We show that the Hausdorff dimension of any proper Teichmüller horocycle flow orbit closure on any irreducible $\textrm {SL}{(2,\textbf {R})}$-invariant subvariety of Abelian or quadratic differentials is bounded away from the dimension of the subvariety in terms of the polynomial mixing rate of the Teichmüller horocycle flow on the subvariety. The proof is based on abstract methods for measurable flows adapted from work of Bourgain and Katz on sparse ergodic theorems.

中文翻译：

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Teichmüller Horocycle 流动轨道闭合的 Hausdorff 维数的界限

我们证明了在任何不可约 $\textrm {SL}{(2,\textbf {R})}$-阿贝尔或二次微分的不变子变体上，任何适当的 Teichmüller horocycle 流轨道闭合的 Hausdorff 维数与以 Teichmüller horocycle 流对子品种的多项式混合率表示的子品种。该证明基于可测量流的抽象方法，该方法改编自 Bourgain 和 Katz 关于稀疏遍历定理的工作。