We study the stability of statistical properties of Anosov maps on tori by examining the stability of the spectrum of an analytically twisted Perron-Frobenius operator on the anisotropic Banach spaces of Gouezel and Liverani. By extending our previous work in [8], we obtain the stability of various statistical properties (the variance of a CLT and the rate function of an LDP) of Anosov maps to general perturbations, including new classes of numerical approximations. In particular, we obtain new results on the stability of the rate function under deterministic perturbations. As a key application, we focus on perturbations arising from numerical schemes and develop two new Fourier-analytic method for efficiently computing approximations of the aforementioned statistical properties. This includes the first example of a rigorous scheme for approximating the peripheral spectral data of the Perron-Frobenius operator of an Anosov map without mollification. Using the two schemes we obtain the first rigorous estimates of the variance and rate function for Anosov maps.

中文翻译：

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Anosov 映射在环面上的统计特性的傅里叶近似

我们通过检查 Gouezel 和 Liverani 的各向异性 Banach 空间上解析扭曲的 Perron-Frobenius 算子的频谱稳定性来研究 Anosov 映射在 tori 上的统计特性的稳定性。通过扩展我们之前在 [8] 中的工作，我们获得了 Anosov 映射到一般扰动的各种统计特性（CLT 的方差和 LDP 的速率函数）的稳定性，包括新类别的数值近似。特别是，我们获得了确定性扰动下速率函数稳定性的新结果。作为一个关键应用，我们专注于数值方案引起的扰动，并开发了两种新的傅立叶分析方法来有效计算上述统计特性的近似值。这包括一个严格方案的第一个例子，用于在没有软化的情况下逼近 Anosov 地图的 Perron-Frobenius 算子的外围光谱数据。使用这两种方案，我们获得了 Anosov 地图方差和速率函数的第一个严格估计。