Cutpoints of invariant subcontinua of polynomial Julia sets

TL;DR

This paper investigates the structure of invariant subcontinua within polynomial Julia sets, establishing fixed point results and properties of Riemann rays, which enhance understanding of the topological and dynamical features of Julia sets.

Contribution

It proves fixed point results for branched coverings and shows how these results imply properties of cutpoints and Riemann rays in Julia sets of complex polynomials.

Findings

Periodic cutpoints of invariant subcontinua are also cutpoints of the Julia set.

Riemann rays landing at certain points are isotopic to rays of the entire Julia set.

Under specific conditions, the structure of invariant subcontinua is well-understood.

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 $Q$.