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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