p-chaoticity and regular action of abelian C¹-diffeomorphisms groups of ℂⁿ fixing a point

2012/08/31 by Yahya N'Dao, N'Dao, Yahya, Ayadi Adlene +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS

paper · pdf · doi:10.48550/arxiv.1208.6395

openalex publication_date 2012/08/31 · arxiv created 2013/03/10 · arxiv updated 2013/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we introduce the notion of regular action of any abelian subgroup G of Diff¹(Cⁿ) on Cⁿ (i.e. the closure of every orbit of G in some open set is a topological sub-manifold of Cⁿ). 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 Cⁿ), where vect(L_G) is the vector space generated by all Df₀, f in G. Moreover, weprove that the action of any abelian lie subgroup of Diff¹(Cⁿ), is regular.

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