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.
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This work was supported by National Science Foundation (Grant no. DMS-1807558).
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Dedicated to Misha Lyubich’s 60-th birthday.
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The second named author was partially supported by NSF grant DMS-1807558. The third named author has been supported by the HSE University Basic Research Program.
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Blokh, A., Oversteegen, L. & Timorin, V. Cutpoints of Invariant Subcontinua of Polynomial Julia Sets. Arnold Math J. 8, 271–284 (2022). https://doi.org/10.1007/s40598-021-00186-8
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DOI: https://doi.org/10.1007/s40598-021-00186-8