Pointwise Equidistribution and Translates of Measures on Homogeneous Spaces

Let $(X,\mathfrak{B},\mu)$ be a Borel probability space. Let $T_n: X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\nu$ be a probability measure on $X$ such that $\frac{1}{N}\sum_{n=1}^N (T_n)_\ast \nu \rightarrow \mu$ in the weak-$\ast$ topology. Under general conditions, we show that for $\nu$ almost every $x\in X$, the measures $\frac{1}{N}\sum_{n=1}^N \delta_{T_n x}$ get equidistributed towards $\mu$ if $N$ is restricted to a set of full upper density. We present applications of these results to translates of closed orbits of Lie groups on homogeneous spaces. As a corollary, we prove equidistribution of exponentially sparse orbits of the horocycle flow on quotients of $SL(2,\mathbb{R})$, starting from every point in almost every direction.