Quasiregular maps of Sierpi\'nski carpet Julia sets

TL;DR

This paper characterizes quasiregular maps that map Sierpiński carpet Julia sets of postcritically finite rational maps onto each other, showing they are restrictions of rational maps and establishing algebraic relations when the maps coincide.

Contribution

It extends previous results by proving that such quasiregular maps are restrictions of rational maps and establishes algebraic relations in the case of identical Julia sets.

Findings

Quasiregular maps between Sierpiński carpet Julia sets are restrictions of rational maps.

When the Julia sets are the same, a specific algebraic relation involving iterates holds.

The results do not extend to Julia sets that are not Sierpiński carpets, such as tree-like or gasket structures.

Abstract

We prove that if $f$ and $g$ are postcritically finite rational maps whose Julia sets $\mathcal{J}(f), \mathcal{J}(g)$, respectively, are Sierpi\'nski carpets, and if $\xi$ is a quasiregular map of the Riemann sphere $\widehat{\mathbb{C}}$ with $\xi^{-1}(\mathcal{J}(g))=\mathcal{J}(f)$, then $\xi$ 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 $l$, $f^k\circ \xi^l=f^{2k}$. These conclusions extend the main results of M. Bonk, M. Lyubich, S. Merenkov, Quasisymmetries of Sierpi\'nski carpet Julia sets, Adv. Math, 301 (2016), 383-422. Finally, we demonstrate that when Julia sets of postcritically finite rational maps are not Sierpi\'nski carpets, say they are tree-like or gaskets, the above conclusions no longer hold.