November  2018, 38(11): 5379-5387. doi: 10.3934/dcds.2018237

Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus

1. 

Departamento de Matemática and CMAFCIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, Piso 2, 1749-016 Lisboa, Portugal

2. 

Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rua Marquês de São Vicente 225, Rio de Janeiro, RJ, 22430-060, Brazil

Received  April 2017 Revised  October 2017 Published  August 2018

Fund Project: The first author was supported by Fundação para a Ciência e a Tecnologia, under the project: UID/MAT/04561/2013. The second author was supported by the Norwegian Research Council project no. 213638, "Discrete Models in Mathematical Analysis"

Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work "Complex one-frequency cocycles" by A. Avila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.

Citation: Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237
References:
[1]

A. Avila, Density of positive Lyapunov exponents for $\text {SL}(2, \mathbb R)$-cocycles, J. Am. Math. Soc., 24 (2011), 999-1014. doi: 10.1090/S0894-0347-2011-00702-9. Google Scholar

[2]

A. AvilaS. Jitomirskaya and C. Sadel, Complex one-frequency cocycles, J. Eur. Math. Soc. (JEMS), 16 (2014), 1915-1935. doi: 10.4171/JEMS/479. Google Scholar

[3]

J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165. Google Scholar

[4]

——, Cocycles of isometries and denseness of domination, Q. J. Math., 66 (2015), 773-798. doi: 10.1093/qmath/hav020. Google Scholar

[5]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math.(2), 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423. Google Scholar

[6]

P. Duarte and S. Klein, Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles, to appear in J. Eur. Math. Soc. (JEMS), https://arXiv.org/abs/1603.06851Google Scholar

[7]

P. Duarte and S. Klein, Lyapunov Exponents of Linear Cocycles; Continuity Via Large Deviations, Atlantis Studies in Dynamical Systems, vol. 3, Atlantis Press, 2016. doi: 10.2991/978-94-6239-124-6. Google Scholar

[8]

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. doi: 10.1002/9781118032527. Google Scholar

[9]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Google Scholar

[10]

L. I. Nicolaescu, An Invitation to Morse Theory, Universitext (Berlin. Print), Springer, 2007. Google Scholar

[11]

M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602. Google Scholar

show all references

References:
[1]

A. Avila, Density of positive Lyapunov exponents for $\text {SL}(2, \mathbb R)$-cocycles, J. Am. Math. Soc., 24 (2011), 999-1014. doi: 10.1090/S0894-0347-2011-00702-9. Google Scholar

[2]

A. AvilaS. Jitomirskaya and C. Sadel, Complex one-frequency cocycles, J. Eur. Math. Soc. (JEMS), 16 (2014), 1915-1935. doi: 10.4171/JEMS/479. Google Scholar

[3]

J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165. Google Scholar

[4]

——, Cocycles of isometries and denseness of domination, Q. J. Math., 66 (2015), 773-798. doi: 10.1093/qmath/hav020. Google Scholar

[5]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math.(2), 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423. Google Scholar

[6]

P. Duarte and S. Klein, Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles, to appear in J. Eur. Math. Soc. (JEMS), https://arXiv.org/abs/1603.06851Google Scholar

[7]

P. Duarte and S. Klein, Lyapunov Exponents of Linear Cocycles; Continuity Via Large Deviations, Atlantis Studies in Dynamical Systems, vol. 3, Atlantis Press, 2016. doi: 10.2991/978-94-6239-124-6. Google Scholar

[8]

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. doi: 10.1002/9781118032527. Google Scholar

[9]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Google Scholar

[10]

L. I. Nicolaescu, An Invitation to Morse Theory, Universitext (Berlin. Print), Springer, 2007. Google Scholar

[11]

M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602. Google Scholar

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