Rodrigo A. Perez, Indiana State University

Abstract

When a holomorphic function $f:C to C$ has a fixed point f(0)=0 with derivative λ=f'(0) of unit size, the question arises of conjugating f to the rotation z ->λ z. This is possible when the argument of λ has good approximation properties; eg, when it is Diophantine. The largest domain of conjugation is known as a Siegel disk.

A famous open problem is to give bounds on the size of Siegel disks. As a concrete case, if $Arg(lambda)$ is the Golden Ratio, does the Siegel disk contain a disk of radius 1/4?

In the talk I will explore a circle of ideas emerging from our approach to this problem: The value 1/4 is connected to the growth of Catalan numbers enumerating binary trees. A consequence of this correspondence is a 2 variable version of the Vandermonde convolution for (deformed) q-binomials. The work is a joint collaboration with M. Aspenberg, Lund University.

Link to video: https://vimeo.com/1013378023?share=copy

Presented Via New Zoom: https://rutgers.zoom.us/j/91865817691

Password: 6564120420

For further information see: https://sites.math.rutgers.edu/~zeilberg/expmath/