[Paper Review] Typical properties of periodic Teichmueller geodesics

Ursula Hamenstaedt|arXiv (Cornell University)|Sep 21, 2014

Mathematical Dynamics and Fractals24 references3 citations

TL;DR

This paper establishes that certain dynamical and arithmetic properties of periodic Teichmueller geodesics are typical in the moduli space of area-one abelian differentials on genus g surfaces. It proves that eigenvalues of the monodromy matrix are arbitrarily close to Lyapunov exponents, the trace field is a totally real field of degree g over Q, and for g > 2, orbits with full SL(2,R)-orbit closure are typical. The work further establishes finiteness of affine invariant manifolds of fixed rank and of algebraically primitive Teichmueller curves.

ABSTRACT

Call a property for periodic orbits of the Teichmueller flow acting on the moduli space Q of area one abelian differentials on a surface of genus g typical if the growth rate of orbits with this property is maximal. We show that the following property is typical. The eigenvalues of the symplectic matrix defined by the orbit are arbitrarily close to the Lyapunov exponents of Q, 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 show that for every $l\geq 2$, there are finitely many affine invariant manifolds in Q of rank l which contain all affine invariant manifolds in Q of rank l, and we establish finiteness of algebraically primitive Teichmueller curves.

Motivation & Objective

• To identify and characterize typical properties of periodic orbits under the Teichmueller flow on the moduli space Q of area-one abelian differentials.
• To determine which dynamical and arithmetic features of periodic Teichmueller geodesics occur with maximal growth rate, defining 'typicality' via growth rate maximality.
• To establish the finiteness of affine invariant manifolds of fixed rank l ≥ 2 in Q, and to prove finiteness of algebraically primitive Teichmueller curves.
• To analyze the monodromy matrices of periodic orbits and their relation to Lyapunov exponents and trace fields.
• To show that for g > 2, periodic orbits whose SL(2,R)-orbit closure equals the entire moduli space Q are typical.

Proposed method

• The authors define 'typicality' as maximal growth rate of periodic orbits with a given property, using asymptotic counting in the moduli space Q.
• They analyze the monodromy matrix associated with each periodic orbit and compare its eigenvalues to the Lyapunov exponents of the Teichmueller flow.
• They use the trace field of the monodromy matrix to show it is a totally real field of degree g over Q, under typical conditions.
• The proof relies on deep results from Teichmueller dynamics, including the structure of SL(2,R)-orbit closures and the classification of affine invariant submanifolds.
• They apply finiteness results for affine invariant manifolds of fixed rank, showing that for each l ≥ 2, only finitely many such manifolds contain all others of rank l.
• The argument for finiteness of algebraically primitive Teichmueller curves combines dynamical constraints with number-theoretic properties of trace fields and Galois actions.

Experimental results

Research questions

• RQ1 Which dynamical and arithmetic properties of periodic Teichmueller geodesics occur with maximal growth rate in the moduli space Q of area-one abelian differentials?
• RQ2 To what extent do the eigenvalues of the monodromy matrix of a periodic orbit approximate the Lyapunov exponents of the Teichmueller flow?
• RQ3 What is the structure of the trace field of the monodromy matrix for typical periodic orbits, and how does it relate to the genus g of the surface?
• RQ4 Are there only finitely many affine invariant submanifolds of fixed rank l ≥ 2 in Q, and do they admit a maximal element under inclusion?
• RQ5 Is the set of algebraically primitive Teichmueller curves in Q finite, and what dynamical or arithmetic constraints enforce this finiteness?

Key findings

• The eigenvalues of the monodromy matrix associated with a typical periodic orbit are arbitrarily close to the Lyapunov exponents of the Teichmueller flow on Q.
• For typical periodic orbits, the trace field of the monodromy matrix is a totally real field of degree g over Q.
• When g > 2, periodic orbits whose SL(2,R)-orbit closure equals the entire moduli space Q are typical, meaning they occur with maximal growth rate.
• For every l ≥ 2, there are only finitely many affine invariant submanifolds of rank l in Q that contain all other such submanifolds of rank l.
• The set of algebraically primitive Teichmueller curves in Q is finite, as a consequence of the finiteness of affine invariant manifolds and dynamical constraints on their structure.