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

On the work of Rodriguez Hertz on rigidity in dynamics (Brin Prize article)

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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