TITLE: Handel's Theorem for surfaces with boundary
QUESTION [6 upvotes]: Handel's Theorem(Entropy and semi-conjugacy in dimension two, 1987): let $M$ denote a closed surface. Let $\vartheta$ be a pseudo-Anosov (orientation-presrv.) homeomorphism of $M$ and $g$ be an (orientation-presrv.) homeomorphism of $M$ isotopic to $\vartheta$. If the topological entropy of $\vartheta$ equals the topological entropy of $g$, then $\vartheta$ is semi-conjugate to $g$.
Is there any similar statement that holds for surfaces with boundary ?

REPLY [2 votes]: 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].