# American Institute of Mathematical Sciences

January  2008, 2(1): 15-42. doi: 10.3934/jmd.2008.2.15

## On the spectrum of a large subgroup of a semisimple group

 1 IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France

Received  September 2006 Revised  September 2007 Published  October 2007

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.
Citation: Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15
 [1] Alexander Gorodnik, Theron Hitchman, Ralf Spatzier. Regularity of conjugacies of algebraic actions of Zariski-dense groups. Journal of Modern Dynamics, 2008, 2 (3) : 509-540. doi: 10.3934/jmd.2008.2.509 [2] Minju Lee, Hee Oh. Topological proof of Benoist-Quint's orbit closure theorem for $\boldsymbol{ \operatorname{SO}(d, 1)}$. Journal of Modern Dynamics, 2019, 15: 263-276. doi: 10.3934/jmd.2019021 [3] Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045 [4] B. Harbourne, P. Pokora, H. Tutaj-Gasińska. On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces. Electronic Research Announcements, 2015, 22: 103-108. doi: 10.3934/era.2015.22.103 [5] Piotr Oprocha. Specification properties and dense distributional chaos. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 821-833. doi: 10.3934/dcds.2007.17.821 [6] F. H. Clarke, Yu. S . Ledyaev, R. J. Stern. Proximal techniques of feedback construction. Conference Publications, 1998, 1998 (Special) : 177-194. doi: 10.3934/proc.1998.1998.177 [7] Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185 [8] Michael Herty, Reinhard Illner. On Stop-and-Go waves in dense traffic. Kinetic & Related Models, 2008, 1 (3) : 437-452. doi: 10.3934/krm.2008.1.437 [9] Martin Frank, Cory D. Hauck, Edgar Olbrant. Perturbed, entropy-based closure for radiative transfer. Kinetic & Related Models, 2013, 6 (3) : 557-587. doi: 10.3934/krm.2013.6.557 [10] Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1 [11] Giuseppe Marino, Hong-Kun Xu. Convergence of generalized proximal point algorithms. Communications on Pure & Applied Analysis, 2004, 3 (4) : 791-808. doi: 10.3934/cpaa.2004.3.791 [12] Hadi Khatibzadeh, Vahid Mohebbi, Mohammad Hossein Alizadeh. On the cyclic pseudomonotonicity and the proximal point algorithm. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 441-449. doi: 10.3934/naco.2018027 [13] Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91 [14] Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477 [15] Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162 [16] Shiqiu Liu, Frédérique Oggier. On applications of orbit codes to storage. Advances in Mathematics of Communications, 2016, 10 (1) : 113-130. doi: 10.3934/amc.2016.10.113 [17] Yves Frederix, Giovanni Samaey, Christophe Vandekerckhove, Ting Li, Erik Nies, Dirk Roose. Lifting in equation-free methods for molecular dynamics simulations of dense fluids. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 855-874. doi: 10.3934/dcdsb.2009.11.855 [18] Mário Bessa, César M. Silva. Dense area-preserving homeomorphisms have zero Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1231-1244. doi: 10.3934/dcds.2012.32.1231 [19] Kei Irie. Dense existence of periodic Reeb orbits and ECH spectral invariants. Journal of Modern Dynamics, 2015, 9: 357-363. doi: 10.3934/jmd.2015.9.357 [20] YunKyong Hyon. Hysteretic behavior of a moment-closure approximation for FENE model. Kinetic & Related Models, 2014, 7 (3) : 493-507. doi: 10.3934/krm.2014.7.493

2020 Impact Factor: 0.848