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Zbl 0661.30034
Masur, Howard
Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential. (English)
[A] Holomorphic functions and moduli I, Proc. Workshop, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 10, 215-228 (1988).

[For the entire collection see Zbl 0646.00004.] \par Suppose X is a closed surface of genus $\ge 2$ with a complex structure, and let $q\sb 0(z)dz\sp 2$ be a nonzero holomorphic quadratic differential on X. Let $N\sb 1(T)$ denote the number of singular trajectories of the differential of length $\le T$ joining two zeroes of $q\sb 0$ and let $N\sb 2(T)$ denote the number of parallel families of closed regular trajectories of length $\le T$. The results in the present paper and in two preceding papers of the author establish the following bounds for the numbers $N\sb 1(T)$ and $N\sb 2(T).$ \par Theorem 1: $$ 0<\underline{\lim}\sb{T\to \infty}(N\sb 1(T)/T\sp 2)\quad \le \quad \overline{\lim}\sb{T\to \infty}(N\sb 1(T)/T\sp 2)<\infty. $$ Theorem 2: There exists $c>0$ such that $$ cN\sb 1(T)\le N\sb 2(T)\le N\sb 1(T). $$
[V.L.Hansen]

MSC 2000:: *30F30 Differentials on Riemann surfaces

Keywords: quadratic differential

Citations: Zbl 0646.00004

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