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  2021, 17: 65-109. doi: 10.3934/jmd.2021003

Local Lyapunov spectrum rigidity of nilmanifold automorphisms

Jonathan DeWitt

Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA

Received  January 22, 2020 Revised  November 04, 2020 Published  February 2021

Fund Project: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1746045

We study the regularity of a conjugacy between an Anosov automorphism $ L $ of a nilmanifold $ N/\Gamma $ and a volume-preserving, $ C^1 $-small perturbation $ f $. We say that $ L $ is locally Lyapunov spectrum rigid if this conjugacy is $ C^{1+} $ whenever $ f $ is $ C^{1+} $ and has the same volume Lyapunov spectrum as $ L $. For $ L $ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $ L $ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.

Keywords: Lyapunov exponents, rigidity, nilmanifolds, coarse geometry.
Mathematics Subject Classification: Primary:37D20, 37D25;Secondary:51F30.
Citation: Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003
References:
[1]

E. Breuillard, Geometry of locally compact groups of polynomial growth and shape of large balls, Groups Geom. Dyn., 8 (2014), 669-732.  doi: 10.4171/GGD/244.  Google Scholar

[2]

A. Brown, Smoothness of stable holonomies inside center-stable manifolds and the $C^2$ hypothesis in Pugh-Shub and Ledrappier-Young theory, preprint, arXiv: 1608.05886. Google Scholar

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C. Butler, Characterizing symmetric spaces by their Lyapunov spectra, preprint, arXiv: 1709.08066. Google Scholar

[4]

J.-P. Conze and J.-C. Marcuard, Conjugaison topologique des automorphismes et des translations ergodiques de nilvariétés, Ann. Inst. H. Poincaré Sect. B (N.S.), 6 (1970), 153-157.   Google Scholar

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R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems. II, Comm. Math. Phys., 109 (1987), 369-378.  doi: 10.1007/BF01206141.  Google Scholar

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R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150 (1992), 289-320.  doi: 10.1007/BF02096662.  Google Scholar

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M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.  Google Scholar

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A. Erchenko, Flexibility of Lyapunov exponents with respect to two classes of measures on the torus, preprint, arXiv: 1909.11457. Google Scholar

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J. Franks, Anosov diffeomorphisms on tori, Trans. Amer. Math. Soc., 145 (1969), 117-124.  doi: 10.1090/S0002-9947-1969-0253352-7.  Google Scholar

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A. Gogolev, Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori, J. Mod. Dyn., 2 (2008), 645-700.  doi: 10.3934/jmd.2008.2.645.  Google Scholar

[13]

A. Gogolev, Rigidity lecture notes, 2019, available at https://people.math.osu.edu/gogolyev.1/index_files/CIRM_notes_all.pdf. Google Scholar

[14]

A. Gogolev, B. Kalinin and V. Sadovskaya, Local rigidity for Anosov automorphisms, Math. Res. Lett., 18 (2011), 843-858.  doi: 10.4310/MRL.2011.v18.n5.a4.  Google Scholar

[15]

A. Gogolev, B. Kalinin and V. Sadovskaya, Local rigidity of Lyapunov spectrum for toral automorphisms, Israel J. Math., 238 (2020), 389-403.  doi: 10.1007/s11856-020-2028-6.  Google Scholar

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A. Gogolev, P. Ontaneda and F. R. Hertz, New partially hyperbolic dynamical systems I, Acta Math., 215 (2015), 363-393.  doi: 10.1007/s11511-016-0135-3.  Google Scholar

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Y. Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France, 101 (1973), 333-379.   Google Scholar

[18]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[19]

M. Jiang, Y. B. Pesin and R. de la Llave, On the integrability of intermediate distributions for Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 15 (1995), 317-331.  doi: 10.1017/S0143385700008397.  Google Scholar

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J.-L. Journé, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4 (1988), 187-193.  doi: 10.4171/RMI/69.  Google Scholar

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B. Kalinin, Livšic theorem for matrix cocycles, Ann. of Math. (2), 173 (2011), 1025-1042.  doi: 10.4007/annals.2011.173.2.11.  Google Scholar

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B. Kalinin, A. Katok and F. R. Hertz, Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data", J. Mod. Dyn., 4 (2010), 207-209.  doi: 10.3934/jmd.2010.4.207.  Google Scholar

[23] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[24]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.  Google Scholar

[25]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539.  doi: 10.2307/1971328.  Google Scholar

[26]

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

[27]

T. L. Payne, Anosov automorphisms of nilpotent Lie algebras, J. Mod. Dyn., 3 (2009), 121-158.  doi: 10.3934/jmd.2009.3.121.  Google Scholar

[28]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546.  doi: 10.1215/S0012-7094-97-08616-6.  Google Scholar

[29]

M. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[30]

V. A. Rokhlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp.  Google Scholar

[31]

R. Saghin and J. Yang, Lyapunov exponents and rigidity of Anosov automorphisms and skew products, Adv. Math., 355 (2019), 45 pp. doi: 10.1016/j.aim.2019.106764.  Google Scholar

[32]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

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J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504. Google Scholar

show all references

References:
[1]

E. Breuillard, Geometry of locally compact groups of polynomial growth and shape of large balls, Groups Geom. Dyn., 8 (2014), 669-732.  doi: 10.4171/GGD/244.  Google Scholar

[2]

A. Brown, Smoothness of stable holonomies inside center-stable manifolds and the $C^2$ hypothesis in Pugh-Shub and Ledrappier-Young theory, preprint, arXiv: 1608.05886. Google Scholar

[3]

C. Butler, Characterizing symmetric spaces by their Lyapunov spectra, preprint, arXiv: 1709.08066. Google Scholar

[4]

J.-P. Conze and J.-C. Marcuard, Conjugaison topologique des automorphismes et des translations ergodiques de nilvariétés, Ann. Inst. H. Poincaré Sect. B (N.S.), 6 (1970), 153-157.   Google Scholar

[5]

Y. Cornulier, Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups, Bull. Soc. Math. France, 144 (2016), 693-744.  doi: 10.24033/bsmf.2725.  Google Scholar

[6]

R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems. II, Comm. Math. Phys., 109 (1987), 369-378.  doi: 10.1007/BF01206141.  Google Scholar

[7]

R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150 (1992), 289-320.  doi: 10.1007/BF02096662.  Google Scholar

[8]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.  Google Scholar

[9] J. Eldering, Normally Hyperbolic Invariant Manifolds: The Noncompact Case, Atlantis Studies in Dynamical Systems, 2, Atlantis Press, Paris, 2013.  doi: 10.2991/978-94-6239-003-4.  Google Scholar
[10]

A. Erchenko, Flexibility of Lyapunov exponents with respect to two classes of measures on the torus, preprint, arXiv: 1909.11457. Google Scholar

[11]

J. Franks, Anosov diffeomorphisms on tori, Trans. Amer. Math. Soc., 145 (1969), 117-124.  doi: 10.1090/S0002-9947-1969-0253352-7.  Google Scholar

[12]

A. Gogolev, Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori, J. Mod. Dyn., 2 (2008), 645-700.  doi: 10.3934/jmd.2008.2.645.  Google Scholar

[13]

A. Gogolev, Rigidity lecture notes, 2019, available at https://people.math.osu.edu/gogolyev.1/index_files/CIRM_notes_all.pdf. Google Scholar

[14]

A. Gogolev, B. Kalinin and V. Sadovskaya, Local rigidity for Anosov automorphisms, Math. Res. Lett., 18 (2011), 843-858.  doi: 10.4310/MRL.2011.v18.n5.a4.  Google Scholar

[15]

A. Gogolev, B. Kalinin and V. Sadovskaya, Local rigidity of Lyapunov spectrum for toral automorphisms, Israel J. Math., 238 (2020), 389-403.  doi: 10.1007/s11856-020-2028-6.  Google Scholar

[16]

A. Gogolev, P. Ontaneda and F. R. Hertz, New partially hyperbolic dynamical systems I, Acta Math., 215 (2015), 363-393.  doi: 10.1007/s11511-016-0135-3.  Google Scholar

[17]

Y. Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France, 101 (1973), 333-379.   Google Scholar

[18]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[19]

M. Jiang, Y. B. Pesin and R. de la Llave, On the integrability of intermediate distributions for Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 15 (1995), 317-331.  doi: 10.1017/S0143385700008397.  Google Scholar

[20]

J.-L. Journé, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4 (1988), 187-193.  doi: 10.4171/RMI/69.  Google Scholar

[21]

B. Kalinin, Livšic theorem for matrix cocycles, Ann. of Math. (2), 173 (2011), 1025-1042.  doi: 10.4007/annals.2011.173.2.11.  Google Scholar

[22]

B. Kalinin, A. Katok and F. R. Hertz, Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data", J. Mod. Dyn., 4 (2010), 207-209.  doi: 10.3934/jmd.2010.4.207.  Google Scholar

[23] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[24]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.  Google Scholar

[25]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539.  doi: 10.2307/1971328.  Google Scholar

[26]

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

[27]

T. L. Payne, Anosov automorphisms of nilpotent Lie algebras, J. Mod. Dyn., 3 (2009), 121-158.  doi: 10.3934/jmd.2009.3.121.  Google Scholar

[28]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, Duke Math. J., 86 (1997), 517-546.  doi: 10.1215/S0012-7094-97-08616-6.  Google Scholar

[29]

M. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[30]

V. A. Rokhlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, 1952 (1952), 55 pp.  Google Scholar

[31]

R. Saghin and J. Yang, Lyapunov exponents and rigidity of Anosov automorphisms and skew products, Adv. Math., 355 (2019), 45 pp. doi: 10.1016/j.aim.2019.106764.  Google Scholar

[32]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[33]

J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504. Google Scholar

Figure 1.  A diagrammatic depiction of the weak and strong foliations illustrating the points involved in the definition of the weak and strong distances. The strong distance between $ q $ and $ r $ is the distance from $ z $ to $ q $ along the $ \mathscr{S}_{i+1}^u $ foliation. The weak distance between $ q $ and $ r $ is the distance between $ z $ and $ r $ along the $ \mathscr{W}_i^u $ foliation
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