No Access

Continuity of Lyapunov exponents is equivalent to continuity of Oseledets subspaces

Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, Porto Alegre, RS, CEP 91509-900, Brazil

E-mail Address: lhbackes@impa.br

Search for more papers by this author

and

Mauricio Poletti

IMPA – Instituto Nacional de Matemática Pura e Aplicada, Estrada D. Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil

E-mail Address: mpoletti@impa.br

Search for more papers by this author

https://doi.org/10.1142/S0219493717500472Cited by:4 (Source: Crossref)

Abstract

We prove that, for semi-invertible continuous cocycles, continuity of Lyapunov exponents is equivalent to continuity, in measure, of Oseledets subspaces.

Keywords:

AMSC: 37H15, 37A20, 37D25

References

1. V. Araújo, A. I. Bufetov and S. Filip, On Hölder-continuity of Oseledets subspaces, J. Lond. Math. Soc. 93 (2016) 194–218. Crossref, Web of Science, Google Scholar
2. A. Avila, A. Eskin and M. Viana, Continuity of Lyapunov exponents of random matrix products, in preparation. Google Scholar
3. L. Backes, A note on the continuity of Oseledets subspaces for fiber-bunched cocycles, Cad. IME – Serie Mat. 9 (2015) 3–8. Google Scholar
4. L. Backes, A. Brown and C. Butler, Continuity of Lyapunov exponents for cocycles with invariant holonomies, arxiv:1507.08978. Google Scholar
5. L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents (Cambridge Univ. Press, 2007). Crossref, Google Scholar
6. M. Bessa and M. Stadlbauer, On the Lyapunov spectrum of relative transfer operators, Stoch. Dyn. 16 (2016) 1650024. Link, Web of Science, Google Scholar
7. J. Bochi, Discontinuity of the Lyapunov exponents for non-hyperbolic cocycles, Preprint www.mat.puc-rio.br/ $\sim$ jairo/. Google Scholar
8. J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. Dynam. Sys. 22 (2002) 1667–1696. Crossref, Web of Science, Google Scholar
9. J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. Math. 161 (2005) 1423–1485. Crossref, Web of Science, Google Scholar
10. C. Bocker and M. Viana, Continuity of Lyapunov exponents for random two-dimensional matrices, Ergod. Th. Dynam. Sys., to appear, doi:10.1017/etds.2015.116. Google Scholar
11. C. Butler, Discontinuity of Lyapunov exponents near fiber bunched cocycles, Ergod. Th. Dynam. Sys., to appear, doi:10.1017/etds.2016.56. Google Scholar
12. D. Dragičević and G. Froyland, Hölder continuity of Oseledets splittings for semi-invertible operator cocycles, Ergod. Th. Dynam. Sys., to appear, doi: 10.1017/etds.2016.55. Google Scholar
13. P. Duarte and S. Klein, Lyapunov Exponents of Linear Cocycles, Atlantis Series in Dynamical Systems (Springer, 2016). Crossref, Google Scholar
14. G. Froyland, S. LLoyd and A. Quas, Coherent structures and isolated spectrum for Perron–Frobenius cocycles, Ergod. Th. Dynam. Sys. 30 (2010) 729–756. Crossref, Web of Science, Google Scholar
15. E. C. Malheiro and M. Viana, Lyapunov exponents of linear cocycles over Markov shifts, Stoch. Dyn. 15 (2015) 1550020. Link, Web of Science, Google Scholar
16. G. Ochs, Stability of Oseledets spaces is equivalent to stability of Lyapunov exponents, Dynam. Stability Sys. 14 (1999) 183–201. Crossref, Web of Science, Google Scholar
17. V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968) 197–231. Web of Science, Google Scholar
18. Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Math., Vol. 1486 (Springer-Verlag, 1991), pp. 64–80. Crossref, Google Scholar
19. D. Ruelle, Analyticity properties of the characteristic exponents of random matrix products, Adv. Math. 32 (1979) 68–80. Crossref, Web of Science, Google Scholar
20. M. Viana, Lectures on Lyapunov Exponents (Cambridge Univ. Press, 2014). Crossref, Google Scholar