The Structure of Almost Stationary Measures

Abstract

Let $G$ be a higher-rank simple Lie group acting on a space $X$. A theorem of Nevo and Zimmer asserts that every ergodic stationary probability measure on $X$ is either $G$-invariant or admits a projective factor $G/Q$ for a proper parabolic subgroup $Q$. We develop a quantitative theory of stationary measures and prove an effective form of this dichotomy. We introduce notions of $\eps$-almost stationarity, $δ$-almost invariance and $δ$-almost projective factor, and show that every $\eps$-almost stationary measure is either $δ$-almost invariant or carries a $δ'$-almost projective factor, with $δ,δ'$ explicit in $\eps$ and depending only on $G$. No ergodicity, arithmeticity or Diophantine hypothesis is imposed, and the bounds are uniform over all $G$-spaces. The proof introduces several tools: the \emph{entropigeonhole method}, an entropy-based pigeonhole principle yielding a quantitative Mautner phenomenon; \emph{factor functions}, quantitative analogues of functions on homogeneous factor spaces; and a \emph{fast generation} dichotomy in the spirit of growth in groups. In a companion paper these are used to show, among other things, that a discrete subgroup of infinite covolume has injectivity radius at least $c\log^{(4)}r$ somewhere in the ball of radius $r$, which is an effective form of a theorem of Frączyk and Gelander.