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A construction of pseudo-Anosov homeomorphisms. (English) Zbl 0706.57008

Summary: 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.

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57R30 Foliations in differential topology; geometric theory
57R50 Differential topological aspects of diffeomorphisms
37D99 Dynamical systems with hyperbolic behavior
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