Typical and atypical properties of periodic Teichmueller geodesics
  Consider a component Q of a stratum in the moduli space of area one abelian differentials on a surface of genus g. Call a property P for periodic orbits of the Teichmueller flow typical if the growth rate of orbits with this property is maximal. Typical are: The logarithms of the eigenvalues of the symplectic matrix defined by the orbit are arbitrarily close to the Lyapunov exponents of Q, and its trace field is a totally real splitting field of degree g over Q. If g>2 then periodic orbits whose SL(2,R)-orbit closure equals Q are typical. We also show that Q contains only finitely many algebraically primitive Teichmueller curves, and only finitely many affine invariant submanifolds of rank at least 2. 
Ursula Hamenstaedt
math.DS