Quasisymmetric geometry of the carpet Julia sets

Let $J_f$ be a Sierpi\'{n}ski carpet which is the Julia set of rational map $f$ and $\mathcal{C}$ the set of all peripheral circles of this carpet. We prove that $J_f$ is quasisymmetrically equivalent to a round carpet if the elements in $\mc{C}$ avoid the $\omega$-limt sets of all critical points of $f$. Suppose that $f$ is semi-hyperbolic, then the elements in $\mathcal{C}$ are uniform quasicircles. Moreover, the elements in $\mathcal{C}$ are uniformly relatively separated if and only if they are disjoint with the $\omega$-limit sets of all critical points.