We consider a counting problem in the setting of hyperbolic dynamics. Let ${\varphi }_t:\mathrm{\Lambda }\to \mathrm{\Lambda }$ be a weak-mixing hyperbolic flow. We count the proportion of prime periodic orbits of ${\varphi }_t$, with length less than T, that satisfy an averaging condition related to a Hölder continuous function $f:\mathrm{\Lambda }\to \mathbb{R}$. We show, assuming an approximability condition on ϕ, that as $T\to \mathrm{\infty }$, we obtain a central limit theorem. The proof uses transfer operator estimates due to Dolgopyat to provide the bounds on complex functions that we need to carry out our analysis. We can then use contour integration to obtain the asymptotic behaviour which gives the central limit theorem.

中文翻译：

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双曲流周期轨道的中心极限定理

我们考虑双曲动力学设置中的一个计数问题。让${\varphi }_{Ť}：\mathrm{\Lambda }\to \mathrm{\Lambda }$是弱混合双曲流。我们计算出的主要周期轨道的比例${\varphi }_{Ť}$长度小于T且满足与Hölder连续函数有关的平均条件$F：\mathrm{\Lambda }\to \mathbb{[R}$。我们证明，假设ϕ的近似条件为$Ť\to \mathrm{\infty }$，我们得到一个中心极限定理。该证明使用由于Dolgopyat引起的转移算子估计来提供我们进行分析所需的复杂函数的界限。然后，我们可以使用轮廓积分获得渐近行为，从而给出中心极限定理。