Advertisement

Boundedness and invariant metrics for diffeomorphism cocycles over hyperbolic systems

  • Victoria SadovskayaEmail author
Original Paper
  • 5 Downloads

Abstract

Let \(\mathcal {A}\) be a Hölder continuous cocycle over a hyperbolic dynamical system with values in the group of diffeomorphisms of a compact manifold \({\mathcal {M}}\). We consider the periodic data of \(\mathcal {A}\), i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of \(\mathcal {A}\) is bounded in \(\text{ Diff }^{\,q}({\mathcal {M}})\), \(q>1\), then the set of values of the cocycle is bounded in \(\text{ Diff }^{\,r}({\mathcal {M}})\) for each \(r<q\). Moreover, such a cocycle is isometric with respect to a Hölder continuous family of Riemannian metrics on \({\mathcal {M}}\).

Keywords

Cocycle Diffeomorphism group Periodic orbit Hyperbolic system Symbolic system 

Mathematics Subject Classification

37D20 54H15 

Notes

References

  1. 1.
    Avila, A., Kocsard, A., Liu, X.: Livšic theorem for diffeomorphism cocycles. Geom. Funct. Anal. 28, 943–964 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Backes, L., Kocsard, A.: Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems. Ergodic Theory Dyn. Syst. 36, 1703–1722 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    de la Llave, R., Obaya, R.: Regularity of the composition operator in spaces of Hölder functions. Discret. Contin. Dyn. Syst. 5(1), 157–184 (1999)zbMATHGoogle Scholar
  4. 4.
    de la Llave, R., Windsor, A.: Livšic theorem for non-commutative groups including groups of diffeomorphisms, and invariant geometric structures. Ergodic Theory Dyn. Syst. 30(4), 1055–1100 (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Guysinsky, M.: Livšic theorem for cocycles with values in the group of diffeomorphisms. PreprintGoogle Scholar
  6. 6.
    Hurtado, S.: Examples of diffeomorphism group cocycles with no periodic approximation. To appear in Proceedings of the AMS. arXiv:1705.06361
  7. 7.
    Kalinin, B.: Livšic theorem for matrix cocycles. Ann. Math. 173(2), 1025–1042 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kocsard, A., Potrie, R.: Livšic theorem for low-dimensional diffeomorphism cocycles. Comment. Math. Helv. 91, 39–64 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kalinin, B., Sadovskaya, V.: Linear cocycles over hyperbolic systems and criteria of conformality. J. Modern Dyn. 4(3), 419–441 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kalinin, B., Sadovskaya, V.: Cocycles with one exponent over partially hyperbolic systems. Geom. Dedicata 167(1), 167–188 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  12. 12.
    Katok, A., Nitica, V.: Rigidity in Higher Rank Abelian Group Actions: Introduction and Cocycle Problem, vol. 1. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Lang, S.: Fundamentals of Differential Geometry. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  14. 14.
    Livšic, A.N.: Homology properties of Y-systems. Math. Zametki 10, 758–763 (1971)Google Scholar
  15. 15.
    Livšic, A.N.: Cohomology of dynamical systems. Math. USSR Izvestija 6, 1278–1301 (1972)CrossRefGoogle Scholar
  16. 16.
    Nitica, V., Török, A.: Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices. Duke Math. J. 79(3), 751–810 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nitica, V., Török, A.: Regularity results for the solutions of the Livsic cohomology equation with values in diffeomorphism groups. Ergodic Theory Dyn. Syst. 16(2), 325–333 (1996)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Nitica, V., Török, A.: Regularity of the transfer map for cohomologous cocycles. Ergodic Theory Dyn. Syst. 18(5), 1187–1209 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sadovskaya, V.: On uniformly quasiconformal Anosov systems. Math. Res. Lett. 12(3), 425–441 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sadovskaya, V.: Cohomology of fiber bunched cocycles over hyperbolic systems. Ergodic Theory Dyn. Syst. 35(8), 2669–2688 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pollicott, M., Walkden, C.P.: Livšic theorems for connected Lie groups. Trans. Am. Math. Soc. 353(7), 2879–2895 (2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    Schmidt, K.: Remarks on Livšic theory for non-Abelian cocycles. Ergodic Theory Dyn. Syst. 19(3), 703–721 (1999)CrossRefzbMATHGoogle Scholar
  23. 23.
    Schreiber, S.J.: On growth rates of subadditive functions for semi-flows. J. Differ. Equ. 148, 334–350 (1998)CrossRefzbMATHGoogle Scholar
  24. 24.
    Taylor, M.: Existence and regularity of isometries. Trans. Am. Math. Soc. 358(6), 2415–2423 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

Personalised recommendations