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

Local Lyapunov spectrum rigidity of nilmanifold automorphisms

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.

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.

[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.

[3]

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

[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. 

[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.

[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.

[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.

[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.

[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.
[10]

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

[11]

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

[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.

[13]

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

[14]

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

[15]

A. GogolevB. 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.

[16]

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

[17]

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

[18]

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

[19]

M. JiangY. 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.

[20]

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

[21]

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

[22]

B. KalininA. 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.

[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.
[24]

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

[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.

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

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.

[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.

[3]

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

[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. 

[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.

[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.

[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.

[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.

[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.
[10]

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

[11]

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

[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.

[13]

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

[14]

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

[15]

A. GogolevB. 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.

[16]

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

[17]

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

[18]

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

[19]

M. JiangY. 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.

[20]

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

[21]

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

[22]

B. KalininA. 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.

[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.
[24]

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

[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.

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[32]

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

[33]

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

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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