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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("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.0285","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-05-01T20:32:19Z","cross_cats_sorted":[],"title_canon_sha256":"ddc24230a5765a50687931999d34edd1cf5d9aa59bcb2a8d4143818cf6e7858b","abstract_canon_sha256":"ad7b0a798d866c28bd00af01f8541616b23542c1a343757ae7c746c8d6a878c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:50.981093Z","signature_b64":"wtjslOTDUBNjWrolSyUNNRoVGJ9Fh1HGTpdkbVcynTS3y1OZdy+EXyRnySs9cfN/m4z4kZiEf950tMVgbbJFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af1c7af0f0316f1a6595833842770341000a643fb638268b3747fbda8aa79e96","last_reissued_at":"2026-05-18T01:32:50.980561Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:50.980561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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