  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.
