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), 597-609. doi: 10.2307/1971331.  Google Scholar

[2]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, With an appendix by Misha Brin, DMV Seminar, 25, Birkhäuser Verlag, Basel, 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), no. 6, 925-954.  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), 185-244.  Google Scholar

[5]

A. Brown, F. Rodriguez Hertz and Z. Wang, Global smooth and topological rigidity of hyperbolic lattice actions, arXiv:1512.06720, 2015. Google Scholar

[6]

K. Burns and A. Katok, Manifolds with nonpositive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317. 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), 35-59.  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), 1805-1858. 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), 4405-4430. doi: 10.1093/imrn/rnq252.  Google Scholar

[10]

P. Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. (2), 111 (1980), 435-476. doi: 10.2307/1971104.  Google Scholar

[11]

P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II, Acta Math., 149 (1982), 41-69. doi: 10.1007/BF02392349.  Google Scholar

[12]

M. Einsiedler and T. Fisher, Differentiable rigidity for hyperbolic toral actions, Israel J. Math., 157 (2007), 347-377. 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, Comm. Pure Appl. Math., 56 (2003), 1184-1221. 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), 169-238. 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, Eur. Math. Soc., Zürich, 2006, 1731-1759.  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), 83-128. doi: 10.3934/jmd.2008.2.83.  Google Scholar

[17]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 155-241.  Google Scholar

[18]

M. Einsiedler and E. Lindenstrauss, On measures invariant under tori on quotients of semisimple groups, Ann. of Math. (2), 181 (2015), 993-1031. doi: 10.4007/annals.2015.181.3.3.  Google Scholar

[19]

D. Fisher, Local rigidity of group actions: Past, present, future, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 45-97. 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, J. Amer. Math. Soc., 26 (2013), 167-198. 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), 67-122. doi: 10.4007/annals.2009.170.67.  Google Scholar

[22]

J. Franks, Anosov diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 61-93.  Google Scholar

[23]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49. doi: 10.1007/BF01692494.  Google Scholar

[24]

B. Farb and S. Weinberger, Isometries, rigidity and universal covers, Ann. of Math. (2), 168 (2008), 915-940. doi: 10.4007/annals.2008.168.915.  Google Scholar

[25]

A. Gogolev, Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 30 (2010), 441-456. doi: 10.1017/S0143385709000169.  Google Scholar

[26]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.  Google Scholar

[27]

A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms, Acta Math., 215 (2015), 127-159. 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), 361-410. doi: 10.2307/2946593.  Google Scholar

[29]

S. Hurder, A survey of rigidity theory for Anosov actions, in Differential Topology, Foliations, and Group Actions (Rio de Janeiro, 1992), Contemp. Math., 161, Amer. Math. Soc., Providence, RI, 1994, 143-173. 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), 137-173.  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, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 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), 361-400. 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), 203-241. 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), 253-280. 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), 27-38. doi: 10.1016/0040-9383(95)00012-7.  Google Scholar

[36]

N. Kopell, Commuting diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 165-184.  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), 609-615. 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), 487-515. 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), 135-172. 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), 751-778. 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), 292-319.  Google Scholar

[42]

B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic Anosov $\mathbbZ^k$ actions, Geom. Topol., 10 (2006), 929-954 (electronic). 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), 651-670. doi: 10.1307/mmj/1197056461.  Google Scholar

[44]

B. Kalinin and R. Spatzier, On the classification of Cartan actions, Geom. Funct. Anal., 17 (2007), 468-490. 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), 421-445. doi: 10.1090/S0002-9947-1991-1058434-X.  Google Scholar

[46]

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[47]

R. Lyons, On measures simultaneously $2$- and $3$-invariant, Israel J. Math., 61 (1988), 219-224. doi: 10.1007/BF02766212.  Google Scholar

[48]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: 10.2307/2373551.  Google Scholar

[49]

R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc., 229 (1977), 351-370. 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, B.C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 21-34.  Google Scholar

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F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms, J. Mod. Dyn., 1 (2007), 425-442. doi: 10.3934/jmd.2007.1.425.  Google Scholar

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F. Rodriguez Hertz and Z. Wang, Global rigidity of higher rank abelian Anosov algebraic actions, Invent. Math., 198 (2014), 165-209. doi: 10.1007/s00222-014-0499-y.  Google Scholar

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show all references

References:
[1]

W. Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2), 122 (1985), 597-609. doi: 10.2307/1971331.  Google Scholar

[2]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, With an appendix by Misha Brin, DMV Seminar, 25, Birkhäuser Verlag, Basel, 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), no. 6, 925-954.  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), 185-244.  Google Scholar

[5]

A. Brown, F. Rodriguez Hertz and Z. Wang, Global smooth and topological rigidity of hyperbolic lattice actions, arXiv:1512.06720, 2015. Google Scholar

[6]

K. Burns and A. Katok, Manifolds with nonpositive curvature, Ergodic Theory Dynam. Systems, 5 (1985), 307-317. 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), 35-59.  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), 1805-1858. 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), 4405-4430. doi: 10.1093/imrn/rnq252.  Google Scholar

[10]

P. Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. (2), 111 (1980), 435-476. doi: 10.2307/1971104.  Google Scholar

[11]

P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II, Acta Math., 149 (1982), 41-69. doi: 10.1007/BF02392349.  Google Scholar

[12]

M. Einsiedler and T. Fisher, Differentiable rigidity for hyperbolic toral actions, Israel J. Math., 157 (2007), 347-377. 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, Comm. Pure Appl. Math., 56 (2003), 1184-1221. 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), 169-238. 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, Eur. Math. Soc., Zürich, 2006, 1731-1759.  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), 83-128. doi: 10.3934/jmd.2008.2.83.  Google Scholar

[17]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 155-241.  Google Scholar

[18]

M. Einsiedler and E. Lindenstrauss, On measures invariant under tori on quotients of semisimple groups, Ann. of Math. (2), 181 (2015), 993-1031. doi: 10.4007/annals.2015.181.3.3.  Google Scholar

[19]

D. Fisher, Local rigidity of group actions: Past, present, future, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 45-97. 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, J. Amer. Math. Soc., 26 (2013), 167-198. 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), 67-122. doi: 10.4007/annals.2009.170.67.  Google Scholar

[22]

J. Franks, Anosov diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 61-93.  Google Scholar

[23]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49. doi: 10.1007/BF01692494.  Google Scholar

[24]

B. Farb and S. Weinberger, Isometries, rigidity and universal covers, Ann. of Math. (2), 168 (2008), 915-940. doi: 10.4007/annals.2008.168.915.  Google Scholar

[25]

A. Gogolev, Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 30 (2010), 441-456. doi: 10.1017/S0143385709000169.  Google Scholar

[26]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.  Google Scholar

[27]

A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms, Acta Math., 215 (2015), 127-159. 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), 361-410. doi: 10.2307/2946593.  Google Scholar

[29]

S. Hurder, A survey of rigidity theory for Anosov actions, in Differential Topology, Foliations, and Group Actions (Rio de Janeiro, 1992), Contemp. Math., 161, Amer. Math. Soc., Providence, RI, 1994, 143-173. 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), 137-173.  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, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 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), 361-400. 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), 203-241. 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), 253-280. 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), 27-38. doi: 10.1016/0040-9383(95)00012-7.  Google Scholar

[36]

N. Kopell, Commuting diffeomorphisms, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 165-184.  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), 609-615. 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), 487-515. 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), 135-172. 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), 751-778. 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), 292-319.  Google Scholar

[42]

B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic Anosov $\mathbbZ^k$ actions, Geom. Topol., 10 (2006), 929-954 (electronic). 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), 651-670. doi: 10.1307/mmj/1197056461.  Google Scholar

[44]

B. Kalinin and R. Spatzier, On the classification of Cartan actions, Geom. Funct. Anal., 17 (2007), 468-490. 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), 421-445. 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), 49-68. doi: 10.1017/S0143385700009573.  Google Scholar

[47]

R. Lyons, On measures simultaneously $2$- and $3$-invariant, Israel J. Math., 61 (1988), 219-224. doi: 10.1007/BF02766212.  Google Scholar

[48]

A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. doi: 10.2307/2373551.  Google Scholar

[49]

R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc., 229 (1977), 351-370. 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, B.C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 21-34.  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)], 17, Springer-Verlag, Berlin, 1991.  Google Scholar

[52]

J. Palis and J.-C. Yoccoz, Centralizers of Anosov diffeomorphisms on tori, Ann. Sci. École Norm. Sup. (4), 22 (1989), 99-108.  Google Scholar

[53]

J. Palis and J.-C. Yoccoz, Rigidity of centralizers of diffeomorphisms, Ann. Sci. École Norm. Sup. (4), 22 (1989), 81-98.  Google Scholar

[54]

F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms, J. Mod. Dyn., 1 (2007), 425-442. doi: 10.3934/jmd.2007.1.425.  Google Scholar

[55]

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