Hausdorff dimension of sets of nonergodic measured foliations. (English) Zbl 0774.58024

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 \(\delta\).


37A99 Ergodic theory
54H20 Topological dynamics (MSC2010)
28A78 Hausdorff and packing measures
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