Visible Measures along $Ω(n)$ and Distribution of Horocycle Orbits

Abstract

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.