Let A be the full diagonal group in \({\text {SL}}_{n}(\mathbb {R})\). We study possible limits of Haar measures on periodic A-orbits in the space of unimodular lattices \(X_n\). We prove the existence of non-ergodic measures which are also weak limits of these compactly supported A-invariant measures. In fact, given any countably many A-invariant ergodic measures, we show that there exists a sequence of Haar measures on periodic A-orbits such that the ergodic decomposition of its weak limit has these measures as factors with positive weight. In particular, we prove that any compactly supported A-invariant and ergodic measure is the weak limit of the restriction of different compactly supported periodic measures to a fixed proportion of the time. In addition, for any \(c\in (0,1]\) we find a sequence of Haar measures on periodic A orbits that converges weakly to \(cm_{X_n}\) where \(m_{X_n}\) denotes the Haar measure on \(X_n\). In particular, we prove the existence of partial escape of mass for Haar measures on periodic A orbits. These results give affirmative answers to questions posed by Shapira in [Sha16]. Our proofs are based on a modification of Shapira’s proof in [Sha16] and on a generalization of a construction of Cassels, as well as on effective equidistribution estimates of Hecke neighbors by Clozel, Oh and Ullmo in [COU01].

设A为\({\text {SL}}_{n}(\mathbb {R})\)中的完整对角群。我们研究了单模晶格空间\(X_n\)中周期性A轨道的 Haar 测度的可能极限。我们证明了非遍历测度的存在，这也是这些紧支持的A不变测度的弱限制。事实上，给定任何可数个A不变遍历测度，我们证明在周期性A轨道上存在一系列 Haar 测度，使得其弱极限的遍历分解将这些测度作为具有正权重的因子。特别是，我们证明任何紧支持的A不变和遍历测度都是将不同紧支持的周期性测度限制到固定时间比例的弱极限。此外，对于任何\(c\in (0,1]\)，我们发现周期性A轨道上的哈尔测度序列弱收敛于\(cm_{X_n}\)，其中\(m_{X_n}\)表示特别是，我们证明了周期性A轨道上的哈尔测度存在部分逃逸质量，这些结果对夏皮拉在 [Sha16] 中提出的问题给出了肯定的答案。对 [Sha16] 中 Shapira 证明的修改，以及对 Cassels 构造的概括，以及 Clozel、Oh 和 Ullmo 在 [COU01] 中对 Hecke 邻居的有效均分布估计的修改。