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Constructing pseudo-Anosov maps with given dilatations

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Abstract

In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.

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Acknowledgments

This paper would not be possible without W. Thurston’s idea of constructing pseudo-Anosov maps using postcritically-finite maps of intervals, and turning them into \(\omega \)-limit sets of 2-dimensional dynamical systems [5]. We would also like to thank John H. Hubbard and Dylan Thurston for discussions and helpful suggestions during the process of writing this paper, Joshua P. Bowman, Andre de Carvalho, Giulio Tiozzo and Danny Calegari for helpful insights, and Fatma Rekik for helping with the drawing and coloring of the figures. Special thanks to the referee for helpful suggestions, comments and corrections. The first author was partially supported by the ERC Grant No. 10160104.

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Authors and Affiliations

  1. Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Hyungryul Baik

  2. Department of Mathematics, Cornell University, 111 Malott Hall, Ithaca, NY, 14853, USA

    Ahmad Rafiqi

  3. Department of Mathematics, Cornell University, 105 Malott Hall, Ithaca, NY, 14853, USA

    Chenxi Wu

Authors
  1. Hyungryul Baik
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  2. Ahmad Rafiqi
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  3. Chenxi Wu
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Correspondence to Ahmad Rafiqi.

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Baik, H., Rafiqi, A. & Wu, C. Constructing pseudo-Anosov maps with given dilatations. Geom Dedicata 180, 39–48 (2016). https://doi.org/10.1007/s10711-015-0089-1

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  • DOI: https://doi.org/10.1007/s10711-015-0089-1

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