Complete semi-conjugacies for psuedo-Anosov homeomorphisms

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Suppose S is a surface of genus ≥ 2, f: S → S is a surface homeomorphism isotopic to a pseudo-Anosov map α and suppose S is the universal cover of S and F and A are lifts of f and α respectively. We show there is a semiconjugacy Θ: S → Łˢ × Łᵘ from F to A, where Łˢ (Łᵘ) is the completion of the R-tree of leaves of the stable (resp. unstable) foliation for A and A is the map induced by A. We also generalize a result of Markovich and show that for any g ∈ Homeo(S) which commutes with f and has identity lift G: S → S and for any (c,w) in the image of Θ each component of Θ⁻¹(c,w) is G-invariant.