The Teichmüller geodesic flow {g t }, first studied by H. Masur [15] and W. Veech [21], acts on the moduli space of Riemann surfaces endowed with a holomorphic differential. More precisely, let S be a closed surface of genus g ≥ 2. One introduces on S a complex structure σ and a holomorphic differential ω. The pair (σ, ω) is considered to be equivalent to another pair of the same nature (σ 1 , ω 1 ) if there is a diffeomorphism of S sending (σ, ω) to (σ 1 , ω 1 ). The moduli space M(g) consists of the equivalence classes, and the flow {g t } on M(g) is induced by the action on the pairs (σ, ω) defined by the formula g t (σ, ω) = (σ ′ , ω ′ ), where ω ′ = e t ℜ(ω) + ie −t ℑ(ω), while the complex structure σ ′ is determined by the requirement that ω ′ be holomorphic. If (σ, ω) and (σ ′ , ω ′ ) are equivalent, then the differentials ω and ω ′ have the same orders of zeros and the same area. Therefore, these orders and area are well-defined on M(g). Moreover, they are preserved by the Teichmüller flow {g t }. Take an arbitrary non-ordered collection κ = (k 1 , . . . , k r ) with k i ∈ N, k 1 + · · · + k r = 2g − 2, and denote by M κ the subspace of M(g) corresponding to the differentials of area 1 (i.e., (i/2) ω ∧ω = 1) with orders of zeros k i , i = 1, . . . , r; M κ is said to be a stratum in M(g). Each stratum
