Dynamics in several complex variables: transcendental dynamics, entropy and wandering components.

Complex dynamical systems is an active research area world wide, with strong connections to real dynamical systems, complex analysis, number theory and ergodic theory. It is a classical field or research, that really came to life in the 1970s, thanks to fundamental contributions of many of the worlds strongest mathematicians, combined with new intuition and inspiration arising from computer pictures. Since the 1990s there has been particular interest in higher dimensional dynamical systems, starting with fundamental contributions of Hubbard and Oberste Vorth, Bedford and Smillie, and Fornaess and Sibony. This new direction of research has lead to new connections with related fields of research such as algebraic and complex geometry. In the last five years there has been great progress in this line of research. One of the most visible breakthroughs is the discovery of wandering Fatou components for polynomial maps, in a joint publication of Han Peters in Annals of Mathematics. Besides settling an old conjecture, this discovery gave rise to a whole range of new open problems, besides giving a new impulse to already existing problems, such as the possible existence of wandering domains for complex Henon maps. These questions are the topic of an ongoing Top 2 grant of Han Peters, which involves 2nd year PhD student Lorenzo Guerini. A recent new direction concerns the study of transcendental dynamical systems in higher dimensional dynamics. In a recent collaboration of the Peters and Fornaess, with Anna Miriam Benini and Leandro Arosio, a new class of maps called transcendental Henon maps was introduced and studied. It was shown that these maps portray a wide range of interesting dynamical behavior, but that they are considerably more rigid than arbitrary holomorphic automorphisms, offering more potential for rigorous description of the dynamics. A particularly interesting open problem is whether these maps always have infinite topological entropy. Peters and Fornaess, in collaboration with Benini, recently made an important step in solving this conjecture, by proving that transcendental maps in one complex variable always have infinite entropy.