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Transactions of the American Mathematical Society

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A construction of pseudo-Anosov homeomorphisms

Author: Robert C. Penner
Journal: Trans. Amer. Math. Soc. 310 (1988), 179-197
MSC: Primary 57N05; Secondary 20F34, 58F15
MathSciNet review: 930079
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Abstract: We describe a generalization of Thurston's original construction of pseudo-Anosov maps on a surface $ F$ of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map $ f:F \to F$ arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by $ f$. To finally prove that $ f$ is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov.

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Keywords: Pseudo-Anosov, measured foliation, train track, mapping class group, Dehn twist
Article copyright: © Copyright 1988 American Mathematical Society

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