Is a typical bi-Perron number a pseudo-Anosov dilatation?
  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. 
Hyungryul Baik, Ahmad Rafiqi, Chenxi Wu
math.GT math.DS