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September  2010, 26(3): 923-947. doi: 10.3934/dcds.2010.26.923

Jordan decomposition and dynamics on flag manifolds

Thiago Ferraiol 1, , Mauro Patrão 2, and  Lucas Seco 2,

1. 

Departamento de Matemática, Universidade Estadual de Campinas, Cx. Postal 6065, Campinas-SP, 13.083-859, Brazil
2. 

Departamento de Matemática, Universidade de Brasília, Campus Darcy Ribeiro, Cx. Postal 4481, Brasília-DF, 70.904-970, Brazil, Brazil

Received  January 2009 Revised  October 2009 Published  December 2009

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: structural stability, recurrence, generalized flag manifolds, Morse decomposition, Floquet theory., Jordan decomposition, Conley index.
Mathematics Subject Classification: Primary: 37B35, 22E46, 37C20; Secondary: 37B30, 37B4.
Citation: Thiago Ferraiol, Mauro Patrão, Lucas Seco. Jordan decomposition and dynamics on flag manifolds. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 923-947. doi: 10.3934/dcds.2010.26.923
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