Advanced Search
Article Contents
Article Contents

Fiber bunching and cohomology for Banach cocycles over hyperbolic systems

The author is supported in part by NSF grant DMS-1301693
Abstract Full Text(HTML) Related Papers Cited by
  • We consider Hölder continuous cocycles over hyperbolic dynamical systems with values in the group of invertible bounded linear operators on a Banach space. We show that two fiber bunched cocycles are Hölder continuously cohomologous if and only if they have Hölder conjugate periodic data. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base system. We show that this condition can be established based on the periodic data of a cocycle. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and one with values in a set which is compact in strong operator topology.

    Mathematics Subject Classification: Primary: 37D20, 37C15, 37H05.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   A. Avila , J. Santamaria  and  M. Viana , Holonomy invariance: Rough regularity and applications to Lyapunov exponents, Astérisque, 358 (2013) , 13-74. 
      L. Backes , Rigidity of fiber bunched cocycles, Bul. Brazilian Math. Soc., 46 (2015) , 163-179.  doi: 10.1007/s00574-015-0089-7.
      L. Backes  and  A. Kocsard , Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems, 36 (2016) , 1703-1722.  doi: 10.1017/etds.2014.149.
      G. Grabarnik  and  M. Guysinsky , Livšic theorem for Banach rings, Discrete and Continuous Dynamical Systems, 37 (2017) , 4379-4390.  doi: 10.3934/dcds.2017187.
      B. Kalinin , Livšic theorem for matrix cocycles, Annals of Math., 173 (2011) , 1025-1042.  doi: 10.4007/annals.2011.173.2.11.
      B. Kalinin  and  V. Sadovskaya , Cocycles with one exponent over partially hyperbolic systems, Geometriae Dedicata, 167 (2013) , 167-188.  doi: 10.1007/s10711-012-9808-z.
      B. Kalinin and V. Sadovskaya, Periodic approximation of Lyapunov exponents for Banach cocycles, To appear in Ergodic Theory Dynam. Systems.
      A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54 Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.
      A. Katok and V. Nitica, Rigidity in Higher Rank Abelian Group Actions: Volume 1, Introduction and Cocycle Problem, Cambridge University Press, 2011. doi: 10.1017/CBO9780511803550.
      A. N. Livšic , Homology properties of Y-systems, Math. Zametki, 10 (1971) , 555-564. 
      A. N. Livšic , Cohomology of dynamical systems, Math. USSR Izvestija, 36 (1972) , 1296-1320. 
      R. de la Llave  and  A. Windsor , Livšic theorem for non-commutative groups including groups of diffeomorphisms, and invariant geometric structures, Ergodic Theory Dynam. Systems, 30 (2010) , 1055-1100.  doi: 10.1017/S014338570900039X.
      V. Nitica  and  A. Török , Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices, Duke Math. J., 79 (1995) , 751-810.  doi: 10.1215/S0012-7094-95-07920-4.
      V. Nitica  and  A. Török , Regularity of the transfer map for cohomologous cocycles, Ergodic Theory Dynam. Systems, 18 (1998) , 1187-1209.  doi: 10.1017/S0143385798117480.
      W. Parry , The Livšic periodic point theorem for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999) , 687-701.  doi: 10.1017/S0143385799146789.
      M. Pollicott  and  C. P. Walkden , Livšic theorems for connected Lie groups, Trans. Amer. Math. Soc., 353 (2001) , 2879-2895.  doi: 10.1090/S0002-9947-01-02708-8.
      V. Sadovskaya , Cohomology of GL(2, $\mathbb{R} $)-valued cocycles over hyperbolic systems, Discrete and Continuous Dynamical Systems, 33 (2013) , 2085-2104.  doi: 10.3934/dcds.2013.33.2085.
      V. Sadovskaya , Cohomology of fiber bunched cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems, 35 (2015) , 2669-2688.  doi: 10.1017/etds.2014.43.
      K. Schmidt , Remarks on Livšic theory for non-Abelian cocycles, Ergodic Theory Dynam. Systems, 19 (1999) , 703-721.  doi: 10.1017/S0143385799146790.
      M. Viana , Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Annals of Math., 167 (2008) , 643-680.  doi: 10.4007/annals.2008.167.643.
  • 加载中

Article Metrics

HTML views(830) PDF downloads(142) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint