Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T22:58:51.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Cohomology of fiber-bunched twisted cocycles over hyperbolic systems

Part of: Ergodic theory Dynamical systems with hyperbolic behavior Random dynamical systems

Published online by Cambridge University Press:  21 July 2020

Lucas Backes
Lucas Backes*
Affiliation:
Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, Rio Grande do Sul, Brazil ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

A twisted cocycle taking values on a Lie group G is a cocycle that is twisted by an automorphism of G in each step. In the case where G = GL(d, ℝ), we prove that if two Hölder continuous twisted cocycles satisfying the so-called fiber-bunching condition have the same periodic data then they are cohomologous.

Keywords

twisted cocyclescohomologyhyperbolic systemsperiodic points

MSC classification

Primary: 37H05: Foundations, general theory of cocycles, algebraic ergodic theory 37A20: Orbit equivalence, cocycles, ergodic equivalence relations
Secondary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 3 , August 2020 , pp. 844 - 860
Check for updates
Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avila, A., Kocsard, A. and Liu, X., Livšic theorem for diffeomorphism cocycles, Geom. Funct. Anal. 28 (2018), 943–964.10.1007/s00039-018-0454-yCrossRefGoogle Scholar
Avila, A. and Viana, M., Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math. 181 (2010), 115–189.CrossRefGoogle Scholar
Backes, L., Rigidity of fiber bunched cocycles, Bull. Braz. Math. Soc. 46 (2015), 163–179.10.1007/s00574-015-0089-7CrossRefGoogle Scholar
Backes, L., Brown, A. and Butler, C., Continuity of Lyapunov exponents for cocycles with invariant holonomies, J. Mod. Dyn. 12 (2018), 223–260.CrossRefGoogle Scholar
Backes, L. and Kocsard, A., Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems 36 (2016), 1703–1722.CrossRefGoogle Scholar
Bonatti, C., Gómez-Mont, X. and Viana, M., Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 579–624.CrossRefGoogle Scholar
Damjanović, D. and Katok, A., Local rigidity of partially hyperbolic actions I. KAM method and ℤk actions on the torus, Ann. of Math. 172 (2010), 1805–1858.CrossRefGoogle Scholar
de la Llave, R., Invariants for smooth conjugacy of hyperbolic dynamical systems. I, Comm. Math. Phys. 109 (1987), 369–378.CrossRefGoogle Scholar
Kalinin, B., Livšic theorem for matrix cocycles, Ann. of Math. 173 (2011), 1025–1042.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B., Introduction to the modern theory of dynamical systems, (Cambridge University Press, London–New York, 1995).CrossRefGoogle Scholar
Katok, A. and Niţică, V., Rigidity in higher rank abelian group actions. Volume I. Introduction and cocycle problem, Cambridge Tracts in Mathematics, Volume 185 (Cambridge University Press, Cambridge, 2011).CrossRefGoogle Scholar
Kononenko, A., Twisted cocycles and rigidity problems, Electron. Res. Announc. 1 (1995), 26–34.10.1090/S1079-6762-95-01004-3CrossRefGoogle Scholar
Livšic, A., Homology properties of Y-systems, Mat. Zametki 10 (1971), 758–763.Google Scholar
Livšic, A., Cohomology of dynamical systems, Math. USSR Izv. 6 (1972), 1278–1301.CrossRefGoogle Scholar
McDonald, R. B., Automorphisms of GL n(R), Trans. Amer. Math. Soc. 246 (1978), 155–171.Google Scholar
Mañé, R., Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Volume 8 (Springer-Verlag, Berlin, 1987). Translated from Portuguese by Silvio Levy.Google Scholar
Niţică, V. and Török, A., Regularity of the transfer map for cohomologous cocycles, Ergodic Theory Dynam. Systems 18 (1998), 1187–1209.CrossRefGoogle Scholar
Parry, W., The Livšic periodic point theorem for non-Abelian cocycles, Ergodic Theory Dynam. Systems 19 (1999), 687–701.CrossRefGoogle Scholar
Sadovskaya, V., Cohomology of fiber bunched cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems 35 (2015), 2669–2688.10.1017/etds.2014.43CrossRefGoogle Scholar
Schmidt, K., Remarks on Livšic theory for non-Abelian cocycles, Ergodic Theory Dynam. Systems 19 (1999), 703–721.CrossRefGoogle Scholar
Viana, M., Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Ann. of Math. 167 (2008), 643–680.CrossRefGoogle Scholar
Walkden, C., Solutions to the twisted cocycle equation over hyperbolic systems, Discrete Contin. Dyn. Syst. 6 (2000), 935–946.CrossRefGoogle Scholar