Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps $f\text{\hspace{0.17em}⁣}:S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is semi-conjugate to $z^d\text{\hspace{0.17em}⁣}:S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $\gamma \text{\hspace{0.17em}⁣}:S^1\to S^2$ (onto), such that $F\circ \gamma (z)=\gamma (z^d)$ (for all $z\in S^1$).