  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 $Q$.
