On the Lagrange spectrum of moduli spaces of Riemann surfaces $M_{g,n}$

We consider the moduli space $M_{g,n}$ of Riemann surfaces of
genus g with n punctures endowed with the Teichmueller metric.
The Teichmueller geodesic flow is a non-uniformly hyperbolic
flow on $M_{g,n}$ (with respect to the Masur-Veech measure). The
Lagrange spectrum of $M_{g,n}$ is a closed subset of the positive
real numbers that measures how closed geodesics escape $(M_{g,n}$ is
not compact). The classical Lagrange spectrum corresponds to the
case of $M_{1,1}$ and is motivated by diophantine approximations.
We will show that some properties of the classical Lagrange
spectrum extends to the Lagrange spectrum of any $M_{g,n}$. But most
of it remains mysterious.