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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References:

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