{"paper":{"title":"Directional recurrence and directional rigidity for infinite measure preserving actions of nilpotent lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alexandre I. Danilenko","submitted_at":"2014-08-08T11:00:29Z","abstract_excerpt":"Let $\\Gamma$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure preserving action $T$ of $\\Gamma$ and a \"direction\" in $G$ (i.e. an element $\\theta$ of the projective space $P(\\goth g)$ of the Lie algebra $\\goth g$ of $G$), some notions of recurrence and rigidity for $T$ along $\\theta$ are introduced. It is shown that the set of recurrent directions $\\Cal R(T)$ and the set of rigid directions for $T$ are both $G_\\delta$. In the case where $G=\\Bbb R^d$ and $\\Gamma=\\Bbb Z^d$, we prove that (a) for each $G_\\delta$-subset $\\Delta$ of $P(\\goth g)$ and a countable "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1815","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}