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

Masur, Howard; Smillie, John

Ann. Math. (2) 134, No. 3, 455-543 (1991).

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 \(\delta\).

Reviewer: H.Crauel (Saarbrücken)

MSC:

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

Keywords:

sets of nonergodic measured foliations; Hausdorff dimension