COMPLEX ANALYSIS AND DYNAMICS

In the use of particle accelerators, particles go around in a tube at high speed. It is important to model the orbits by investigating how they cross a transversal plane. This has been modelled in a paper by Løw, Pereira, Peters and Wold and published in Mathematische Annalen. In science, events depend often on many parameters. However most often only a few of these are measured. The question arises whether such partial information can actually say something precise about the system. This problem is investigated in a model case in a paper by Fornaess and Peters, published in Ergodic Theory and Dynamical Systems. In an evolving system, the behaviour over time can in the long term be orderly or chaotic. It depends on the initial data. The orderly behaviour is investigated in complex dynamics as Fatou sets. These are the topic of two papers, one by Fornaess and Wold, published in the Proceedings of that American Mathematical Society and one by Boc-Thaler, Fornaess and Peters, published in Ergodic Theory and Dynamical Systems. In complex analysis and other branches of mathematics, the disc and the ball are fundamental objects. But then, given an object, how do you recognise whether this is (possibly after allowable deformations or so called squeezings) a disc or a ball? This is the topic of two papers, one by Diederich, Fornaess and Wold published in the International Journal of Mathematics and one by Wold, published in Mathematische Zeitschrift. Another result on squeezing, focusing on metrics, was also published by Fornaess and Wold, in Complex Analysis and Geometry. A fundamental difference between complex analysis in one and several dimensions is that in several dimensions, analytic functions cannot have a pole at an isolated point. So functions extend across isolated points. A basic topic in complex analysis is what other objects can be extended. This is the topic of the paper by Andrist, Shcherbina and Wold, published in Ark. Mat. Convex objects are very important both in practical situations and in science. Therefore it is important to know whether an object can be deformed within allowed rules to become convex at least on small pieces. This is the topic in an article published in the Proc. AMS by Deng, Fornæss and Wold. There this is studied in singular spaces. Squeezing has been studied in two more articles, by Fornæss and Rong in Mathematische Annalen and by Fornæss and Shcherbina in J. of Geom. Anal. Resonance is an important phenomenon, for example, a bridge can collapse of soldiers are marching across. David Hahn has published an article in Complex. Anal. Oper. Theory about resonance. Fornæss and Juije Wu has published an article in J. Geom. Anal. which investigates whether complicated objects can be approximated by less complicated objects. This is an important topic in applications and in science. Kutschebauch and Wold has published an article in J. Reine Angew. Math which also concerns approximation. Bracci, Fornæss and Wold have published an article about metrics in Mathematische Zeitschrift. The article has a special focus on worm domains. Arosio, Benini, Fornæss and Peters have started a new project in dynamical systems, namely the dynamics of transcendental Henon maps. This generalizes dynamics of transcendental functions in one complex variable and the dynamics of polynomial Henon maps in two complex variables. The article is published in Mathematische Annalen. Fornæss and Wold have made contributions to the theory of the squeezing function in an article published in the Pacific Journal of Mathematics. Convexity is a theme in an article by Arosio and Wold published in the Indiana Journal of Mathematics. The topic is embeddings of totally real manifolds and their polynomially convex hulls. Simon and Stensønes have published an article in the Journal of Geometric Analysis about homogeneous plurisubharmonic polynomials. This is an important topic related to the understanding og the geometry of weakly pseudoconvex boundaries. Entropy is an important concept in dynamics. Entropy is used as a measure of how chaotic a system is. In an article by Benini, Fornæss and Peters, published in Acta Math. Vietnam this is studied for transcendental functions in the complex plane. The approximation theory of Fornæss and Wu is continued in two articles, one by Fornæss and Wu published in Mathematische Zeitschrift and one by Biard, Fornæss and Wu published in Transactions of the American Mathematical Society. It is important to have many concrete examples of phenomena. In the case of complex manifolds, several new ones were found by Deng and Fornæss, published in an article in the Journal of Geometric Analysis. A ball is an example of a set with simple geometry. If a set has flat points, then it can be difficult to analyze the properties of the set. This is studied in a

During this grant period, we have progressed very well towards the goals put forward in the application. We have achieved closer connections between the Norwegian SCV groups in Oslo, Trondheim and Stavanger. This has lead to many joint papers. The collaboration with leading Chinese Universities has blossomed. One of the group has a Professor 2 position at Tsinghua University, the leading academic institution in China. Also we have strong connections with the Chinese Academy of Science in Beijing including hiring a postdoc from there and Shanghai Jiao Tong University where one of the group has a former PhD student who is now professor and collaborator there. We also have strengthened the ties with several central institutions in Europe and the USA. The mathematical results as is seen in the list of papers coming out of this project is quite diverse and involves many collaborations between members of the different Norwegian groups and with various international groups.

In this project we want to study problems in Several Complex Variables and Holomorphic Dynamical Systems in Several Variables. The two fields have many connections and have had fruitful interactions over the recent years, in particular solutions with good estimates for the d-bar and dd^c equations have been used to solve equidistribution problems in dynamics. We want to address old and new problems both by improving already existing methods and developing new tools in order to solve these problems. The project contains several classical problems in Several Complex Variables (existence of peak points for A(\Omega), descriptions of envelopes of holomorphy, embeddings of Riemann surfaces), some of which we want to address by using methods developed in Dynamical Systems. A substantial part is devoted to well known and difficult problems arising in the study of dynamical systems (foliations, existence of exceptional minimal sets, equidistribution and connections with Nevanlinna theory) The project is a joint venture between the NTNU- and UiO-groups in SCV and dynamics, and a key goal is to support the collaboration between them. We will also include the remaining national nodes (Stavanger and Bergen), and we have a strong focus on international collaboration, especially on building cooperation with China.