Following the works of S. Mozes and N. Shah [Ergodic Theory Dyn. Syst. 15, No. 1, 149–159 (1995; Zbl 0818.58028)] and A. Eskin et al. [Ann. Math. (2) 143, No. 2, 253–299 (1996; Zbl 0852.11054); Geom. Funct. Anal. 7, No. 1, 48–80 (1997; Zbl 0872.22009)], given a sequence \((\mu_n)_{n \in \mathbb{N}} = (\mu_{H_n, g_n})_{n \in \mathbb{N}}\) of homogeneous measures associated with sequences \((H_n)_{n\in \mathbb{N}}\) of subgroups of \(G\) and sequences \((g_n)_{n \in \mathbb{N}}\) of elements of \(G\), under natural assumptions on \((H_n)_{n\in \mathbb{N}}\) and \((g_n)_{n \in \mathbb{N}}\), the weak limit of \((\mu_n)_n\), in the space of probability measures, it is homogeneous itself. If \(\mu\) is the weak limit of \((\mu_n)_n\) in the space of probability measures on the one-point-compactification \(\Gamma\setminus G \cup \{\infty\}\) then \(\mu\) is either a homogeneous probability measure on \(\Gamma\setminus G\), or equal to the Dirac delta measure at infinity.
In this paper, the authors conjecture that given a sequence of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space \(S= \Gamma \setminus G /K\), any weak limit is homogeneous with support contained in precisely one of the boundary components (including \(S\) itself). They study this conjecture and they prove it in a number of cases.