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 [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], 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 the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], 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, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

中文翻译：

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极值和动态定义的测量，第二部分：来自保形动力系统的测量

我们提出了一种证明各种动态定义度量的丢番图极值的新方法，极大地扩展了已知极值度量的类别。这概括并改进了著名的 Kleinbock 和 Margulis 定理 [齐次空间流动的对数定律。发明。数学。138(3) (1999), 451–494] 解决 Sprindžuk 猜想，以及 Kleinbock、Lindenstrauss 和 Weiss 对其的扩展 [关于分形测度和丢番图近似。选择数学。10(2004), 479–523]，以下简称 KLW。作为应用，我们证明了具有足够大 Hausdorff 维数的光滑动力系统的所有双曲线测度的极值性，以及所有非平面几何有限群的 Patterson-Sullivan 测度的极值性。导致大量新应用的关键技术思想是显着削弱 KLW 的极值充分条件。在本系列论文的第一篇[极值和动态定义的措施，第一部分：准衰减措施的丢番图性质。选择数学。24(3) (2018), 2165–2206]，我们引入并发展了对两类措施的系统描述，我们称之为准腐烂和弱准衰减. 我们证明了弱准衰减意味着矩阵逼近框架中的强极值性，并证明了该定理的“非理性继承指数”版本。在本系列的第二篇论文中，我们建立了各类共形动力系统的充分条件，以使其测度准衰减。特别地，我们证明了上述关于 Patterson-Sullivan 测度的结果，并且我们证明了非平面无限迭代函数系统（包括那些不满足开集条件的系统）和有理函数的平衡状态（包括保形测度）是拟定的。 -腐烂。