LAUSR

Let $(X,\mathfrak{B},\unicode[STIX]{x1D707})$ be a Borel probability space. Let $T_{n}:X\rightarrow X$ be a sequence of continuous transformations on $X$ . Let $\unicode[STIX]{x1D708}$ be a probability measure on $X$ such that $(1/N)\sum _{n=1}^{N}(T_{n})_{\ast }\unicode[STIX]{x1D708}\rightarrow \unicode[STIX]{x1D707}$ in the weak- $\ast$ topology. Under general conditions, we show that for $\unicode[STIX]{x1D708}$ almost every $x\in X$ , the measures $(1/N)\sum _{n=1}^{N}\unicode[STIX]{x1D6FF}_{T_{n}x}$ become equidistributed towards $\unicode[STIX]{x1D707}$ 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 $\text{SL}(2,\mathbb{R})$ , starting from every point in almost every direction.