We show that the ergodic integrals for the horocycle flow on the two-torus associated by Giulietti and Liverani with an Anosov diffeomorphism either grow linearly or are bounded; in other words, there are no deviations. For this, we use the topological invariance of the Artin–Mazur zeta function to exclude resonances outside the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools used in the proof. As a bonus, we show that for any $C^\infty $ Anosov diffeomorphism F on the two-torus, the correlations for the measure of maximal entropy and $C^\infty $ observables decay with a rate strictly smaller than $e^{-h_{\mathrm {top}}(F)}$ . We compare our results with very recent related work of Forni.

中文翻译：

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两个环面上的 Giulietti-Liverani horocycle 流的遍历平均值没有偏差

我们证明了 Giulietti 和 Liverani 与 Anosov 微分同胚相关的双环面上的 horocycle 流的遍历积分要么是线性增长的，要么是有界的；换句话说，没有偏差。为此，我们使用 Artin-Mazur zeta 函数的拓扑不变性来排除开放单位圆盘之外的共振。作用于各向异性分布的合适空间的转移算子及其 Ruelle 行列式是证明中使用的关键工具。作为奖励，我们表明对于任何$C^\infty $阿诺索夫微分同胚F在两个环面上，最大熵度量的相关性和$C^\infty $可观测量的衰减率严格小于$e^{-h_{\mathrm {top}}(F)}$. 我们将我们的结果与 Forni 最近的相关工作进行了比较。