October  2018, 38(10): 5163-5188. doi: 10.3934/dcds.2018228

Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms

LAGA - Université Paris 13, 99 Av. Jean-Baptiste Clément, 93430 Villetaneus, France

Received  January 2018 Published  July 2018

We consider $ \mathit{Sp}\left( 2\mathit{d},\mathbb{R} \right)$ cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the Hölder topology.

Citation: Mauricio Poletti. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5163-5188. doi: 10.3934/dcds.2018228
References:
[1]

Vitor Araújo and Maria José Pacifico, Three Dimensional Flows, volume 114. Apr 2012.

[2]

A. Avila and R. Krikorian, Monotonic cocycles, Inventiones mathematicae, 202 (2015), 271– 331, arXiv: 1310.0703v1 doi: 10.1007/s00222-014-0572-6.

[3]

A. Avila, J. Santamaria and M. Viana, Cocycles over partially hyperbolic maps, Preprint, https://institucional.impa.br/preprint/index.action, 2008.

[4]

A. AvilaJ. Santamaria and M. Viana, Holonomy invariance: Rough regularity and applications to Lyapunov exponents, Astérisque, 358 (2013), 13-74. 

[5]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189.  doi: 10.1007/s00222-010-0243-1.

[6]

A. AvilaM. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity Ⅰ: Geodesic flows, J. Eur. Math. Soc. (JEMS), 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534.

[7]

A. Avila, Density of positive Lyapunov exponents for sl(2, r)-cocycles, J. Amer. Math. Soc., 24 (2011), 999-1014.  doi: 10.1090/S0894-0347-2011-00702-9.

[8]

L. Backes, A. Brown and C. Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies, arXiv: 1507.08978v2

[9]

L. Backes and M. Poletti, Continuity of lyapunov exponents is equivalent to continuity of oseledets subspaces, Stochastics and Dynamics, 17 (2017), 1750047, 18pp. doi: 10.1142/S0219493717500472.

[10]

M. Bessa, J. Bochi, M. Cambrainha, C. Matheus, P. Varandas and D. Xu. Positivity of the top lyapunov exponent for cocycles on semisimple lie groups over hyperbolic bases, Bulletin of the Brazilian Mathematical Society, New Series, 49 (2018), 73–87, arXiv: 1611.10158 doi: 10.1007/s00574-017-0048-6.

[11]

J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. & Dynam. Sys., 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.

[12]

M. Cambrainha, Generic Symplectic Cocycles are Hyperbolic, PhD thesis, IMPA, 2013.

[13]

C. Liang, K. Marin and J. Yang, Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018, arXiv: 1604.05987 doi: 10.1016/j.anihpc.2018.01.007.

[14]

V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-210. 

[15]

V. A. Rokhlin, On the fundamental ideas of measure theory, A. M. S. Transl., 10 (1962), 1–54, Transl. from Mat. Sbornik, 25 (1949), 107–150. First published by the A. M. S. in 1952 as Translation Number 71.

[16]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math., 167 (2008), 643-680.  doi: 10.4007/annals.2008.167.643.

[17]

M. Viana and K. Oliveira, Fundamentos da Teoria Ergódica, Coleção Fronteiras da Matemática. Sociedade Brasileira de Matemática, 2014.

[18]

M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press, 2016. doi: 10.1017/CBO9781316422601.

[19]

M. Viana and J. Yang, Physical measures and absolute continuity for one-dimensional center direction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 845-877.  doi: 10.1016/j.anihpc.2012.11.002.

[20]

Y. Wang and J. You, Examples of discontinuity of lyapunov exponent in smooth quasiperiodic cocycles, Duke Math. J., 162 (2013), 2363-2412.  doi: 10.1215/00127094-2371528.

[21]

D. Xu, Density of positive lyapunov exponents for symplectic cocycles, arXiv: 1506.05403

show all references

References:
[1]

Vitor Araújo and Maria José Pacifico, Three Dimensional Flows, volume 114. Apr 2012.

[2]

A. Avila and R. Krikorian, Monotonic cocycles, Inventiones mathematicae, 202 (2015), 271– 331, arXiv: 1310.0703v1 doi: 10.1007/s00222-014-0572-6.

[3]

A. Avila, J. Santamaria and M. Viana, Cocycles over partially hyperbolic maps, Preprint, https://institucional.impa.br/preprint/index.action, 2008.

[4]

A. AvilaJ. Santamaria and M. Viana, Holonomy invariance: Rough regularity and applications to Lyapunov exponents, Astérisque, 358 (2013), 13-74. 

[5]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189.  doi: 10.1007/s00222-010-0243-1.

[6]

A. AvilaM. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity Ⅰ: Geodesic flows, J. Eur. Math. Soc. (JEMS), 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534.

[7]

A. Avila, Density of positive Lyapunov exponents for sl(2, r)-cocycles, J. Amer. Math. Soc., 24 (2011), 999-1014.  doi: 10.1090/S0894-0347-2011-00702-9.

[8]

L. Backes, A. Brown and C. Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies, arXiv: 1507.08978v2

[9]

L. Backes and M. Poletti, Continuity of lyapunov exponents is equivalent to continuity of oseledets subspaces, Stochastics and Dynamics, 17 (2017), 1750047, 18pp. doi: 10.1142/S0219493717500472.

[10]

M. Bessa, J. Bochi, M. Cambrainha, C. Matheus, P. Varandas and D. Xu. Positivity of the top lyapunov exponent for cocycles on semisimple lie groups over hyperbolic bases, Bulletin of the Brazilian Mathematical Society, New Series, 49 (2018), 73–87, arXiv: 1611.10158 doi: 10.1007/s00574-017-0048-6.

[11]

J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. & Dynam. Sys., 22 (2002), 1667-1696.  doi: 10.1017/S0143385702001165.

[12]

M. Cambrainha, Generic Symplectic Cocycles are Hyperbolic, PhD thesis, IMPA, 2013.

[13]

C. Liang, K. Marin and J. Yang, Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2018, arXiv: 1604.05987 doi: 10.1016/j.anihpc.2018.01.007.

[14]

V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-210. 

[15]

V. A. Rokhlin, On the fundamental ideas of measure theory, A. M. S. Transl., 10 (1962), 1–54, Transl. from Mat. Sbornik, 25 (1949), 107–150. First published by the A. M. S. in 1952 as Translation Number 71.

[16]

M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math., 167 (2008), 643-680.  doi: 10.4007/annals.2008.167.643.

[17]

M. Viana and K. Oliveira, Fundamentos da Teoria Ergódica, Coleção Fronteiras da Matemática. Sociedade Brasileira de Matemática, 2014.

[18]

M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press, 2016. doi: 10.1017/CBO9781316422601.

[19]

M. Viana and J. Yang, Physical measures and absolute continuity for one-dimensional center direction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 845-877.  doi: 10.1016/j.anihpc.2012.11.002.

[20]

Y. Wang and J. You, Examples of discontinuity of lyapunov exponent in smooth quasiperiodic cocycles, Duke Math. J., 162 (2013), 2363-2412.  doi: 10.1215/00127094-2371528.

[21]

D. Xu, Density of positive lyapunov exponents for symplectic cocycles, arXiv: 1506.05403

Figure 1.  definition of $z_n$ and $x'$
Figure 2.  definition $h:\mathcal{W}^c(p)\to \mathcal{W}^c(p)$ for class ${\bf{A}}$
Figure 3.  definition $h:\mathcal{W}^c(p)\to \mathcal{W}^c(p)$ for class ${\bf{B}}$
Figure 4.  perturbation of $H$
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