Dynamical Systems Seminar: Baker domains in one and two variables

Dates

Wednesday, February 24, 2021 - 14:00 to 15:00

Location

Teams Meeting

Speaker: Anna Miriam Benini  (University of Parma)

Title: Baker domains in one and two variables

 For polynomial maps on the complex plane $\mathbb{C}$ there is only one way in which orbits can converge to infinity: they have to belong to the attracting basin of infinity, which can be seen as a superattracting fixed point.  For transcendental maps infinity is an essential singularity, and there are many ways in which orbits can converge to infinity while belonging to a periodic Fatou component.  Such components are called Baker domains and can sport several different dynamical features. 

In $\mathbb{C}^2$ we call a Fatou component escaping if the orbits within converge to the line at infinity (we consider the projective space as compactification of $\mathbb{C}^2$ ). The picture is somewhat similar as the one-dimensional case: for polynomial automorphisms of $\mathbb{C}^2$,  escaping orbits belong to a Fatou component which can be seen as an attracting basin of a point on the line at infinity, while for transcendental automorphisms, many more possibilities for escaping Fatou components open up both from the geometrical and from the dynamical point of view. We will describe a few examples and present some of the many possible open questions that arise. This talk includes results obtained with Arosio, Fornaess, Peters and with Saracco, Zedda.