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 $\text{ti}S$ 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 $g \in Homeo(S)$whichcommuteswith$f$andhasidentitylift$G : \ti S \to \ti S$andforany$(c,w)$intheimageof$\Theta$eachcomponentof$\Theta^{-1}(c,w)$is$G$-invariant.