Somewhere dense orbit of abelian subgroup of diffeomorphisms maps acting on C^n

TL;DR

This paper characterizes when an abelian subgroup of diffeomorphisms acting on C^n has a somewhere dense orbit, linking it to the density of a subgroup generated by certain exponential maps and their vector space span.

Contribution

It provides a precise criterion involving exponential maps and vector space spans for the density of orbits of abelian diffeomorphism subgroups on C^n.

Findings

Orbit G(x) is somewhere dense iff certain exponential maps generate a dense subgroup.

Characterization involves the existence of specific elements in the exponential preimage of G.

Density of the subgroup generated by vector fields determines orbit density.

Abstract

In this paper, we give a characterization for any abelian subgroup G of a lie group of diffeomorphisms maps of C^n, having a somewhere dense orbit G(x), x in C^n: G(x) is somewhere dense in C^n if and only if there are f_{1},....,f_{2n+1 in exp^{-1}(G) such that f_{2n+1} in vect(f_{1},...,f_{2n}) and Z.f_{1}(x)+....+Z.f_{2n+1}(x) is dense subgroup of C^n, where vect(f_{1},....,f_{2n}) is the vector space over R generated by f_{1},....,f_{2n}.