Abstract
The set of directions from a finite area quadratic differential on a Riemann surface of finite type that diverge on average under Teichmüller geodesic flow has Hausdorff dimension exactly equal to one-half.
Highlights
Suppose gt is a flow on a topological space S
Let Mg,n be the moduli space of closed genus g Riemann surfaces with n punctures
The cotangent space of Mg,n coincides with the moduli space Qg,n of holomorphic quadratic differentials on surfaces in Mg,n and admits an SL(2, R) action generated by complex scalar multiplication rθ(q) = eiθq and Teichmuller geodesic flow gt
Summary
Suppose gt is a flow on a topological space S. For any holomorphic quadratic differential q it is a consequence of Chaika-Eskin [CE15, Theorem 1.1] that the set of directions θ such that rθq diverges on average under the Teichmuller flow in its stratum has measure zero. For a quadratic or Abelian differential q the set of directions θ such that rθq diverges on average In [MS91], the main theorem is that outside finitely many exceptional strata of quadratic differentials (the exceptions being the ones where every flat structure induced by a holomorphic quadratic differential has a holonomy double cover that is a translation covering of a flat torus) , there is a constant δ > 0 depending on the stratum so that for almostevery quadratic differential q in the stratum the set of directions with non-ergodic flow - NE(q) - has Hausdorff dimension exactly δ. The fact that D(q) is positive dimensional for a full measure set of q and zero-dimensional for a dense set of q shows that an analogue of Theorem 1 for divergent directions does not exist in general
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