Journal of Modern Dynamics (JMD)

On the spectrum of a large subgroup of a semisimple group

Pages: 15 - 42, Issue 1, January 2008      doi:10.3934/jmd.2008.2.15

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Yves Guivarc'h - IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France (email)

Abstract: We consider a semi-simple algebraic group $\mathbf G$ defined over a local field of zero characteristic and we denote by $G$ the group of its $k$-rational points. For $\Gamma$ a "large" sub-semigroup of $G$ we define a closed subgroup 〈Spec$\Gamma$〉 associated with $\Gamma$, and we show that 〈Spec$\Gamma$〉 is large in a certain sense. This allows us to study the $\Gamma$-orbit closures for certain $\Gamma$-actions. The analytic structure of closed subgroups of $G$, over $\mathbb R$ or $\mathbb Q_{p}$, allows to use the Lie algebras techniques. The properties of the limit set of $\Gamma$ are developed ; they play an important role in the proofs.

Keywords:  Zariski-dense, boundary, proximal, randomwalk, orbit closure.
Mathematics Subject Classification:  Primary: 37H15; Secondary: 22E30,35.

Received: September 2006;      Revised: September 2007;      Available Online: October 2007.