Is a typical bi-Perron number a pseudo-Anosov dilatation?

Hyungryul Baik; Ahmad Rafiqi; Chenxi Wu

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arXiv · 1612.03084

Is a typical bi-Perron number a pseudo-Anosov dilatation?

Hyungryul Baik·Ahmad Rafiqi·Chenxi Wu

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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 unit 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.

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Hyungryul Baik, Ahmad Rafiqi, Chenxi Wu. 2016-12-09. Is a typical bi-Perron number a pseudo-Anosov dilatation?. https://arxiv.org/abs/1612.03084

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