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On spectra of Koopman, groupoid and quasi-regular representations
Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups
Imperial College London, South, Kensington Campus, 180 Queen’s Gate, Huxley Building, London SW7 2AZ, UK |
We study close-to-constants quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $d \in \mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is $C^{\infty}$-conjugate to it, and the KAM scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems.
References:
| [1] |
A. Avila and R. Krikorian,
Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. of Math. (2), 164 (2006), 911-940.
doi: 10.4007/annals.2006.164.911. |
| [2] |
A. Avila, B. Fayad and R. Krikorian,
A KAM scheme for SL(2.$\mathbb{R} $) cocycles with Liouvillean frequencies, Geom. Funct. Anal., 21 (2011), 1001-1019.
doi: 10.1007/s00039-011-0135-6. |
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D. Bump, Lie Groups, Graduate Texts in Mathematics, 225, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-4094-3. |
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J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, Berlin, 2000.
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L. H. Eliasson,
Ergodic skew-systems on Td × SO(3.R), Ergodic Theory Dynam. Systems, 22 (2002), 1429-1449.
doi: 10.1017/S0143385702000998. |
| [7] |
B. Fayad and R. Krikorian,
Rigidity results for quasiperiodic SL(2.$\mathbb{R} $)-cocycles, J. Mod. Dyn., 3 (2009), 497-510.
doi: 10.3934/jmd.2009.3.479. |
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H. Furstenberg,
Strict ergodicity and transformations of the torus, Amer. J. Math., 83 (1961), 573-601.
doi: 10.2307/2372899. |
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S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962. |
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X. Hou and J. You,
Local rigidity of reducibility of analytic quasi-periodic cocycles on U(n), Discrete Contin. Dyn. Syst., 24 (2009), 441-454.
doi: 10.3934/dcds.2009.24.441. |
| [11] |
N. Karaliolios, Invariant distributions for quasiperiodic cocycles in $\mathbb{T}^d $ × SU(2), arXiv: 1407.4763, 2014. |
| [12] |
N. Karaliolios, Continuous spectrum or measurable reducibility for quasiperiodic cocycles in $\mathbb{T}^d $ × SU(2), arXiv: 1512.00057, 2015. |
| [13] |
N. Karaliolios, Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups, Mém. Soc. Math. Fr. (N.S.), No. 146 (2016), 4+ⅱ+200 pp. |
| [14] |
A. Kolmogoroff,
On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Amer. Math. Soc. Translation, 1949 (1949), 19 pp..
|
| [15] |
R. Krikorian, Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque, (1999), ⅵ+216 pp. |
| [16] |
R. Krikorian,
Global density of reducible quasi-periodic cocycles on T1 × SU(2), Ann. of Math. (2), 154 (2001), 269-326.
doi: 10.2307/3062098. |
show all references
References:
| [1] |
A. Avila and R. Krikorian,
Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. of Math. (2), 164 (2006), 911-940.
doi: 10.4007/annals.2006.164.911. |
| [2] |
A. Avila, B. Fayad and R. Krikorian,
A KAM scheme for SL(2.$\mathbb{R} $) cocycles with Liouvillean frequencies, Geom. Funct. Anal., 21 (2011), 1001-1019.
doi: 10.1007/s00039-011-0135-6. |
| [3] |
D. Bump, Lie Groups, Graduate Texts in Mathematics, 225, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-4094-3. |
| [4] |
J. Dieudonné, Eléments d'Analyse, 5, Gauthier-Villars, 1975. |
| [5] |
J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, Berlin, 2000.
doi: 10.1007/978-3-642-56936-4. |
| [6] |
L. H. Eliasson,
Ergodic skew-systems on Td × SO(3.R), Ergodic Theory Dynam. Systems, 22 (2002), 1429-1449.
doi: 10.1017/S0143385702000998. |
| [7] |
B. Fayad and R. Krikorian,
Rigidity results for quasiperiodic SL(2.$\mathbb{R} $)-cocycles, J. Mod. Dyn., 3 (2009), 497-510.
doi: 10.3934/jmd.2009.3.479. |
| [8] |
H. Furstenberg,
Strict ergodicity and transformations of the torus, Amer. J. Math., 83 (1961), 573-601.
doi: 10.2307/2372899. |
| [9] |
S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962. |
| [10] |
X. Hou and J. You,
Local rigidity of reducibility of analytic quasi-periodic cocycles on U(n), Discrete Contin. Dyn. Syst., 24 (2009), 441-454.
doi: 10.3934/dcds.2009.24.441. |
| [11] |
N. Karaliolios, Invariant distributions for quasiperiodic cocycles in $\mathbb{T}^d $ × SU(2), arXiv: 1407.4763, 2014. |
| [12] |
N. Karaliolios, Continuous spectrum or measurable reducibility for quasiperiodic cocycles in $\mathbb{T}^d $ × SU(2), arXiv: 1512.00057, 2015. |
| [13] |
N. Karaliolios, Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups, Mém. Soc. Math. Fr. (N.S.), No. 146 (2016), 4+ⅱ+200 pp. |
| [14] |
A. Kolmogoroff,
On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Amer. Math. Soc. Translation, 1949 (1949), 19 pp..
|
| [15] |
R. Krikorian, Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque, (1999), ⅵ+216 pp. |
| [16] |
R. Krikorian,
Global density of reducible quasi-periodic cocycles on T1 × SU(2), Ann. of Math. (2), 154 (2001), 269-326.
doi: 10.2307/3062098. |
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