Cutpoints of Invariant Subcontinua of Polynomial Julia Sets

Abstract

We prove fixed point results for branched covering maps f of the plane. For complex polynomials P with Julia set \(J_{P}\) these imply that periodic cutpoints of some invariant subcontinua of \(J_{P}\) are also cutpoints of \(J_{P}\). We deduce that, under certain assumptions on invariant subcontinua Q of \(J_{P}\), every Riemann ray to Q landing at a periodic repelling/parabolic point \(x\in Q\) is isotopic to a Riemann ray to \(J_{P}\) relative to Q.