Hausdorff dimension of sets of nonergodic measured foliations (Q1185439)

Let \(F_ \theta\), \(\theta\in S^ 1\), be a one-parameter family of measured foliations of a surface \(M\) with flat structure \(q\). Put NE\((q)=\{\theta\in S^ 1:F_ \theta\) is not ergodic\}. The set of flat structures on a given surface forms a moduli space which is called a stratum. The stratum of the flat torus and the stratum of spheres with four singular points are called exceptional. The following is the main Theorem: For each component \(C\) of each nonexceptional stratum there exists \(\delta>0\) such that for almost every \(q\in C\) (with respect to the natural measure class of the stratum) the Hausdorff dimension of NE\((q)\) is \(\delta\).