Cutpoints of invariant subcontinua of polynomial Julia sets

Alexander Blokh; Lex Oversteegen; Vladlen Timorin

paper

Cutpoints of invariant subcontinua of polynomial Julia sets

arXiv:2010.00419

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 $Q$ .

17 pages