American Institute of Mathematical Sciences

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

On the spectrum of a large subgroup of a semisimple group

Yves Guivarc'h 1,

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.
Keywords: orbit closure., Zariski-dense, proximal, randomwalk, boundary.
Mathematics Subject Classification: Primary: 37H15; Secondary: 22E30,3.
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, , () : -. 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 - A, 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 - A, 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]

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
[10]

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
[11]

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
[12]

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
[13]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[14]

Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
[15]

Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 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]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565
[18]

Yan Gu, Nobuo Yamashita. A proximal ADMM with the Broyden family for convex optimization problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020091
[19]

Igor Griva, Roman A. Polyak. Proximal point nonlinear rescaling method for convex optimization. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 283-299. doi: 10.3934/naco.2011.1.283
[20]

Ram U. Verma. On the generalized proximal point algorithm with applications to inclusion problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 381-390. doi: 10.3934/jimo.2009.5.381

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences