Quasiregular maps of Sierpiński carpet Julia sets
Sergei Merenkov, Letian Shen
TL;DR
The paper proves a strong rigidity phenomenon for quasiregular maps between carpet Julia sets of postcritically finite rational maps, showing that any $\xi$ with $\xi^{-1}(\mathcal{J}(g))=\mathcal{J}(f)$ extends to a rational map on $\widehat{\mathbb{C}}$, and, in the case $f=g$, satisfies a dynamical relation $f^k\circ \xi^l=f^{2k}$. The approach blends Schottky-map local rigidity, conformal elevator techniques, and entropy-based arguments to produce a global functional equation on $\mathcal{J}(f)$, extend it to a rational map, and deduce postcritical finiteness and iterates relations. The work extends prior results by Bonk–Lyubich–Merenkov and clarifies the role of carpet topology; it also shows that these rigidity phenomena can fail for Julia sets of other topologies such as tree-like or gasket types. Overall, the paper advances understanding of how quasiregular maps are constrained by carpet Julia-set dynamics and highlights sharp contrasts with non-carpet cases.
Abstract
We prove that if $f$ and $g$ are postcritically finite rational maps whose Julia sets $\mathcal{J}(f), \mathcal{J}(g)$, respectively, are Sierpiński carpets, and if $ξ$ is a quasiregular map of the Riemann sphere $\widehat{\mathbb{C}}$ with $ξ^{-1}(\mathcal{J}(g))=\mathcal{J}(f)$, then $ξ$ is the restriction of a rational map to the Julia set $\mathcal{J}(f)$. Moreover, when $g=f$ we prove that, for some positive integers $k$ and $l$, $f^k\circ ξ^l=f^{2k}$. These conclusions extend the main results of M. Bonk, M. Lyubich, S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, Adv. Math, 301 (2016), 383-422. Finally, we demonstrate that when Julia sets of postcritically finite rational maps are not Sierpiński carpets, say they are tree-like or gaskets, the above conclusions no longer hold.