Periodic Reporting for period 1 - CoTraDy (Combinatorics in Transcendental Dynamics)
Período documentado: 2017-09-01 hasta 2019-08-31
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.
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.
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.