Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA |
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.
| [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. 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. |
| [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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. Pugh, M. 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
| [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. 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. |
| [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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. Pugh, M. 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. |
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