# American Institute of Mathematical Sciences

2016, 10: 135-172. doi: 10.3934/jmd.2016.10.135

## Arithmeticity and topology of smooth actions of higher rank abelian groups

 1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States, United States

Received  November 2014 Revised  April 2016 Published  May 2016

We prove that any smooth action of $\mathbb{Z}^{m-1}$, $m\ge 3$, on an $m$-dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e., isomorphic up to a finite permutation to an affine action on the torus or on its factor by $\pm\mathrm{Id}$. Furthermore this isomorphism has nice geometric properties; in particular, it is smooth in the sense of Whitney on a set whose complement has arbitrarily small measure. We further derive restrictions on topology of manifolds that may admit such actions, for example, excluding spheres and obtaining lower estimate on the first Betti number in the odd-dimensional case.
Citation: Anatole Katok, Federico Rodriguez Hertz. Arithmeticity and topology of smooth actions of higher rank abelian groups. Journal of Modern Dynamics, 2016, 10: 135-172. doi: 10.3934/jmd.2016.10.135
##### References:
 [1] J. Alexander, On the subdivision of space by a polyhedron,, Proc. Nat. Acad. Sci. USA, 10 (1924), 6. Google Scholar [2] D. Berend, Multi-invariant sets on tori,, Trans. Amer. Math. Soc., 280 (1983), 509. doi: 10.1090/S0002-9947-1983-0716835-6. Google Scholar [3] L. Barreira and Y. Pesin, Nonuniform hyperbolicity. Dynamics of systems with nonzero Lyapunov exponents,, Encyclopedia of Mathematics and its Applications, (2007). doi: 10.1017/CBO9781107326026. Google Scholar [4] M. Brown, A proof of the generalized Schoenflies theorem,, Bull. Amer. Math. Soc., 66 (1960), 74. doi: 10.1090/S0002-9904-1960-10400-4. Google Scholar [5] R. de la Llave, Smooth conjugacy and SRB measures for uniformly and non-uniformly hyperbolic systems,, Comm. Math. Phys., 150 (1992), 289. doi: 10.1007/BF02096662. Google Scholar [6] 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 [7] D. Fisher and G. Margulis, Almost isometric actions, property (T), and local rigidity,, Invent. Math., 162 (1993), 19. doi: 10.1007/s00222-004-0437-5. Google Scholar [8] G. Höhn and N.-P. Skoruppa, Un résultat de Schinzel,, J. Théor. Nombres Bordeaux, 5 (1993). Google Scholar [9] H. Hopf, Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Räume,, Vierteljschr. Naturforsch. Ges. Zürich, 85 (1940), 165. Google Scholar [10] H. Hu, Some ergodic properties of commuting diffeomorphisms,, Ergodic Theory Dynam. Systems, 13 (1993), 73. doi: 10.1017/S0143385700007215. Google Scholar [11] B. Kalinin, Theory of non-stationary normal forms,, to appear., (). Google Scholar [12] B. Kalinin and A. Katok, Invariant measures for actions of higher rank abelian groups,, in Smooth Ergodic Theory and its applications, (2001), 593. doi: 10.1090/pspum/069/1858547. Google Scholar [13] B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant Measures for Cartan actions on tori,, Journal of Modern Dynamics, 1 (2007), 123. doi: 10.3934/jmd.2007.1.123. Google Scholar [14] 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 [15] A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [16] A. Katok, S. Katok and F. Rodriguez Hertz, The Fried average entropy and slow entropy for actions of higher rank abelian groups,, Geom. Funct. Anal., 24 (2014), 1204. doi: 10.1007/s00039-014-0284-5. Google Scholar [17] A. Katok, S. Katok and K. Schmidt, Rigidity of measurable structure for $Z^d$-actions by automorphisms of a torus,, Comm. Math. Helvetici, 77 (2002), 718. doi: 10.1007/PL00012439. Google Scholar [18] 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 [19] A. Katok and V. Niţică, Rigidity in Higher Rank Abelian Actions. Volume 1. Introduction and Cocycle Problem,, Cambridge Tracts in Mathematics, (2011). doi: 10.1017/CBO9780511803550. Google Scholar [20] A. Katok and F. Rodriguez Hertz, Uniqueness of large invariant measures for $\mathbbZ^k$ actions with Cartan homotopy data,, J. Mod. Dyn., 1 (2007), 287. doi: 10.3934/jmd.2007.1.287. Google Scholar [21] A. Katok and F. Rodriguez Hertz, Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups,, J. Mod. Dyn., 4 (2010), 487. doi: 10.3934/jmd.2010.4.487. Google Scholar [22] A. Katok and F. Rodriguez Hertz, Rigidity of real-analytic actions of $SL(n,\mathbb Z)$ on $\mathbb T^n$: A case of realization of Zimmer program,, Discrete Contin. Dyn. Syst., 27 (2010), 609. doi: 10.3934/dcds.2010.27.609. Google Scholar [23] M. Kervaire and J. Milnor, Groups of homotopy spheres. I,, Ann. of Math. (2), 77 (1963), 504. doi: 10.2307/1970128. Google Scholar [24] J. Milnor, On manifolds homeomorphic to the 7-sphere,, Ann. of Math. (2), 64 (1956), 399. doi: 10.2307/1969983. Google Scholar [25] J. Milnor, Differential topology forty-six years later,, Notices Amer. Math. Soc., 58 (2011), 804. Google Scholar [26] M. Morse, A reduction of the Schoenflies extension problem},, Bull. Amer. Math. Soc., 66 (1960), 113. doi: 10.1090/S0002-9904-1960-10420-X. Google Scholar [27] B. Mazur, On embeddings of spheres,, Bull. Amer. Math. Soc., 65 (1959), 59. doi: 10.1090/S0002-9904-1959-10274-3. Google Scholar [28] W. Parry, Synchronization of canonical measures for hyperbolic attractors,, Comm. Math. Phys., 106 (1986), 267. doi: 10.1007/BF01454975. Google Scholar [29] Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russin Math. Surveys, 32 (1977), 55. doi: 10.1070/RM1977v032n04ABEH001639. Google Scholar [30] F. Rodriguez Hertz, J. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and nonuniform hyperbolicity,, Duke Math. J., 160 (2011), 599. doi: 10.1215/00127094-1444314. Google Scholar [31] 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 [32] 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 [33] A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number,, Acta Arith., 24 (1973), 385. Google Scholar

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##### References:
 [1] J. Alexander, On the subdivision of space by a polyhedron,, Proc. Nat. Acad. Sci. USA, 10 (1924), 6. Google Scholar [2] D. Berend, Multi-invariant sets on tori,, Trans. Amer. Math. Soc., 280 (1983), 509. doi: 10.1090/S0002-9947-1983-0716835-6. Google Scholar [3] L. Barreira and Y. Pesin, Nonuniform hyperbolicity. Dynamics of systems with nonzero Lyapunov exponents,, Encyclopedia of Mathematics and its Applications, (2007). doi: 10.1017/CBO9781107326026. Google Scholar [4] M. Brown, A proof of the generalized Schoenflies theorem,, Bull. Amer. Math. Soc., 66 (1960), 74. doi: 10.1090/S0002-9904-1960-10400-4. Google Scholar [5] R. de la Llave, Smooth conjugacy and SRB measures for uniformly and non-uniformly hyperbolic systems,, Comm. Math. Phys., 150 (1992), 289. doi: 10.1007/BF02096662. Google Scholar [6] 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 [7] D. Fisher and G. Margulis, Almost isometric actions, property (T), and local rigidity,, Invent. Math., 162 (1993), 19. doi: 10.1007/s00222-004-0437-5. Google Scholar [8] G. Höhn and N.-P. Skoruppa, Un résultat de Schinzel,, J. Théor. Nombres Bordeaux, 5 (1993). Google Scholar [9] H. Hopf, Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Räume,, Vierteljschr. Naturforsch. Ges. Zürich, 85 (1940), 165. Google Scholar [10] H. Hu, Some ergodic properties of commuting diffeomorphisms,, Ergodic Theory Dynam. Systems, 13 (1993), 73. doi: 10.1017/S0143385700007215. Google Scholar [11] B. Kalinin, Theory of non-stationary normal forms,, to appear., (). Google Scholar [12] B. Kalinin and A. Katok, Invariant measures for actions of higher rank abelian groups,, in Smooth Ergodic Theory and its applications, (2001), 593. doi: 10.1090/pspum/069/1858547. Google Scholar [13] B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant Measures for Cartan actions on tori,, Journal of Modern Dynamics, 1 (2007), 123. doi: 10.3934/jmd.2007.1.123. Google Scholar [14] 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 [15] A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [16] A. Katok, S. Katok and F. Rodriguez Hertz, The Fried average entropy and slow entropy for actions of higher rank abelian groups,, Geom. Funct. Anal., 24 (2014), 1204. doi: 10.1007/s00039-014-0284-5. Google Scholar [17] A. Katok, S. Katok and K. Schmidt, Rigidity of measurable structure for $Z^d$-actions by automorphisms of a torus,, Comm. Math. Helvetici, 77 (2002), 718. doi: 10.1007/PL00012439. Google Scholar [18] 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 [19] A. Katok and V. Niţică, Rigidity in Higher Rank Abelian Actions. Volume 1. Introduction and Cocycle Problem,, Cambridge Tracts in Mathematics, (2011). doi: 10.1017/CBO9780511803550. Google Scholar [20] A. Katok and F. Rodriguez Hertz, Uniqueness of large invariant measures for $\mathbbZ^k$ actions with Cartan homotopy data,, J. Mod. Dyn., 1 (2007), 287. doi: 10.3934/jmd.2007.1.287. Google Scholar [21] A. Katok and F. Rodriguez Hertz, Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups,, J. Mod. Dyn., 4 (2010), 487. doi: 10.3934/jmd.2010.4.487. Google Scholar [22] A. Katok and F. Rodriguez Hertz, Rigidity of real-analytic actions of $SL(n,\mathbb Z)$ on $\mathbb T^n$: A case of realization of Zimmer program,, Discrete Contin. Dyn. Syst., 27 (2010), 609. doi: 10.3934/dcds.2010.27.609. Google Scholar [23] M. Kervaire and J. Milnor, Groups of homotopy spheres. I,, Ann. of Math. (2), 77 (1963), 504. doi: 10.2307/1970128. Google Scholar [24] J. Milnor, On manifolds homeomorphic to the 7-sphere,, Ann. of Math. (2), 64 (1956), 399. doi: 10.2307/1969983. Google Scholar [25] J. Milnor, Differential topology forty-six years later,, Notices Amer. Math. Soc., 58 (2011), 804. Google Scholar [26] M. Morse, A reduction of the Schoenflies extension problem},, Bull. Amer. Math. Soc., 66 (1960), 113. doi: 10.1090/S0002-9904-1960-10420-X. Google Scholar [27] B. Mazur, On embeddings of spheres,, Bull. Amer. Math. Soc., 65 (1959), 59. doi: 10.1090/S0002-9904-1959-10274-3. Google Scholar [28] W. Parry, Synchronization of canonical measures for hyperbolic attractors,, Comm. Math. Phys., 106 (1986), 267. doi: 10.1007/BF01454975. Google Scholar [29] Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russin Math. Surveys, 32 (1977), 55. doi: 10.1070/RM1977v032n04ABEH001639. Google Scholar [30] F. Rodriguez Hertz, J. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and nonuniform hyperbolicity,, Duke Math. J., 160 (2011), 599. doi: 10.1215/00127094-1444314. Google Scholar [31] 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 [32] 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 [33] A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number,, Acta Arith., 24 (1973), 385. Google Scholar
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