Typical and atypical properties of periodic Teichmueller geodesics

TL;DR

This paper investigates the properties of periodic Teichmueller geodesics, identifying typical behaviors related to eigenvalues, trace fields, and orbit closures, and establishes finiteness results for certain invariant submanifolds in the moduli space.

Contribution

It characterizes typical properties of periodic orbits in Teichmueller flow and proves finiteness of algebraically primitive Teichmueller curves and higher-rank affine invariant submanifolds.

Findings

Eigenvalues of symplectic matrices approximate Lyapunov exponents

Trace fields are totally real of degree g

Finiteness of algebraically primitive Teichmueller curves and rank ≥ 2 submanifolds

Abstract

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.