Quasiregular maps of Sierpi\'nski carpet Julia sets

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.