2026/08/02 by Ilya Gekhtman, Simon Machado, Omri Solan +1
Mathematics · #math.DS #math.PR #msc:22F10 #msc:22D40 #msc:60B15
arxiv created 2026/08/02 · arxiv updated 2026/08/04
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 entropigeonhole method, an entropy-based pigeonhole principle yielding a quantitative Mautner phenomenon; factor functions, quantitative analogues of functions on homogeneous factor spaces; and a 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 clog(4)r somewhere in the ball of radius r, which is an effective form of a theorem of Frączyk and Gelander.