# American Institute of Mathematical Sciences

July  2013, 7(3): 429-460. doi: 10.3934/jmd.2013.7.429

## Modified Schmidt games and a conjecture of Margulis

 1 Goldsmith 207, Brandeis University, Waltham, MA 02454-9110 2 Ben Gurion University, Be'er Sheva, Israel 84105

Received  December 2012 Published  December 2013

We prove a conjecture of G.A. Margulis on the abundance of certain exceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizing a theory of modified Schmidt games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in mid-1960s.
Citation: Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429-460. doi: 10.3934/jmd.2013.7.429
##### References:

show all references

##### References:
 [1] Weisheng Wu. Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3463-3481. doi: 10.3934/dcds.2016.36.3463 [2] Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341 [3] Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164 [4] George Daniel Mostow. Margulis. Journal of Modern Dynamics, 2008, 2 (1) : 1-5. doi: 10.3934/jmd.2008.2.1 [5] James Tanis. Exponential multiple mixing for some partially hyperbolic flows on products of ${\rm{PSL}}(2, \mathbb{R})$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 989-1006. doi: 10.3934/dcds.2018042 [6] Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435 [7] Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037 [8] Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419 [9] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271 [10] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68 [11] Ze Li, Lifeng Zhao. Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 607-638. doi: 10.3934/dcds.2019025 [12] Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114 [13] Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195 [14] Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27 [15] Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549 [16] Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193 [17] Boris Kalinin, Victoria Sadovskaya. Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 245-259. doi: 10.3934/dcds.2016.36.245 [18] Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469 [19] Andrey Gogolev. Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 2011, 5 (4) : 747-769. doi: 10.3934/jmd.2011.5.747 [20] Andy Hammerlindl, Rafael Potrie, Mario Shannon. Seifert manifolds admitting partially hyperbolic diffeomorphisms. Journal of Modern Dynamics, 2018, 12: 193-222. doi: 10.3934/jmd.2018008

2018 Impact Factor: 0.295