# American Institute of Mathematical Sciences

2016, 10: 191-207. doi: 10.3934/jmd.2016.10.191

## On the work of Rodriguez Hertz on rigidity in dynamics

 1 Department of Mathematics, 2074 East Hall, 530 Church Street, University of Michigan, Ann Arbor, MI 48109-1043

Received  March 2016 Published  June 2016

This paper is a survey about recent progress in measure rigidity and global rigidity of Anosov actions, and celebrates the profound contributions by Federico Rodriguez Hertz to rigidity in dynamical systems.
Citation: Ralf Spatzier. On the work of Rodriguez Hertz on rigidity in dynamics. Journal of Modern Dynamics, 2016, 10: 191-207. doi: 10.3934/jmd.2016.10.191
##### References:
 [1] W. Ballmann, Nonpositively curved manifolds of higher rank,, Ann. of Math. (2), 122 (1985), 597. doi: 10.2307/1971331. Google Scholar [2] W. Ballmann, Lectures on Spaces of Nonpositive Curvature,, With an appendix by Misha Brin, (1995). doi: 10.1007/978-3-0348-9240-7. Google Scholar [3] C. Bonatti, S. Crovisier, G. Vago and A. Wilkinson, Local density of diffeomorphisms with large centralizers,, Ann. Sci. École Norm. Sup. (4), 41 (2008), 925. Google Scholar [4] C. Bonatti, S. Crovisier and A. Wilkinson, The $C^1$ generic diffeomorphism has trivial centralizer,, Inst. Hautes Études Sci. Publ. Math., 109 (2009), 185. Google Scholar [5] A. Brown, F. Rodriguez Hertz and Z. Wang, Global smooth and topological rigidity of hyperbolic lattice actions,, , (2015). Google Scholar [6] K. Burns and A. Katok, Manifolds with nonpositive curvature,, Ergodic Theory Dynam. Systems, 5 (1985), 307. doi: 10.1017/S0143385700002935. Google Scholar [7] K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings,, Inst. Hautes Études Sci. Publ. Math., 65 (1987), 35. Google Scholar [8] D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. KAM method and $\mathbbZ^k$ actions on the torus,, Ann. of Math. (2), 172 (2010), 1805. doi: 10.4007/annals.2010.172.1805. Google Scholar [9] D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. II: The geometric method and restrictions of Weyl chamber flows on $SL(n,\mathbbR)/\Gamma$,, Int. Math. Res. Not. IMRN, 19 (2011), 4405. doi: 10.1093/imrn/rnq252. Google Scholar [10] P. Eberlein, Lattices in spaces of nonpositive curvature,, Ann. of Math. (2), 111 (1980), 435. doi: 10.2307/1971104. Google Scholar [11] P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II,, Acta Math., 149 (1982), 41. doi: 10.1007/BF02392349. Google Scholar [12] M. Einsiedler and T. Fisher, Differentiable rigidity for hyperbolic toral actions,, Israel J. Math., 157 (2007), 347. doi: 10.1007/s11856-006-0016-0. Google Scholar [13] M. Einsiedler and A. Katok, Invariant measures on $G/\Gamma$ for split simple Lie groups $G$,, Dedicated to the memory of Jürgen K. Moser, 56 (2003), 1184. doi: 10.1002/cpa.10092. Google Scholar [14] M. Einsiedler and A. Katok, Rigidity of measures-The high entropy case and non-commuting foliations,, Israel J. Math., 148 (2005), 169. doi: 10.1007/BF02775436. Google Scholar [15] M. Einsiedler and E. Lindenstrauss, Diagonalizable flows on locally homogeneous spaces and number theory,, in International Congress of Mathematicians. Vol. II, (2006), 1731. Google Scholar [16] M. Einsiedler and E. Lindenstrauss, On measures invariant under diagonalizable actions: The rank-one case and the general low-entropy method,, J. Mod. Dyn., 2 (2008), 83. doi: 10.3934/jmd.2008.2.83. Google Scholar [17] M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces,, in Homogeneous Flows, (2010), 155. Google Scholar [18] M. Einsiedler and E. Lindenstrauss, On measures invariant under tori on quotients of semisimple groups,, Ann. of Math. (2), 181 (2015), 993. doi: 10.4007/annals.2015.181.3.3. Google Scholar [19] D. Fisher, Local rigidity of group actions: Past, present, future,, in Dynamics, (2007), 45. doi: 10.1017/CBO9780511755187.003. Google Scholar [20] D. Fisher, B. Kalinin and R. Spatzier, Global rigidity of higher rank Anosov actions on tori and nilmanifolds,, With an appendix by James F. Davis, 26 (2013), 167. doi: 10.1090/S0894-0347-2012-00751-6. Google Scholar [21] D. Fisher and G. Margulis, Local rigidity of affine actions of higher rank groups and lattices,, Ann. of Math. (2), 170 (2009), 67. doi: 10.4007/annals.2009.170.67. Google Scholar [22] J. Franks, Anosov diffeomorphisms,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 61. Google Scholar [23] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory, 1 (1967), 1. doi: 10.1007/BF01692494. Google Scholar [24] B. Farb and S. Weinberger, Isometries, rigidity and universal covers,, Ann. of Math. (2), 168 (2008), 915. doi: 10.4007/annals.2008.168.915. Google Scholar [25] A. Gogolev, Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms,, Ergodic Theory Dynam. Systems, 30 (2010), 441. doi: 10.1017/S0143385709000169. Google Scholar [26] M. Gromov, Groups of polynomial growth and expanding maps,, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53. Google Scholar [27] A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms,, Acta Math., 215 (2015), 127. doi: 10.1007/s11511-015-0130-0. Google Scholar [28] S. Hurder, Rigidity for Anosov actions of higher rank lattices,, Ann. of Math. (2), 135 (1992), 361. doi: 10.2307/2946593. Google Scholar [29] S. Hurder, A survey of rigidity theory for Anosov actions,, in Differential Topology, (1992), 143. doi: 10.1090/conm/161. Google Scholar [30] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [31] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995). doi: 10.1017/CBO9780511809187. Google Scholar [32] B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity,, Ann. of Math. (2), 174 (2011), 361. doi: 10.4007/annals.2011.174.1.10. Google Scholar [33] A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms,, Israel J. Math., 75 (1991), 203. doi: 10.1007/BF02776025. Google Scholar [34] A. Katok and J. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions,, Israel J. Math., 93 (1996), 253. doi: 10.1007/BF02761106. Google Scholar [35] A. Katok, J. Lewis and R. Zimmer, Cocycle superrigidity and rigidity for lattice actions on tori,, Topology, 35 (1996), 27. doi: 10.1016/0040-9383(95)00012-7. Google Scholar [36] N. Kopell, Commuting diffeomorphisms,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 165. Google Scholar [37] A. Katok and F. Rodriguez Hertz, Rigidity of real-analytic actions of $\text{\rm SL}(n,\mathbbZ)$ on $\mathbbT^n$: A case of realization of Zimmer program,, Discrete Contin. Dyn. Syst., 27 (2010), 609. doi: 10.3934/dcds.2010.27.609. Google Scholar [38] A. Katok and F. Rodriguez Hertz, Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher-rank abelian groups,, J. Mod. Dyn., 4 (2010), 487. doi: 10.3934/jmd.2010.4.487. Google Scholar [39] A. Katok and F. Rodriguez Hertz, Arithmeticity and topology of smooth actions of higher rank abelian groups,, J. Mod. Dyn., 10 (2016), 135. doi: 10.3934/jmd.2016.10.135. Google Scholar [40] A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions,, Ergodic Theory Dynam. Systems, 16 (1996), 751. doi: 10.1017/S0143385700009081. Google Scholar [41] A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions,, Tr. Mat. Inst. Steklova, 216 (1997), 292. Google Scholar [42] B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic Anosov $\mathbbZ^k$ actions,, Geom. Topol., 10 (2006), 929. doi: 10.2140/gt.2006.10.929. Google Scholar [43] B. Kalinin and V. Sadovskaya, On the classification of resonance-free Anosov $\mathbbZ^k$ actions,, Michigan Math. J., 55 (2007), 651. doi: 10.1307/mmj/1197056461. Google Scholar [44] B. Kalinin and R. Spatzier, On the classification of Cartan actions,, Geom. Funct. Anal., 17 (2007), 468. doi: 10.1007/s00039-007-0602-2. Google Scholar [45] J. W. Lewis, Infinitesimal rigidity for the action of $\text{\rm SL}(n,\mathbbZ)$ on $\mathbbT^n$,, Trans. Amer. Math. Soc., 324 (1991), 421. doi: 10.1090/S0002-9947-1991-1058434-X. Google Scholar [46] D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms,, Ergodic Theory Dynamical Systems, 2 (1982), 49. doi: 10.1017/S0143385700009573. Google Scholar [47] R. Lyons, On measures simultaneously $2$- and $3$-invariant,, Israel J. Math., 61 (1988), 219. doi: 10.1007/BF02766212. Google Scholar [48] A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422. doi: 10.2307/2373551. Google Scholar [49] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds,, Trans. Amer. Math. Soc., 229 (1977), 351. doi: 10.1090/S0002-9947-1977-0482849-4. Google Scholar [50] G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature,, in Proceedings of the International Congress of Mathematicians (Vancouver, (1974), 21. Google Scholar [51] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1991). Google Scholar [52] J. Palis and J.-C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori,, Ann. Sci. École Norm. Sup. (4), 22 (1989), 99. Google Scholar [53] J. Palis and J.-C. Yoccoz, Rigidity of centralizers of diffeomorphisms,, Ann. Sci. École Norm. Sup. (4), 22 (1989), 81. Google Scholar [54] F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms,, J. Mod. Dyn., 1 (2007), 425. doi: 10.3934/jmd.2007.1.425. Google Scholar [55] F. Rodriguez Hertz and Z. Wang, Global rigidity of higher rank abelian Anosov algebraic actions,, Invent. Math., 198 (2014), 165. doi: 10.1007/s00222-014-0499-y. Google Scholar [56] D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy,, Ergodic Theory Dynam. Systems, 10 (1990), 395. doi: 10.1017/S0143385700005629. Google Scholar [57] S. J. Schreiber, On growth rates of subadditive functions for semiflows,, J. Differential Equations, 148 (1998), 334. doi: 10.1006/jdeq.1998.3471. Google Scholar [58] M. Shub, Expanding maps,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 273. Google Scholar [59] S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar [60] R. J. Spatzier, An invitation to rigidity theory,, in Modern Dynamical Systems and Applications, (2004), 211. Google Scholar [61] K. Vinhage, On the rigidity of Weyl chamber flows and Schur multipliers as topological groups,, J. Mod. Dyn., 9 (2015), 25. doi: 10.3934/jmd.2015.9.25. Google Scholar [62] W. van Limbeek, Riemannian manifolds with local symmetry,, J. Topol. Anal., 6 (2014), 211. doi: 10.1142/S179352531450006X. Google Scholar [63] W. van Limbeek, Symmetry gaps in Riemannian geometry and minimal orbifolds,, , (2014). Google Scholar [64] K. Vinhage and Z. J. Wang, Local rigidity of higher rank homogeneous abelian actions: A complete solution via the geometric method,, , (2015). Google Scholar [65] Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits,, Trans. Amer. Math. Soc., 362 (2010), 4267. doi: 10.1090/S0002-9947-10-04947-0. Google Scholar [66] R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups,, Ann. of Math. (2), 112 (1980), 511. doi: 10.2307/1971090. Google Scholar [67] R. J. Zimmer, Actions of semisimple groups and discrete subgroups,, in Proceedings of the International Congress of Mathematicians, (1986), 1247. Google Scholar

show all references

##### References:
 [1] W. Ballmann, Nonpositively curved manifolds of higher rank,, Ann. of Math. (2), 122 (1985), 597. doi: 10.2307/1971331. Google Scholar [2] W. Ballmann, Lectures on Spaces of Nonpositive Curvature,, With an appendix by Misha Brin, (1995). doi: 10.1007/978-3-0348-9240-7. Google Scholar [3] C. Bonatti, S. Crovisier, G. Vago and A. Wilkinson, Local density of diffeomorphisms with large centralizers,, Ann. Sci. École Norm. Sup. (4), 41 (2008), 925. Google Scholar [4] C. Bonatti, S. Crovisier and A. Wilkinson, The $C^1$ generic diffeomorphism has trivial centralizer,, Inst. Hautes Études Sci. Publ. Math., 109 (2009), 185. Google Scholar [5] A. Brown, F. Rodriguez Hertz and Z. Wang, Global smooth and topological rigidity of hyperbolic lattice actions,, , (2015). Google Scholar [6] K. Burns and A. Katok, Manifolds with nonpositive curvature,, Ergodic Theory Dynam. Systems, 5 (1985), 307. doi: 10.1017/S0143385700002935. Google Scholar [7] K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings,, Inst. Hautes Études Sci. Publ. Math., 65 (1987), 35. Google Scholar [8] D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. KAM method and $\mathbbZ^k$ actions on the torus,, Ann. of Math. (2), 172 (2010), 1805. doi: 10.4007/annals.2010.172.1805. Google Scholar [9] D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. II: The geometric method and restrictions of Weyl chamber flows on $SL(n,\mathbbR)/\Gamma$,, Int. Math. Res. Not. IMRN, 19 (2011), 4405. doi: 10.1093/imrn/rnq252. Google Scholar [10] P. Eberlein, Lattices in spaces of nonpositive curvature,, Ann. of Math. (2), 111 (1980), 435. doi: 10.2307/1971104. Google Scholar [11] P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II,, Acta Math., 149 (1982), 41. doi: 10.1007/BF02392349. Google Scholar [12] M. Einsiedler and T. Fisher, Differentiable rigidity for hyperbolic toral actions,, Israel J. Math., 157 (2007), 347. doi: 10.1007/s11856-006-0016-0. Google Scholar [13] M. Einsiedler and A. Katok, Invariant measures on $G/\Gamma$ for split simple Lie groups $G$,, Dedicated to the memory of Jürgen K. Moser, 56 (2003), 1184. doi: 10.1002/cpa.10092. Google Scholar [14] M. Einsiedler and A. Katok, Rigidity of measures-The high entropy case and non-commuting foliations,, Israel J. Math., 148 (2005), 169. doi: 10.1007/BF02775436. Google Scholar [15] M. Einsiedler and E. Lindenstrauss, Diagonalizable flows on locally homogeneous spaces and number theory,, in International Congress of Mathematicians. Vol. II, (2006), 1731. Google Scholar [16] M. Einsiedler and E. Lindenstrauss, On measures invariant under diagonalizable actions: The rank-one case and the general low-entropy method,, J. Mod. Dyn., 2 (2008), 83. doi: 10.3934/jmd.2008.2.83. Google Scholar [17] M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces,, in Homogeneous Flows, (2010), 155. Google Scholar [18] M. Einsiedler and E. Lindenstrauss, On measures invariant under tori on quotients of semisimple groups,, Ann. of Math. (2), 181 (2015), 993. doi: 10.4007/annals.2015.181.3.3. Google Scholar [19] D. Fisher, Local rigidity of group actions: Past, present, future,, in Dynamics, (2007), 45. doi: 10.1017/CBO9780511755187.003. Google Scholar [20] D. Fisher, B. Kalinin and R. Spatzier, Global rigidity of higher rank Anosov actions on tori and nilmanifolds,, With an appendix by James F. Davis, 26 (2013), 167. doi: 10.1090/S0894-0347-2012-00751-6. Google Scholar [21] D. Fisher and G. Margulis, Local rigidity of affine actions of higher rank groups and lattices,, Ann. of Math. (2), 170 (2009), 67. doi: 10.4007/annals.2009.170.67. Google Scholar [22] J. Franks, Anosov diffeomorphisms,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 61. Google Scholar [23] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory, 1 (1967), 1. doi: 10.1007/BF01692494. Google Scholar [24] B. Farb and S. Weinberger, Isometries, rigidity and universal covers,, Ann. of Math. (2), 168 (2008), 915. doi: 10.4007/annals.2008.168.915. Google Scholar [25] A. Gogolev, Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms,, Ergodic Theory Dynam. Systems, 30 (2010), 441. doi: 10.1017/S0143385709000169. Google Scholar [26] M. Gromov, Groups of polynomial growth and expanding maps,, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53. Google Scholar [27] A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms,, Acta Math., 215 (2015), 127. doi: 10.1007/s11511-015-0130-0. Google Scholar [28] S. Hurder, Rigidity for Anosov actions of higher rank lattices,, Ann. of Math. (2), 135 (1992), 361. doi: 10.2307/2946593. Google Scholar [29] S. Hurder, A survey of rigidity theory for Anosov actions,, in Differential Topology, (1992), 143. doi: 10.1090/conm/161. Google Scholar [30] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [31] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995). doi: 10.1017/CBO9780511809187. Google Scholar [32] B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity,, Ann. of Math. (2), 174 (2011), 361. doi: 10.4007/annals.2011.174.1.10. Google Scholar [33] A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms,, Israel J. Math., 75 (1991), 203. doi: 10.1007/BF02776025. Google Scholar [34] A. Katok and J. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions,, Israel J. Math., 93 (1996), 253. doi: 10.1007/BF02761106. Google Scholar [35] A. Katok, J. Lewis and R. Zimmer, Cocycle superrigidity and rigidity for lattice actions on tori,, Topology, 35 (1996), 27. doi: 10.1016/0040-9383(95)00012-7. Google Scholar [36] N. Kopell, Commuting diffeomorphisms,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 165. Google Scholar [37] A. Katok and F. Rodriguez Hertz, Rigidity of real-analytic actions of $\text{\rm SL}(n,\mathbbZ)$ on $\mathbbT^n$: A case of realization of Zimmer program,, Discrete Contin. Dyn. Syst., 27 (2010), 609. doi: 10.3934/dcds.2010.27.609. Google Scholar [38] A. Katok and F. Rodriguez Hertz, Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher-rank abelian groups,, J. Mod. Dyn., 4 (2010), 487. doi: 10.3934/jmd.2010.4.487. Google Scholar [39] A. Katok and F. Rodriguez Hertz, Arithmeticity and topology of smooth actions of higher rank abelian groups,, J. Mod. Dyn., 10 (2016), 135. doi: 10.3934/jmd.2016.10.135. Google Scholar [40] A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions,, Ergodic Theory Dynam. Systems, 16 (1996), 751. doi: 10.1017/S0143385700009081. Google Scholar [41] A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions,, Tr. Mat. Inst. Steklova, 216 (1997), 292. Google Scholar [42] B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic Anosov $\mathbbZ^k$ actions,, Geom. Topol., 10 (2006), 929. doi: 10.2140/gt.2006.10.929. Google Scholar [43] B. Kalinin and V. Sadovskaya, On the classification of resonance-free Anosov $\mathbbZ^k$ actions,, Michigan Math. J., 55 (2007), 651. doi: 10.1307/mmj/1197056461. Google Scholar [44] B. Kalinin and R. Spatzier, On the classification of Cartan actions,, Geom. Funct. Anal., 17 (2007), 468. doi: 10.1007/s00039-007-0602-2. Google Scholar [45] J. W. Lewis, Infinitesimal rigidity for the action of $\text{\rm SL}(n,\mathbbZ)$ on $\mathbbT^n$,, Trans. Amer. Math. Soc., 324 (1991), 421. doi: 10.1090/S0002-9947-1991-1058434-X. Google Scholar [46] D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms,, Ergodic Theory Dynamical Systems, 2 (1982), 49. doi: 10.1017/S0143385700009573. Google Scholar [47] R. Lyons, On measures simultaneously $2$- and $3$-invariant,, Israel J. Math., 61 (1988), 219. doi: 10.1007/BF02766212. Google Scholar [48] A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96 (1974), 422. doi: 10.2307/2373551. Google Scholar [49] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds,, Trans. Amer. Math. Soc., 229 (1977), 351. doi: 10.1090/S0002-9947-1977-0482849-4. Google Scholar [50] G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature,, in Proceedings of the International Congress of Mathematicians (Vancouver, (1974), 21. Google Scholar [51] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1991). Google Scholar [52] J. Palis and J.-C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori,, Ann. Sci. École Norm. Sup. (4), 22 (1989), 99. Google Scholar [53] J. Palis and J.-C. Yoccoz, Rigidity of centralizers of diffeomorphisms,, Ann. Sci. École Norm. Sup. (4), 22 (1989), 81. Google Scholar [54] F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms,, J. Mod. Dyn., 1 (2007), 425. doi: 10.3934/jmd.2007.1.425. Google Scholar [55] F. Rodriguez Hertz and Z. Wang, Global rigidity of higher rank abelian Anosov algebraic actions,, Invent. Math., 198 (2014), 165. doi: 10.1007/s00222-014-0499-y. Google Scholar [56] D. J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy,, Ergodic Theory Dynam. Systems, 10 (1990), 395. doi: 10.1017/S0143385700005629. Google Scholar [57] S. J. Schreiber, On growth rates of subadditive functions for semiflows,, J. Differential Equations, 148 (1998), 334. doi: 10.1006/jdeq.1998.3471. Google Scholar [58] M. Shub, Expanding maps,, in Global Analysis (Proc. Sympos. Pure Math., (1968), 273. Google Scholar [59] S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar [60] R. J. Spatzier, An invitation to rigidity theory,, in Modern Dynamical Systems and Applications, (2004), 211. Google Scholar [61] K. Vinhage, On the rigidity of Weyl chamber flows and Schur multipliers as topological groups,, J. Mod. Dyn., 9 (2015), 25. doi: 10.3934/jmd.2015.9.25. Google Scholar [62] W. van Limbeek, Riemannian manifolds with local symmetry,, J. Topol. Anal., 6 (2014), 211. doi: 10.1142/S179352531450006X. Google Scholar [63] W. van Limbeek, Symmetry gaps in Riemannian geometry and minimal orbifolds,, , (2014). Google Scholar [64] K. Vinhage and Z. J. Wang, Local rigidity of higher rank homogeneous abelian actions: A complete solution via the geometric method,, , (2015). Google Scholar [65] Z. Wang and W. Sun, Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits,, Trans. Amer. Math. Soc., 362 (2010), 4267. doi: 10.1090/S0002-9947-10-04947-0. Google Scholar [66] R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups,, Ann. of Math. (2), 112 (1980), 511. doi: 10.2307/1971090. Google Scholar [67] R. J. Zimmer, Actions of semisimple groups and discrete subgroups,, in Proceedings of the International Congress of Mathematicians, (1986), 1247. Google Scholar
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