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The definition of pseudo-Anosov homeomorphisms on a surface-with-boundary is not as well established as it is on a closed surface. Your question exhibits one reason for this: however you might want to formulate it, this semiconjugacy statement is false on a surface with boundary. Here's why. Let $S$ be the surface, and list its boundary components as $\... 4 In fact, the argument by Fried that you describe is incomplete. It is unclear that such a scheme can be made to work and there is evidence that it may not work (why should the blow down be smooth, and if it were, why would it be a smooth Anosov flow rather than a smooth topological Anosov flow). However, the result has been clarified and rigourously proved ... 2 The fact that a stable manifold is diffeomorphic to a (real) Euclidean space is a consequence of the Stable Manifold Theorem, see for instance [Katok&Hasselblatt, Introduction to the modern theory of dynamical systems. chap 6 sec 4]: By the Stable Manifold Theorem for every$p\in M$, there is a local stable manifold$W^s_{loc}(p)$which is diffeomorphic ... 2 This question is addressed in detail in Isotopy Stability of Dynamics on Surfaces, Contemp. Math., 246, 17-46, 1999 or arXiv:math/9904160 [math.DS]. 2 In general, for the geodesic flow of a compact negatively curved manifold for example, the strong stable foliation is Holder continuous; in particular, even if each leaf is (locally) a smooth manifold, transversally, the foliation (and therefore the map$x\to\delta_x\$) is no more than Hölder continuous. In the case of geodesic flows in constant negative ...