沿Ω(n)的可见测度与水平圆周轨道的分布

中文摘要 AI 辅助

设Ω(n)表示n的素因子个数（按重数计数）。我们研究σ-紧动力系统(X,T)中序列(1/N)∑_{n≤N}δ_{T^{Ω(n)}x}的弱-*极限集Acc^Ω(x)，证明若x对遍历测度μ是拟泛型的，则μ属于Acc^Ω(x)。这推广了Bergelson与Richter的结果，他们此前在唯一遍历系统框架下研究该问题。我们对SL(2,ℝ)非紧商空间上的水平圆周流，给出Acc^Ω(x)更精确的刻画：对每个非周期点x∈X，除哈尔测度外，存在序列(sₙ),(cₙ)⊆ℝ，使得(1/√(2π))∫_{-∞}^∞ e^{-r²/2}ν^i_{sₙ-2log|1+cₙr|}dr属于Acc^Ω(x)，其中{ν^i_s}_{i≤k}是k个不等价尖点中每个的单参数周期测度族。根据非周期点x的丢番图性质，Acc^Ω(x)包含这类周期测度的完整双参数族，以及每个尖点处的狄拉克测度；特别地，这些结果表明非紧水平圆周流沿Ω(n)的逐点平均几乎处处发散。

英文摘要

Let $Ω(n)$ denote the number of prime factors of $n$, counted with multiplicities. We study the set $Acc^Ω(x)$ of weak-$^*$ limits of the sequence $\frac{1}{N}\sum_{n\leq N}δ_{T^{Ω(n)}x}$ in $σ$-compact dynamical systems $ (X,T)$, demonstrating that if $x \in X$ is quasi-generic for an ergodic measure $μ$, then $μ\in Acc^Ω(x)$. This extends a result of Bergelson and Richter, who studied the problem in the setting of uniquely ergodic systems. We give a more precise description of the set $Acc^Ω(x)$ in the case of the horocycle flow on non-compact quotients of $SL(2,\mathbb{R})$. We show that for every non-periodic $x\in X$, in addition to Haar measure, there exists sequences $(s_n), (c_n) \subseteq \mathbb{R}$ such that $$ \frac{1}{\sqrt{2π}}\int_{-\infty}^{\infty}e^{-\frac{r^2}{2}}ν^{i}_{s_n-2\log|1+c_nr|} dr\in Acc^Ω(x), $$ where $\{ ν^{i}_{s} \}_{i \leq k}$ denotes the one parameter family of periodic measures in each of the $k$ inequivalent cusps. Depending on Diophantine properties of the non-periodic point $x$, we show that $Acc^Ω(x)$ contains a full two parameter family of such periodic measures, as well as the Dirac measure at each cusp. In particular, these results yield almost-everywhere divergence of pointwise averages along $Ω(n)$ for the non-compact horocycle flow.

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30 pages