Research Interests

My dissertation research was in extending theorems from analytic self maps of the complex open unit disk, to analytic self maps of the open polydisk, with specific focus on the Denjoy-Wolff fixed point theorem. The single variable version of this theorem establishes that, in the cases where an analytic function (denote: f) doesn't have a fixed point within the open disk, that there is an `attracting point' on the boundary (denote: T) such that; for any value of the disk (denote: z) the nth composition of f applied to z, as n tends towards infinity, will converge (in the classic limit sense) to T.

The general goal of my dissertation was to investigate the analogous result on the polydisk. There were some very limited results obtained in the case of the two dimensional disk case, but the methodology was not generalizable. My original goal was to determine what, if any, topological properties the set of such points must have in the bidisk, and to find a way to generalize the approach to arbitrary (finite) dimension if possible.

In my dissertation I created a number of counterexamples to demonstrate that most of the anticipated topological properties fail to hold. In particular I built counter examples to my original conjecture; that the set of such points, which I call the Denjoy-Wolff set, is either open (in any coordinate that was not a singleton), or else the entire set was closed. Some topological properties do hold however; I proved that the Denjoy Wolff set of an analytic self-map of the polydisk is connected and I also proved the closest result I could determine to the original open or closed conjecture, which was a kind of converse. All my results were done using techniques that immediately generalize to arbitrary finite dimension.

• Connectedness of Denjoy-Wolff Sets on the polydisk. (In Progress 2021)

• Determining openness or closedness of preimages of Subsets of the Denjoy Wolff Set of a polydisk. (In Progress 2021)