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Pseudo-Anosov Homeomorphisms on Translation Surfaces in Hyperelliptic Components Have Large Entropy

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Abstract

We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belongs to a hyperelliptic component is bounded from below uniformly by \({\sqrt{2}}\) . This is in contrast to Penner’s asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surface of genus g tends to zero at rate 1/g (as g goes to infinity).

We also show that our uniform lower bound \({\sqrt{2}}\) is sharp. More precisely, the least dilatation of a pseudo-Anosov on a genus g > 1 translation surface in a hyperelliptic component belongs to the interval \({]\sqrt{2}, \sqrt{2}+2^{1-g}[}\). The proof uses the Rauzy–Veech induction.

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Authors and Affiliations

  1. Laboratoire d’Analyse, Topologie et Probabilité, Université Paul Cézanne, Faculté de Saint Jérôme, Avenue Escadrille Normandie-Niemen, case cour A, 13397, Marseille cedex 20, France

    Corentin Boissy

  2. Centre de Physique Théorique (CPT), UMR CNRS 6207 Université du Sud Toulon-Var and Fédération de Recherches des Unités de Mathématiques de Marseille Luminy, Case 907, F-13288, Marseille Cedex 9, France

    Erwan Lanneau

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  1. Corentin Boissy
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  2. Erwan Lanneau
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Boissy, C., Lanneau, E. Pseudo-Anosov Homeomorphisms on Translation Surfaces in Hyperelliptic Components Have Large Entropy. Geom. Funct. Anal. 22, 74–106 (2012). https://doi.org/10.1007/s00039-012-0152-0

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  • DOI: https://doi.org/10.1007/s00039-012-0152-0

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