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

Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups

Nikolaos Karaliolios

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.

Keywords: Rigidity, quasi-periodic cocycles, compact Lie groups, KAM theory.
Mathematics Subject Classification: Primary: 37C55.
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.  Google Scholar

[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.  Google Scholar

[3]

D. Bump, Lie Groups, Graduate Texts in Mathematics, 225, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4094-3.  Google Scholar

[4]

J. Dieudonné, Eléments d'Analyse, 5, Gauthier-Villars, 1975. Google Scholar

[5]

J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[6]

L. H. Eliasson, Ergodic skew-systems on Td × SO(3.R), Ergodic Theory Dynam. Systems, 22 (2002), 1429-1449.  doi: 10.1017/S0143385702000998.  Google Scholar

[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.  Google Scholar

[8]

H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.  Google Scholar

[9]

S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962. Google Scholar

[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.  Google Scholar

[11]

N. Karaliolios, Invariant distributions for quasiperiodic cocycles in $\mathbb{T}^d $ × SU(2), arXiv: 1407.4763, 2014. Google Scholar

[12]

N. Karaliolios, Continuous spectrum or measurable reducibility for quasiperiodic cocycles in $\mathbb{T}^d $ × SU(2), arXiv: 1512.00057, 2015. Google Scholar

[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.  Google Scholar

[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..   Google Scholar

[15]

R. Krikorian, Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque, (1999), ⅵ+216 pp.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

D. Bump, Lie Groups, Graduate Texts in Mathematics, 225, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4094-3.  Google Scholar

[4]

J. Dieudonné, Eléments d'Analyse, 5, Gauthier-Villars, 1975. Google Scholar

[5]

J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[6]

L. H. Eliasson, Ergodic skew-systems on Td × SO(3.R), Ergodic Theory Dynam. Systems, 22 (2002), 1429-1449.  doi: 10.1017/S0143385702000998.  Google Scholar

[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.  Google Scholar

[8]

H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.  Google Scholar

[9]

S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962. Google Scholar

[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.  Google Scholar

[11]

N. Karaliolios, Invariant distributions for quasiperiodic cocycles in $\mathbb{T}^d $ × SU(2), arXiv: 1407.4763, 2014. Google Scholar

[12]

N. Karaliolios, Continuous spectrum or measurable reducibility for quasiperiodic cocycles in $\mathbb{T}^d $ × SU(2), arXiv: 1512.00057, 2015. Google Scholar

[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.  Google Scholar

[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..   Google Scholar

[15]

R. Krikorian, Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque, (1999), ⅵ+216 pp.  Google Scholar

[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.  Google Scholar

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