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.

[2]

D. Berend, Multi-invariant sets on tori,, Trans. Amer. Math. Soc., 280 (1983), 509. doi: 10.1090/S0002-9947-1983-0716835-6.

[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.

[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.

[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.

[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.

[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.

[8]

G. Höhn and N.-P. Skoruppa, Un résultat de Schinzel,, J. Théor. Nombres Bordeaux, 5 (1993).

[9]

H. Hopf, Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Räume,, Vierteljschr. Naturforsch. Ges. Zürich, 85 (1940), 165.

[10]

H. Hu, Some ergodic properties of commuting diffeomorphisms,, Ergodic Theory Dynam. Systems, 13 (1993), 73. doi: 10.1017/S0143385700007215.

[11]

B. Kalinin, Theory of non-stationary normal forms,, to appear., ().

[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.

[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.

[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.

[15]

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

M. Kervaire and J. Milnor, Groups of homotopy spheres. I,, Ann. of Math. (2), 77 (1963), 504. doi: 10.2307/1970128.

[24]

J. Milnor, On manifolds homeomorphic to the 7-sphere,, Ann. of Math. (2), 64 (1956), 399. doi: 10.2307/1969983.

[25]

J. Milnor, Differential topology forty-six years later,, Notices Amer. Math. Soc., 58 (2011), 804.

[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.

[27]

B. Mazur, On embeddings of spheres,, Bull. Amer. Math. Soc., 65 (1959), 59. doi: 10.1090/S0002-9904-1959-10274-3.

[28]

W. Parry, Synchronization of canonical measures for hyperbolic attractors,, Comm. Math. Phys., 106 (1986), 267. doi: 10.1007/BF01454975.

[29]

Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russin Math. Surveys, 32 (1977), 55. doi: 10.1070/RM1977v032n04ABEH001639.

[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.

[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.

[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.

[33]

A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number,, Acta Arith., 24 (1973), 385.

show all references

References:
[1]

J. Alexander, On the subdivision of space by a polyhedron,, Proc. Nat. Acad. Sci. USA, 10 (1924), 6.

[2]

D. Berend, Multi-invariant sets on tori,, Trans. Amer. Math. Soc., 280 (1983), 509. doi: 10.1090/S0002-9947-1983-0716835-6.

[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.

[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.

[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.

[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.

[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.

[8]

G. Höhn and N.-P. Skoruppa, Un résultat de Schinzel,, J. Théor. Nombres Bordeaux, 5 (1993).

[9]

H. Hopf, Systeme symmetrischer Bilinearformen und euklidische Modelle der projektiven Räume,, Vierteljschr. Naturforsch. Ges. Zürich, 85 (1940), 165.

[10]

H. Hu, Some ergodic properties of commuting diffeomorphisms,, Ergodic Theory Dynam. Systems, 13 (1993), 73. doi: 10.1017/S0143385700007215.

[11]

B. Kalinin, Theory of non-stationary normal forms,, to appear., ().

[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.

[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.

[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.

[15]

A. Katok, Lyapunov exponents, entropy and periodic points of diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

M. Kervaire and J. Milnor, Groups of homotopy spheres. I,, Ann. of Math. (2), 77 (1963), 504. doi: 10.2307/1970128.

[24]

J. Milnor, On manifolds homeomorphic to the 7-sphere,, Ann. of Math. (2), 64 (1956), 399. doi: 10.2307/1969983.

[25]

J. Milnor, Differential topology forty-six years later,, Notices Amer. Math. Soc., 58 (2011), 804.

[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.

[27]

B. Mazur, On embeddings of spheres,, Bull. Amer. Math. Soc., 65 (1959), 59. doi: 10.1090/S0002-9904-1959-10274-3.

[28]

W. Parry, Synchronization of canonical measures for hyperbolic attractors,, Comm. Math. Phys., 106 (1986), 267. doi: 10.1007/BF01454975.

[29]

Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russin Math. Surveys, 32 (1977), 55. doi: 10.1070/RM1977v032n04ABEH001639.

[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.

[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.

[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.

[33]

A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number,, Acta Arith., 24 (1973), 385.

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