On dynamics of vertices of locally connected polynomial Julia sets

Abstract

Let $P$ be a polynomial whose Julia set $J$ is locally connected. Then a non-preperiodic non-precritical vertex of $J$ must have the limit set which coincides with the limit set of an appropriately chosen recurrent critical point of $P$. In particular, if all critical points of $P$ are non-recurrent then all vertices of $J$ are preperiodic or precritical.