Horocycle flow at product of two primes

Abstract

We show that if $\Gamma$ is a co-compact arithmetic lattice in $SL(2,\mathbb{R})$ or $\Gamma =SL(2,\mathbb{Z})$ then the horocycle orbit of every non-periodic point $x\in SL(2,\mathbb{R})/\Gamma$ equidistributes (with respect to Haar measure) when sampled at integers having exactly two prime factors.