Complete semi-conjugacies for psuedo-Anosov homeomorphisms

Suppose $S$ is a surface of genus $\ge 2 $, $f: S \to S$ is a surface homeomorphism isotopic to a pseudo-Anosov map $\alpha$ and suppose $\ti S$ is the universal cover of $S$ and $F$ and $A$ are lifts of $f$ and $\alpha$ respectively. We show there is a semiconjugacy $\Theta : \ti S \to \bar \L^s \times \bar \L^u$ from $F$ to $\bar A$, where $\bar \L^s$ ($\bar \L^u$) is the completion of the $R$-tree of leaves of the stable (resp. unstable) foliation for $A$ and $\bar A$ is the map induced by $A$. We also generalize a result of Markovich and show that for any $g \in Homeo(S)$ which commutes with $f$ and has identity lift $G : \ti S \to \ti S$ and for any $(c,w)$ in the image of $\Theta$ each component of $\Theta^{-1}(c,w)$ is $G$-invariant.