A construction of pseudo-Anosov homeomorphisms

Penner, Robert C.

doi:10.1090/S0002-9947-1988-0930079-9

A construction of pseudo-Anosov homeomorphisms
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Trans. Amer. Math. Soc. 310 (1988), 179-197 Request permission

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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