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Is a typical bi-Perron algebraic unit a pseudo-Anosov dilatation?

Published online by Cambridge University Press:  28 November 2017

HYUNGRYUL BAIK ,
AHMAD RAFIQI  and
CHENXI WU
HYUNGRYUL BAIK
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Daejeon, South Korea email hrbaik@kaist.ac.kr
AHMAD RAFIQI
Affiliation:
Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853, USA email ar776@cornell.edu
CHENXI WU
Affiliation:
Department of Mathematics, Rutgers University, Hill Center – Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA email wchenxi2013@gmail.com
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Abstract

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic units whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$. As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area-one abelian differentials for low-genus cases.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1745 - 1750
Copyright
© Cambridge University Press, 2017 

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