{"paper":{"title":"Odometer actions of the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alexandre I. Danilenko, Mariusz Lemanczyk","submitted_at":"2013-05-01T20:32:19Z","abstract_excerpt":"Let $H_3(\\Bbb R)$ denote the 3-dimensional real Heisenberg group. Given a family of lattices $\\Gamma_1\\supset\\Gamma_2\\supset\\cdots$ in it, let $T$ stand for the associated uniquely ergodic $H_3(\\Bbb R)$-{\\it odometer}, i.e. the inverse limit of the $H_3(\\Bbb R)$-actions by rotations on the homogeneous spaces $H_3(\\Bbb R)/\\Gamma_j$, $j\\in\\Bbb N$. The decomposition of the underlying Koopman unitary representation of $H_3(\\Bbb R)$ into a countable direct sum of irreducible components is explicitly described. The ergodic 2-fold self-joinings of $T$ are found. It is shown that in general, the $H_3("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.0285","kind":"arxiv","version":2},"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"}