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, $\delta$-almost invariance and $\delta$-almost projective factor, and show that every $\eps$-almost stationary measure is either $\delta$-almost invariant or carries a ${\delta }^{\prime }$-almost projective factor, with $\delta ,{\delta }^{\prime }$ 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. Additionally, we prove a decomposition version of this theorem where each measure is decomposed into an almost invariant part and an almost projective part. This comes at the cost of worse constants. 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. We also prove rates for Benjamini--Schramm convergence of quotients of higher-rank lattices.