New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant Public Deposited

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/8s45qd508

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  • Translation surfaces can be viewed as polygons with parallel and equal sides identified. An affine homeomorphism φ from a translation surface to itself is called pseudo-Anosov when its derivative is a constant matrix in SL₂(R) whose trace is larger than 2 in absolute value. In this setting, the eigendirections of this matrix defines the stable and unstable flow on the translation surface. Taking a transversal to the stable flows, the first return map of the flow induces an interval exchange transformation T. The Sah-Arnoux-Fathi invariant of φ is the sum of the wedge product between the lengths of the subintervals of T and their translations. This wedge product does not depend on the choice of transversal. We apply Veech’s construction of pseudo-Anosov homeomorphisms to produce infinite families of pseudo-Anosov maps in the stratum H(2, 2) with vanishing Sah-Arnoux-Fathi invariant, as well as sporadic examples in other strata.
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