Local Connectivity of Polynomial Julia sets at Bounded Type Siegel Boundaries

Abstract

Consider a polynomial $f$ of degree $d \geq 2$ that has a Siegel disk $\Delta_f$ with a rotation number of bounded type. We prove that there does not exist a hedgehog containing $\Delta_f$. Moreover, if the Julia set $J_f$ of $f$ is connected, then it is locally connected at the Siegel boundary $\partial \Delta_f$.