×

On the work of Rodriguez Hertz on rigidity in dynamics. (English) Zbl 1366.37076

Summary: 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.

MSC:

37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
53C24 Rigidity results

Biographic References:

Rodriguez Hertz, Federico
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] W. Ballmann, Nonpositively curved manifolds of higher rank,, Ann. of Math. (2), 122, 597 (1985) · Zbl 0585.53031 · doi:10.2307/1971331
[2] W. Ballmann, <em>Lectures on Spaces of Nonpositive Curvature</em>,, With an appendix by Misha Brin (1995) · Zbl 1031.53069 · doi:10.1007/978-3-0348-9240-7
[3] C. Bonatti, Local density of diffeomorphisms with large centralizers,, Ann. Sci. École Norm. Sup. (4), 41, 925 (2008) · Zbl 1163.58003
[4] C. Bonatti, The \(C^1\) generic diffeomorphism has trivial centralizer,, Inst. Hautes Études Sci. Publ. Math., 109, 185 (2009) · Zbl 1177.37025
[5] A. Brown, Global smooth and topological rigidity of hyperbolic lattice actions,, <a href= (2015)
[6] K. Burns, Manifolds with nonpositive curvature,, Ergodic Theory Dynam. Systems, 5, 307 (1985) · Zbl 0572.58019 · doi:10.1017/S0143385700002935
[7] K. Burns, Manifolds of nonpositive curvature and their buildings,, Inst. Hautes Études Sci. Publ. Math., 65, 35 (1987) · Zbl 0643.53037
[8] D. Damjanović, Local rigidity of partially hyperbolic actions I. KAM method and \(\mathbbZ^k\) actions on the torus,, Ann. of Math. (2), 172, 1805 (2010) · Zbl 1209.37017 · doi:10.4007/annals.2010.172.1805
[9] D. Damjanović, 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, 4405 (2011) · Zbl 1291.37027 · doi:10.1093/imrn/rnq252
[10] P. Eberlein, Lattices in spaces of nonpositive curvature,, Ann. of Math. (2), 111, 435 (1980) · Zbl 0401.53015 · doi:10.2307/1971104
[11] P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II,, Acta Math., 149, 41 (1982) · Zbl 0511.53048 · doi:10.1007/BF02392349
[12] M. Einsiedler, Differentiable rigidity for hyperbolic toral actions,, Israel J. Math., 157, 347 (2007) · Zbl 1112.37022 · doi:10.1007/s11856-006-0016-0
[13] M. Einsiedler, Invariant measures on \(G/\Gamma\) for split simple Lie groups \(G\),, Dedicated to the memory of Jürgen K. Moser, 56, 1184 (2003) · Zbl 1022.22023 · doi:10.1002/cpa.10092
[14] M. Einsiedler, Rigidity of measures-The high entropy case and non-commuting foliations,, Israel J. Math., 148, 169 (2005) · Zbl 1097.37017 · doi:10.1007/BF02775436
[15] M. Einsiedler, Diagonalizable flows on locally homogeneous spaces and number theory,, in International Congress of Mathematicians. Vol. II, 1731 (2006) · Zbl 1121.37028
[16] M. Einsiedler, On measures invariant under diagonalizable actions: The rank-one case and the general low-entropy method,, J. Mod. Dyn., 2, 83 (2008) · Zbl 1146.37006 · doi:10.3934/jmd.2008.2.83
[17] M. Einsiedler, Diagonal actions on locally homogeneous spaces,, in Homogeneous Flows, 155 (2010) · Zbl 1225.37005
[18] M. Einsiedler, On measures invariant under tori on quotients of semisimple groups,, Ann. of Math. (2), 181, 993 (2015) · Zbl 1316.22009 · doi:10.4007/annals.2015.181.3.3
[19] D. Fisher, Local rigidity of group actions: Past, present, future,, in Dynamics, 45 (2007) · Zbl 1144.22008 · doi:10.1017/CBO9780511755187.003
[20] D. Fisher, Global rigidity of higher rank Anosov actions on tori and nilmanifolds,, With an appendix by James F. Davis, 26, 167 (2013) · Zbl 1338.37040 · doi:10.1090/S0894-0347-2012-00751-6
[21] D. Fisher, Local rigidity of affine actions of higher rank groups and lattices,, Ann. of Math. (2), 170, 67 (2009) · Zbl 1186.22015 · doi:10.4007/annals.2009.170.67
[22] J. Franks, Anosov diffeomorphisms,, in Global Analysis (Proc. Sympos. Pure Math., 61 (1968) · Zbl 0207.54304
[23] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory, 1, 1 (1967) · Zbl 0146.28502 · doi:10.1007/BF01692494
[24] B. Farb, Isometries, rigidity and universal covers,, Ann. of Math. (2), 168, 915 (2008) · Zbl 1175.53055 · doi:10.4007/annals.2008.168.915
[25] A. Gogolev, Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms,, Ergodic Theory Dynam. Systems, 30, 441 (2010) · Zbl 1185.37040 · doi:10.1017/S0143385709000169
[26] M. Gromov, Groups of polynomial growth and expanding maps,, Inst. Hautes Études Sci. Publ. Math., 53, 53 (1981) · Zbl 0474.20018
[27] A. Gorodnik, Mixing properties of commuting nilmanifold automorphisms,, Acta Math., 215, 127 (2015) · Zbl 1360.37010 · doi:10.1007/s11511-015-0130-0
[28] S. Hurder, Rigidity for Anosov actions of higher rank lattices,, Ann. of Math. (2), 135, 361 (1992) · Zbl 0754.58029 · doi:10.2307/2946593
[29] S. Hurder, A survey of rigidity theory for Anosov actions,, in Differential Topology, 143 (1992) · Zbl 0846.58044 · doi:10.1090/conm/161
[30] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51, 137 (1980) · Zbl 0445.58015
[31] A. Katok, <em>Introduction to the Modern Theory of Dynamical Systems</em>,, With a supplementary chapter by Katok and L. Mendoza (1995) · Zbl 0878.58020 · doi:10.1017/CBO9780511809187
[32] B. Kalinin, Nonuniform measure rigidity,, Ann. of Math. (2), 174, 361 (2011) · Zbl 1368.37031 · doi:10.4007/annals.2011.174.1.10
[33] A. Katok, Local rigidity for certain groups of toral automorphisms,, Israel J. Math., 75, 203 (1991) · Zbl 0785.22012 · doi:10.1007/BF02776025
[34] A. Katok, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions,, Israel J. Math., 93, 253 (1996) · Zbl 0857.57038 · doi:10.1007/BF02761106
[35] A. Katok, Cocycle superrigidity and rigidity for lattice actions on tori,, Topology, 35, 27 (1996) · Zbl 0857.57037 · doi:10.1016/0040-9383(95)00012-7
[36] N. Kopell, Commuting diffeomorphisms,, in Global Analysis (Proc. Sympos. Pure Math., 165 (1968)
[37] A. Katok, 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, 609 (2010) · Zbl 1192.37025 · doi:10.3934/dcds.2010.27.609
[38] A. Katok, Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher-rank abelian groups,, J. Mod. Dyn., 4, 487 (2010) · Zbl 1213.37037 · doi:10.3934/jmd.2010.4.487
[39] A. Katok, Arithmeticity and topology of smooth actions of higher rank abelian groups,, J. Mod. Dyn., 10, 135 (2016) · Zbl 1370.37048 · doi:10.3934/jmd.2016.10.135
[40] A. Katok, Invariant measures for higher-rank hyperbolic abelian actions,, Ergodic Theory Dynam. Systems, 16, 751 (1996) · Zbl 0859.58021 · doi:10.1017/S0143385700009081
[41] A. Katok, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions,, Tr. Mat. Inst. Steklova, 216, 292 (1997) · Zbl 0938.37010
[42] B. Kalinin, Global rigidity for totally nonsymplectic Anosov \(\mathbbZ^k\) actions,, Geom. Topol., 10, 929 (2006) · Zbl 1126.37015 · doi:10.2140/gt.2006.10.929
[43] B. Kalinin, On the classification of resonance-free Anosov \(\mathbbZ^k\) actions,, Michigan Math. J., 55, 651 (2007) · Zbl 1233.37013 · doi:10.1307/mmj/1197056461
[44] B. Kalinin, On the classification of Cartan actions,, Geom. Funct. Anal., 17, 468 (2007) · Zbl 1121.37026 · doi:10.1007/s00039-007-0602-2
[45] J. W. Lewis, Infinitesimal rigidity for the action of \(\text{\rm SL}(n,\mathbbZ)\) on \(\mathbbT^n\),, Trans. Amer. Math. Soc., 324, 421 (1991) · Zbl 0726.57028 · doi:10.1090/S0002-9947-1991-1058434-X
[46] D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms,, Ergodic Theory Dynamical Systems, 2, 49 (1982) · Zbl 0507.58034 · doi:10.1017/S0143385700009573
[47] R. Lyons, On measures simultaneously \(2\)- and \(3\)-invariant,, Israel J. Math., 61, 219 (1988) · Zbl 0654.28010 · doi:10.1007/BF02766212
[48] A. Manning, There are no new Anosov diffeomorphisms on tori,, Amer. J. Math., 96, 422 (1974) · Zbl 0242.58003 · doi:10.2307/2373551
[49] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds,, Trans. Amer. Math. Soc., 229, 351 (1977) · Zbl 0356.58009 · doi:10.1090/S0002-9947-1977-0482849-4
[50] G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature,, in Proceedings of the International Congress of Mathematicians (Vancouver, 21 (1974) · Zbl 0367.57012
[51] G. A. Margulis, <em>Discrete Subgroups of Semisimple Lie Groups</em>,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (1991) · Zbl 0732.22008
[52] J. Palis, Centralizers of Anosov diffeomorphisms on tori,, Ann. Sci. École Norm. Sup. (4), 22, 99 (1989) · Zbl 0675.58029
[53] J. Palis, Rigidity of centralizers of diffeomorphisms,, Ann. Sci. École Norm. Sup. (4), 22, 81 (1989) · Zbl 0709.58022
[54] F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms,, J. Mod. Dyn., 1, 425 (2007) · Zbl 1130.37013 · doi:10.3934/jmd.2007.1.425
[55] F. Rodriguez Hertz, Global rigidity of higher rank abelian Anosov algebraic actions,, Invent. Math., 198, 165 (2014) · Zbl 1312.37028 · doi:10.1007/s00222-014-0499-y
[56] D. J. Rudolph, \( \times 2\) and \(\times 3\) invariant measures and entropy,, Ergodic Theory Dynam. Systems, 10, 395 (1990) · Zbl 0709.28013 · doi:10.1017/S0143385700005629
[57] S. J. Schreiber, On growth rates of subadditive functions for semiflows,, J. Differential Equations, 148, 334 (1998) · Zbl 0940.37007 · doi:10.1006/jdeq.1998.3471
[58] M. Shub, Expanding maps,, in Global Analysis (Proc. Sympos. Pure Math., 273 (1968) · Zbl 0213.50302
[59] S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73, 747 (1967) · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1
[60] R. J. Spatzier, An invitation to rigidity theory,, in Modern Dynamical Systems and Applications, 211 (2004) · Zbl 1151.53039
[61] K. Vinhage, On the rigidity of Weyl chamber flows and Schur multipliers as topological groups,, J. Mod. Dyn., 9, 25 (2015) · Zbl 1358.37060 · doi:10.3934/jmd.2015.9.25
[62] W. van Limbeek, Riemannian manifolds with local symmetry,, J. Topol. Anal., 6, 211 (2014) · Zbl 1290.53050 · doi:10.1142/S179352531450006X
[63] W. van Limbeek, Symmetry gaps in Riemannian geometry and minimal orbifolds,, <a href= (2014) · Zbl 1364.53042
[64] K. Vinhage, Local rigidity of higher rank homogeneous abelian actions: A complete solution via the geometric method,, <a href= (2015)
[65] Z. Wang, Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits,, Trans. Amer. Math. Soc., 362, 4267 (2010) · Zbl 1201.37025 · doi:10.1090/S0002-9947-10-04947-0
[66] R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups,, Ann. of Math. (2), 112, 511 (1980) · Zbl 0468.22011 · doi:10.2307/1971090
[67] R. J. Zimmer, Actions of semisimple groups and discrete subgroups,, in Proceedings of the International Congress of Mathematicians, 1247 (1986) · Zbl 0671.57028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.