Let's consider a Riemann surface $X$ of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. A natural parameter on $X$ is a chart for which $q=dz^2$. A $\theta$-trajectory is a maximal arc which, in the natural parameter, is a straight line with slope $\theta$. A $\theta$-trajectory is a saddle connection if both of its ends go into a zero of $q$. I have two questions for which I don't know if there is a known result, so I'll ask them and I hope that you can give me an answer and a reference:
1) For a fixed $\theta\in \mathbb{RP}^1$, how many saddle connections with direction $\theta$ are there?
2) Fixed a point $x\in X$. For which $\theta\in \mathbb{RP}^1$ is there a saddle connection with direction $\theta$?
Is it different if instead of $q$ I consider a 1-form $\omega$?
