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Consider the Pontryagin dual $X_f$ of the cyclic $\\Z[\\Gamma]$-module $\\Z[\\Gamma]/\\Z[\\Gamma] f$ and suppose that the natural action of $\\Gamma$ on $X_f$ is expansive and that $X_f$ is connected. We prove that if $\\tau \\colon X_f \\to X_f$ is a $\\Gamma$-equivariant continuous map, then $\\tau$ is surjective if and only if the restriction of $\\tau$ to each $\\Gamma$-homoclinicity class is injective. This is an analogue of the classical Garden of Eden theorem of Moore and "},"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":"1706.06548","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-06-20T17:03:12Z","cross_cats_sorted":["math.FA","math.GR"],"title_canon_sha256":"c29febf113e5a462298f2ed9266a9f9ff90aac0166fce192ffff59d5ee900559","abstract_canon_sha256":"b5e9093b9525910ba622cb28a3e503e4326aa1a501156c079296a2ada5071086"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:02.957290Z","signature_b64":"/cMwTo/W2J8a52mQE9eSIsDWi4ystTXX6pj9/FNFfdyxhgsFKzXV6BTun24+7SS9ClRRA8gUjgseh5Avzm9TBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4bf79ffb357808266f9a26bdd587effc44597b8ac0ea7a2a4c5cd5cc795357f","last_reissued_at":"2026-05-18T00:42:02.956784Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:02.956784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Garden of Eden theorem for principal algebraic actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GR"],"primary_cat":"math.DS","authors_text":"Michel Coornaert, Tullio Ceccherini-Silberstein","submitted_at":"2017-06-20T17:03:12Z","abstract_excerpt":"Let $\\Gamma$ be a countable abelian group and $f \\in \\Z[\\Gamma]$, where $\\Z[\\Gamma]$ denotes the integral group ring of $\\Gamma$. Consider the Pontryagin dual $X_f$ of the cyclic $\\Z[\\Gamma]$-module $\\Z[\\Gamma]/\\Z[\\Gamma] f$ and suppose that the natural action of $\\Gamma$ on $X_f$ is expansive and that $X_f$ is connected. We prove that if $\\tau \\colon X_f \\to X_f$ is a $\\Gamma$-equivariant continuous map, then $\\tau$ is surjective if and only if the restriction of $\\tau$ to each $\\Gamma$-homoclinicity class is injective. 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