Speaker: Georgios Tsikalas (Vanderbilt University)

Title: Denjoy-Wolff points on the bidisk

Abstract: Let f denote a holomorphic self-map of the unit disk without any interior fixed points. A classical 1926 theorem of Denjoy and Wolff then asserts that the sequence of iterates 

\[f^{[n]}:=f\circ f\circ \cdots \circ f.\]

converges locally uniformly to a boundary fixed point of f, termed the Denjoy-Wolff point. 

The situation changes dramatically when one considers holomorphic fixed-point-free self-maps of the bidisk; the presence of large ``flat" boundary components will, in general, prevent the iterates from converging. The cluster set of the sequence of iterates in this setting was described in a 1954 paper of Herv\'{e}. 

In this talk, we discuss extensions of the notion of a Denjoy-Wolff point to the bidisk. While this is a topic that has already been studied by several researchers, our approach introduces work of Agler, McCarthy and Young (2012) on boundary regularity into the mix. This will allow us to obtain certain refinements of Herv\'{e}'s results. This is joint work with Michael Jury.