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

Abstract

Let $\Omega (n)$ denote the number of prime factors of $n$, counted with multiplicities. We study the set $Ac{c}^{\Omega }(x)$ of weak-${}^{\ast }$ limits of the sequence $\frac{1}{N}{\sum }_{n\le N}{\delta }_{T^{\Omega (n)}x}$ in $\sigma$-compact dynamical systems $(X,T)$, demonstrating that if $x\in X$ is quasi-generic for an ergodic measure $\mu$, then $\mu \in Ac{c}^{\Omega }(x)$. 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 $Ac{c}^{\Omega }(x)$ 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 $$\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{-\frac{r^2}{2}}{\nu }_{s_n-2\log |1+c_{n}r|}^{i}dr\in Ac{c}^{\Omega }(x),$$ where $\{{\nu }_s^i{\}}_{i\le k}$ 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 $Ac{c}^{\Omega }(x)$ 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 $\Omega (n)$ for the non-compact horocycle flow.

30 pages