Dynamics of abelian subgroups of GL(n, C): a structure's Theorem

In this paper, we characterize the dynamic of every abelian subgroups $\mathcal{G}$ of GL($n$, $\mathbb{K}$), $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We show that there exists a $\mathcal{G}$-invariant, dense open set $U$ in $\mathbb{K}^{n}$ saturated by minimal orbits with $\mathbb{K}^{n}- U$ a union of at most $n$ $\mathcal{G}$-invariant vectorial subspaces of $\mathbb{K}^{n}$ of dimension $n-1$ or $n-2$ on $\mathbb{K}$. As a consequence, $\mathcal{G}$ has height at most $n$ and in particular it admits a minimal set in $\mathbb{K}^{n}-\{0\}$.Comment: 16 page

Area preserving analytic flows with dense orbits

The aim of this paper is to give sufficient conditions on area-preserving flows that guarantee the existence of dense orbits. We also answer a question by M.D. Hirsch [M.D. Hirsch, Dense recurrence in area-preserving flows on surfaces, Nonlinearity 12 (1999) 1545–1553]. The results of this work are a generalization of the ones in [M.D. Hirsch, Dense recurrence in area-preserving flows on surfaces, Nonlinearity 12 (1999) 1545–1553] and [H. Marzougui, Area preserving flows with a dense orbit, Nonlinearity 15 (2002) 1379– 1384].The first author was supported by MERST(Ministery of Higher Education,Scientifc Research and Technology,Tunisia)under the Spanish-Tunisian project,grant A/8322/07.The second one was partially supported by MEC(Ministerio de Educación y Ciencia,Spain)and FEDER(Fondo Europeo de Desarrollo Regional),grant MTM2005- 03868,and Fundación Séneca(Comunidad Autónoma de la Región de Murcia,Spain),grant 00684/PI/04