2017, 11: 125-142. doi: 10.3934/jmd.2017006

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

Received  November 17, 2014 Revised  June 19, 2016 Published  January 2017

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.

Citation: Nikolaos Karaliolios. Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups. Journal of Modern Dynamics, 2017, 11: 125-142. doi: 10.3934/jmd.2017006
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. AvilaB. 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.

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. AvilaB. 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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