赫克对应关系的统计性质
Statistical properties of Hecke correspondences
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中文总结 AI 辅助
研究赫克对应关系在\(\mathbb{C}\)和\(\mathbb{C}_p\)上模曲线生成动力系统的统计性质，在\(\mathbb{C}\)上证明等分布速率，在\(\mathbb{C}_p\)上分析两种行为及相关性质，还扩展强化了戈伦和卡萨伊关于随机游走的结果。
中文摘要 AI 辅助
本文研究了由赫克对应关系在复数域\(\mathbb{C}\)上以及对每个素数\(p\)在\(p\)进数域\(\mathbb{C}_p\)上的模曲线上生成的动力系统的统计性质。在\(\mathbb{C}\)上，证明了克洛泽尔和奥塔尔建立的向双曲测度的等分布以指数速率发生，并且在假设拉马努金 - 彼得森猜想有肯定解的情况下确定了精确速率。在\(\mathbb{C}_p\)上出现两种不同类型的行为。第一种是每个轨道在贝科夫斯基仿射直线上收敛到高斯点，并建立了精确的指数收敛速率。第二种是在每个轨道闭包上具有一种唯一遍历性，建立了谱隙性质及由此得到的中心极限定理，这补充了康塔和勒博尔涅在\(\mathbb{C}\)上证明的中心极限定理。最后，本文扩展并强化了戈伦和卡萨伊关于相关随机游走的结果。
英文摘要
This paper studies the statistical properties of the dynamical system generated by a Hecke correspondence on the modular curve over $\mathbb{C}$ and, for every prime number $p$, over $\mathbb{C}_p$. Over $\mathbb{C}$, it proves that the equidistribution to the hyperbolic measure established by Clozel and Otal occurs at an exponential rate. Moreover, it determines the sharp rate, assuming an affirmative solution to the Ramanujan-Petersson conjecture. Over $\mathbb{C}_p$, two distinct types of behavior arise. In the first, every orbit converges towards the Gauss point in the Berkovich affine line, and this paper establishes the sharp exponential convergence rate. In the second, a form of unique ergodicity holds on each orbit closure, and this paper establishes a spectral gap property and, as a consequence, a central limit theorem. This complements the central limit theorem proved by Cantat and Le Borgne over $\mathbb{C}$. Finally, the paper extends and strengthens the results of Goren and Kassaei on the associated random walks.
