An avoidance principle and Margulis functions for expanding translates of unipotent orbits (Q6617232)

An \textit{avoidance principle} in dynamics refers to a result quantifying how much time trajectories avoid certain subsets of the ambient space. Classical examples include non-divergence results for horocycle flows on unit tangent bundles of hyperbolic surfaces and more generally unipotent flows on homogeneous spaces or moduli spaces of translation surfaces. In the paper under review, the authors prove an avoidance principle for \textit{expanding translates} of unipotent orbits for some quotients of semisimple Lie groups, and in addition, a quantitative isolation result for closed unipotent orbits and give an upper bound on the number of closed orbits of bounded volume. The key tool is the construction of an appropriate \textit{Margulis function}, a function which measures, in a rough sense, closeness to the subsets one wants to avoid, and which satsify a kind of sub-harmonicity which allows one to apply classical ideas from the theory of random walks to obtain recurrence results, as pioneered by \textit{A. Eskin} and \textit{G. Margulis} [in: Random walks and geometry. Proceedings of a workshop at the Erwin Schrödinger Institute, Vienna, June 18 -- July 13, 2001. In collaboration with Klaus Schmidt and Wolfgang Woess. Collected papers. Berlin: de Gruyter. 431--444 (2004; Zbl 1064.60092)].