Abstract:

If $z=0$ is a fixed point of multiplier $\lambda \; =\; ei\; 2\; \pi \; \theta$ and θ is a Brjuno number, we know that there exists a Siegel disk D at the origin. We ask under which arithmetic conditions for θ we can assure that the boundary is a quasicircle and/or contains a critical point.

In this talk we will review the existing partial answers to this question, and the methods in the proofs. If time permits, we will explain in detail the surgery construction that proves that for the function λ $sin\; z$, if θ is of constant type then the boundary of the Siegel disk is a quasicircle and contains both critical points. This result is proved by Gaogei Zhang in "On the dynamics of $e2\; \pi \; i\; \theta sin(z)$", Illinois Journal of Mathematics, Vol. 49, 2005.