{"paper":{"title":"On the Nonexistence of Continuous Immersions for Discrete-time Systems","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Discrete-time systems with countably many but more than one omega-limit sets cannot be continuously and injectively immersed into finite-dimensional linear systems.","cross_cats":["cs.SY","math.DS"],"primary_cat":"eess.SY","authors_text":"Eduardo Sontag, Eron Ristich, Necmiye Ozay","submitted_at":"2026-05-14T17:53:56Z","abstract_excerpt":"Understanding when linear immersions of nonlinear dynamical systems exist is important since such immersions allow us to leverage the rich tools of linear system theory to analyze nonlinear dynamics. Recently, Liu et al. (2023) showed that continuous-time dynamical systems that admit countably many but more than one omega-limit sets cannot be immersed into finite dimensional linear systems with a one-to-one and continuous mapping. In this paper, we extend these results to discrete-time dynamics and show that similar obstructions exist also in discrete time. We further consider a generalization"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we extend these results to discrete-time dynamics and show that similar obstructions exist also in discrete time.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The immersion mapping must be continuous and one-to-one while the system has countably many but more than one omega-limit sets.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Discrete-time systems with countably many but more than one omega-limit sets admit no continuous one-to-one immersion into finite-dimensional linear systems.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Discrete-time systems with countably many but more than one omega-limit sets cannot be continuously and injectively immersed into finite-dimensional linear systems.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"adbe2b4d224a300ef56a2462b17abcf20bcab05f7b83d84c9c7303b297c66536"},"source":{"id":"2605.15161","kind":"arxiv","version":1},"verdict":{"id":"26a8c4c2-df78-490e-8124-cd008d11dba5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:58:48.194508Z","strongest_claim":"we extend these results to discrete-time dynamics and show that similar obstructions exist also in discrete time.","one_line_summary":"Discrete-time systems with countably many but more than one omega-limit sets admit no continuous one-to-one immersion into finite-dimensional linear systems.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The immersion mapping must be continuous and one-to-one while the system has countably many but more than one omega-limit sets.","pith_extraction_headline":"Discrete-time systems with countably many but more than one omega-limit sets cannot be continuously and injectively immersed into finite-dimensional linear systems."},"references":{"count":42,"sample":[{"doi":"","year":null,"title":"On the Nonexistence of Continuous Immersions for Discrete-time Systems , booktitle =","work_id":"4f25c89e-6bb2-496c-a4ca-73ba18e1e3f5","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1960,"title":"Proceedings of the American Mathematical Society , volume=","work_id":"889415a9-7c5c-4b9e-bed8-3edb9a914f69","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Doklady Akademii Nauk SSSR , volume=","work_id":"a25430ea-aefa-4b08-876d-966a579e1614","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=","work_id":"ba7c1a16-a84b-4184-b6dd-9baac9b5bf7b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Koopman operator in systems and control , author=. 2020 , publisher=","work_id":"c32abeb4-1aaf-4bd9-bf40-3f556f33a16e","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":42,"snapshot_sha256":"1f4dfd7af8401afb3b12e54014e42825aaf91c97121c73009fe4292d8213ed57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"fcb0f558e85122e752999fe30bc785c422e47ec5091f8b00217533aeb8c643ce"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}