$p$-chaoticity and regular action of abelian $C^{1}$-diffeomorphisms groups of $\mathbb{C}^{n}$ fixing a point

TL;DR

This paper studies the dynamics of abelian groups of $C^{1}$-diffeomorphisms on complex n-space, introducing regular actions and showing conditions under which p-chaoticity cannot occur, with a focus on fixed points and orbit structures.

Contribution

It introduces the concept of regular action for abelian diffeomorphism groups and proves that such actions cannot be p-chaotic under certain conditions, extending understanding of orbit regularity.

Findings

Regular actions imply non-p-chaotic behavior when fixing a point.

Abelian $C^{1}$-diffeomorphism groups have inherently regular actions.

p-chaoticity is incompatible with the conditions studied for these groups.

Abstract

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). 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.