{"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. This is an analogue of the classical Garden of Eden theorem of Moore and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.06548","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"}