Siegel Measures

The goals of this paper are first to describe and then to apply an ergodictheoretic generalization of the Siegel integral formula from the geometry of numbers. The general formula will be seen to serve both as a guide and as a tool for questions concerning the distribution, in senses to be made precise, of the set of closed leaves of measured foliations subordinate to meromorphic quadratic differentials on closed Riemann surfaces. In preparation of a discussion of the main results we recall two earlier theorems. The first of these, by H. Masur, has been a starting point for the present work. Let q be a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface X. For a certain countable set of θ ∈ R the horizontal foliation associated to e−2iθq has one or more maximal cylinders of closed leaves. Each cylinder determines a pair of vectors v = ±reiθ, where r is the common |q|-length of closed leaves in the cylinder. Let Π(q) be the set of vectors, with multiplicities, which arise from closed cylinders as θ varies. Finally, let N(q,R) = Card{v ∈ Π(q) | |v| < R} be the growth function of Π(q).