线性与动态定义测度，第一部分：拟衰减测度的丢番图性质
Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures
摘要：
我们提出了一种新的方法证明丢番图极值的各种动态定义的措施，大大扩展了类的措施被称为极值。这推广和改进了Kleinbock和Margulis（'98）解决Sprindžuk猜想的著名定理，以及Kleinbock，Lindenstrauss和Weiss（' 04）的扩展，以下简称KLW。作为应用，我们证明了具有足够大的豪斯多夫维数的光滑动力系统的所有双曲测度的极值性，以及所有非平面几何有限群的Patterson-Sullivan测度的极值性.关键的技术思想，这导致了大量的新应用，是一个显着削弱KLW的充分条件的极值。在第一部分中，我们介绍并发展了两类度量的系统说明，我们称之为$quasi$ - $decaying$和$weakly$ $quasi$ - $decaying$。我们证明了弱拟衰减蕴含强极值的矩阵近似框架（这是近年来备受关注），从而证明了KLW的猜想。我们还证明了这个定理的“继承指数的非理性”的版本，描述的丢番图属性的空间的矩阵和措施支持这些子空间之间的关系。在随后的论文中，我们展示了大量的准衰减措施的例子，在支持的论文，“几乎任何措施，从动力学和/或分形几何是准衰减”。除了上述例子，我们还证明了（例如）无限迭代函数系统的Gibbs测度（包括共形测度）是拟衰减的，即使所讨论的系统不满足开集条件.我们还讨论了来自动力学的非极值措施的例子，说明理论必须停止。
英文摘要：
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis ('98) resolving Sprindžuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss ('04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson--Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW's sufficient conditions for extremality. In Part I, we introduce and develop a systematic account of two classes of measures, which we call $quasi$-$decaying$ and $weakly$ $quasi$-$decaying$. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework (which has received much attention in recent years), thus proving a conjecture of KLW. We also prove the "inherited exponent of irrationality" version of this theorem, describing the relationship between the Diophantine properties of certain subspaces of the space of matrices and measures supported on these subspaces. In subsequent papers, we exhibit numerous examples of quasi-decaying measures, in support of the thesis that "almost any measure from dynamics and/or fractal geometry is quasi-decaying". In addition to the examples described above, we also prove (for example) that Gibbs measures (including conformal measures) of infinite iterated function systems are quasi-decaying, even if the systems in question do not satisfy the open set condition. We also discuss examples of non-extremal measures coming from dynamics, illustrating where the theory must halt.
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