Consider a polynomial with a Siegel disc of bounded type rotation number. It is known that the Siegel boundary is a quasi-circle that contains at least one critical point. In the quadratic case, this means that the entire post-critical set is trapped within the Siegel boundary, where the theory of real analytic circle maps provides us with excellent control. However, in the higher degree case, there exist multiple critical points. A priori, these “free” critical points may accumulate on the Siegel boundary in a complicated way, causing extreme distortions in the geometry nearby. In my talk, I show that in fact, this does not happen, and that the Julia set is locally connected at the Siegel boundary.
In the remaining time, I will describe the global structure of such Julia sets. Finally, I will discuss the application of these results to the parameter space of cubic Siegel polynomials, showing, in particular, that the Zakeri curve is locally connected.