通过几乎所有旋转的加权伯克霍夫平均的最优一致收敛速率
Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations
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中文总结 AI 辅助
研究与紧支集加权函数相关的环面平移的加权伯克霍夫平均，引入新分析技术，为几乎所有旋转和特定初始点建立最优一致收敛速率，确立其最优性，表明可观测量正则性决定收敛速率，刻画了加权伯克霍夫平均的快速收敛。
中文摘要 AI 辅助
在本文中，我们研究了与紧支集加权函数相关的环面平移的加权伯克霍夫平均。通过引入几种新的分析技术，我们为几乎所有旋转和特定（甚至所有）初始点建立了最优一致收敛速率。与经典遍历理论中最好能达到的\(\mathcal{O}(N^{-1})\)速率不同，我们表明这些加权平均呈现多项式甚至指数收敛。我们在多个方面确立了这些收敛速率的最优性，特别是关于跨越四种不同情况的正则性指标：有限可微性、\(C^\infty\)类、对数\(C^\infty\)类和热夫雷类。我们的结果表明可观测量的正则性基本上决定了收敛速率；此外，我们证明在一般情况下，没有其他加权函数能产生更快的一致速率。与标准时间平均通常的慢收敛形成对比，这项工作为加权伯克霍夫平均的快速收敛提供了最优且几乎完整的刻画。
英文摘要
In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions. By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points. Unlike the $\mathcal{O}(N^{-1})$ rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence. We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the $C^\infty$ class, logarithmic $C^\infty$ classes, and Gevrey classes. Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that no admissible choice of weighting function can, in general, overcome the lower bounds imposed by this regularity. In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.
发表机构
- School of Mathematics, Jilin University（吉林大学数学学院）
- Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University（东北师范大学数学与交叉科学研究院）
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