Jordan decomposition and dynamics on flag manifolds

Pages: 923 - 947, Volume 26, Issue 3, March 2010

doi:10.3934/dcds.2010.26.923       Abstract        Full Text (305.9K)       Related Articles

Thiago Ferraiol - Departamento de Matemática, Universidade Estadual de Campinas, Cx. Postal 6065, Campinas-SP, 13.083-859, Brazil (email)
Mauro Patrão - Departamento de Matemática, Universidade de Brasília, Campus Darcy Ribeiro, Cx. Postal 4481, Brasília-DF, 70.904-970, Brazil (email)
Lucas Seco - Departamento de Matemática, Universidade de Brasília, Campus Darcy Ribeiro, Cx. Postal 4481, Brasília-DF, 70.904-970, Brazil (email)

Abstract: Let $\g$ be a real semisimple Lie algebra and $G = \Int(\g)$. In this article, we relate the Jordan decomposition of $X \in \g$ (or $g \in G$) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by $X$ (or the discrete-time flow generated by $g$). We characterize the recurrent set and the finest Morse decomposition (including its stable sets) of these flows and show that its entropy always vanishes. We characterize the structurally stable ones and compute the Conley index of the attractor Morse component. When the nilpotent part of $X$ is trivial, we compute the Conley indexes of all Morse components. Finally, we consider the dynamical aspects of linear differential equations with periodic coefficients in $\g$, which can be regarded as an extension of the dynamics generated by an element $X \in \g$. In this context, we generalize Floquet theory and extend our previous results to this case.

Keywords:  Jordan decomposition, recurrence, Morse decomposition, generalized flag manifolds, structural stability, Conley index, Floquet theory.
Mathematics Subject Classification:  Primary: 37B35, 22E46, 37C20; Secondary: 37B30, 37B40.

Received: January 2009;      Revised: October 2009;      Published: December 2009.