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In this note, we deduce a partial answer to the question in the title. Inparticular, we show that asymptotically almost all bi-Perron algebraic unitwhose characteristic polynomial has degree at most $2n$ do not correspond todilatations 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 statementon the number of closed geodesics of the same length in the moduli space ofarea one abelian differentials for low genus cases.

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