concerning the dynamics of entire functions have proven useful in improving Newton's Method, a widely used algorithm to find roots of polynomials which has applications in several areas of science.
As outreach activity we have planned and executed several conferences about fractals for high school students, which have been very successful. We believe that presenting female researchers in science to this particular type of public is important to promote female role models in science to teenagers, and encourages female students to pursue careers in STEM.

The scientific objectives were to investigate the patterns arising from the aforementioned subsystems (the curves escaping points, or rays) and their relation to periodic points, that is, equilibrium states of the systems.
For entire transcendental maps, periodic rays are special curves in the plane consisting of points whose orbits converge to infinity, and are one of the main objects of investigation of this project. Singular values are special values near which the function is not locally invertible, and periodic points are points that are invariant under some iterate of the function. With N. Fagella we proved that under general hypothesis periodic rays divide the plane into regions which, in a precise sense, encode the orbits of singular values, and that the latter are forced to interact with the nonrepelling periodic points present in such regions. This led to a new proof of the Fatou-Shishikura inequality, which in addition gives results for functions with infinitely many singular values. We have also been able to show that periodic point which are not the landing point of a periodic ray must interact in a precise way with singular orbits.
With L. Rempe-Gillen we have been able to prove that, if the orbits of singular values are bounded, then all repelling periodic points are landing points of rays.
With H. Peters and JE Fornaess we studied the entropy of entire transcendental functions, and with both of them and L. Arosio we extendend some results to a special class of transcendental automorphisms of C^2.
With N. Fagella, Gwyneth Stallard, Phil Rippon and Vasso Evdoridou we produced a rather complete classification of bounded simply connected wandering domains as well as a series of original examples illustrating the classification.
We submitted 4 preprints and published 2 additional papers. The results were presented at several international conferences and dynamical systems seminars.
We studied several new aspects of the dynamics of transcendental functions, especially concerning the relation between the set of escaping points and the set of periodic points and of singular values. We introduced two main new techniques in the field of transcendental dynamics: the concept of fundamental tails and a new way to obtain information about the dynamics of a transcendental function via the landing structure of its periodic rays. This technique has already found application in two preprints [EFJS], [PRS]. The other type of techniques which have been developed exploit results from [BF15]. This project extended unexpectedly also to dynamics in several complex variables, leading to the completion of a work on transcendental Henon maps of C^2 [ABFP] and a new work in progress which deals more specifically with escaping points of transcendental Henon maps.
At the networking level, this project contributed to strengthen the collaboration between the research group from the Universitat de Barcelona and the research group of the Open University (UK), increasing international collaboration. It also contributed to strengthening the interaction between the field of dynamics in one complex variable and the field of dynamics in several complex variable, an interaction which we believe should be much stronger than it currently is.