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We prove that if G fixes 0 and dim(vect(L_{G}) =n, then the action of G, can not be p-chaotic for every 0<= p <=n-1. (i.e. If G has a dense orbit then the set of all regular orbit with order p can not be dense in C^{n}), where vect(L_{G}) is the vector space generated by all Df_{0}, f in G. Moreover, weprove that the action of any abelian lie subgroup of Diff^{1}(C^{n}), is regular."},"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":"1208.6395","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-31T06:38:50Z","cross_cats_sorted":[],"title_canon_sha256":"07b672ab7a1f395e3afbb6db154005e8f7e6cbc4e182f3a7a4f1649da08c1fdd","abstract_canon_sha256":"a8358b162a11d7f8f2d2cb6d5b07eaa37c4f8f221e1d25c765ebcf1566d26e8f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:31:29.501255Z","signature_b64":"pnCETT2j/xx4gllbpjdPefzEcZhYYDVrsfj45u8iKQT8mEdbhJ+j0OyjjalPEVt0OEb/p3+IMLLz0lsjUXc6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4bada07f447ab60ee3a21eb302f5136cde99fef4cb350408a71796b7ef1cb2e1","last_reissued_at":"2026-05-18T03:31:29.500467Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:31:29.500467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$p$-chaoticity and regular action of abelian $C^{1}$-diffeomorphisms groups of $\\mathbb{C}^{n}$ fixing a point","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ayadi Adlene, Yahya N'Dao","submitted_at":"2012-08-31T06:38:50Z","abstract_excerpt":"In this paper, we introduce the notion of regular action of any abelian subgroup G of $Diff^{1}(C^n) on C^n (i.e. the closure of every orbit of G in some open set is a topological sub-manifold of C^n). 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