Pointwise equidistribution and translates of measures on homogeneous spaces

Abstract

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.