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.
