{"paper":{"title":"Height arguments toward the dynamical Mordell-Lang problem in arbitrary characteristic","license":"http://creativecommons.org/licenses/by/4.0/","headline":"If no cohomological Lyapunov multiplier of an endomorphism iteration is an integer, then the return set of a dense orbit into a curve is finite.","cross_cats":["math.AG","math.NT"],"primary_cat":"math.DS","authors_text":"Junyi Xie, She Yang","submitted_at":"2025-04-02T10:04:14Z","abstract_excerpt":"We use height arguments to prove two results about the dynamical Mordell-Lang problem.\n  (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any iteration is not an integer.\n  (ii) Let $f\\times g:X\\times C\\rightarrow X\\times C$ be an endomorphism, where $f$ and $g$ are surjective endomorphisms of a projective variety $X$ and a projective curve $C$, respectively. If the degree of $g$ is greater than the first dynamical degree of $f$, then the return sets of the system $(X\\times C,f\\times g)$ have the"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any iteration is not an integer. (Abstract, result (i))","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The paper depends on the standard functoriality and Northcott-type properties of height functions continuing to hold for endomorphisms of projective varieties over fields of arbitrary characteristic, allowing the derivation of the required height inequalities from the non-integer multiplier condition. (Invoked throughout the height arguments for both (i) and (ii))","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Height arguments establish finiteness and structural results for return sets in the dynamical Mordell-Lang problem for endomorphisms of projective varieties and products, plus examples of complex behavior on zero-entropy tori.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If no cohomological Lyapunov multiplier of an endomorphism iteration is an integer, then the return set of a dense orbit into a curve is finite.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ba06041666b108f300af66bc1f8e14302b933a697b9d6df81d21e01ec8665773"},"source":{"id":"2504.01563","kind":"arxiv","version":4},"verdict":{"id":"ffb5bc64-a256-434e-9dcd-e8c38bcb04c0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-22T22:15:22.455631Z","strongest_claim":"For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any iteration is not an integer. (Abstract, result (i))","one_line_summary":"Height arguments establish finiteness and structural results for return sets in the dynamical Mordell-Lang problem for endomorphisms of projective varieties and products, plus examples of complex behavior on zero-entropy tori.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The paper depends on the standard functoriality and Northcott-type properties of height functions continuing to hold for endomorphisms of projective varieties over fields of arbitrary characteristic, allowing the derivation of the required height inequalities from the non-integer multiplier condition. (Invoked throughout the height arguments for both (i) and (ii))","pith_extraction_headline":"If no cohomological Lyapunov multiplier of an endomorphism iteration is an integer, then the return set of a dense orbit into a curve is finite."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.01563/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}