An important problem in combinatorial noncommutative algebra is to give an asymptotic characterization the growth functions of finitely generated algebras (equivalently, semigroups, or hereditary languages). The function every generated, infinite-dimensional increasing and submultiplicative. question what extent these natural necessary conditions are also sufficient—and particular, whether they...

Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball radius $R$ in Teichm\"{u}ller space is asymptotic to $e^{hR}$, where $h$ dimension space. We show for any pseudo-Anosov $f$, there exists power $n$, such that $f^n$ conjugacy coarsely $e^{\frac{h}{2}R}$.