Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Semi-uniform sub-additive ergodic theorems for discontinuous skew-product transformations


Authors: Meirong Zhang, Zuohuan Zheng and Zhe Zhou
Journal: Proc. Amer. Math. Soc. 141 (2013), 3195-3206
MSC (2010): Primary 37A20; Secondary 28A35, 58D05
DOI: https://doi.org/10.1090/S0002-9939-2013-11580-7
Published electronically: May 29, 2013
MathSciNet review: 3068972
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we will establish some semi-uniform ergodic theorems for skew-product transformations with discontinuity from the point of view of topology. The main assumptions are that the discontinuity sets of transformations are neglected in some measure-theoretical sense. The theorems have extended the classical results which have been established for continuous dynamical systems.


References [Enhancements On Off] (What's this?)

  • 1. J. Alves, V. Araujo and B. Saussol, On the uniform hyperbolicity of some nonuniformly hyperbolic systems, Proc. Amer. Math. Soc. 131 (2003), 1303-1309. MR 1948124 (2003k:37046)
  • 2. A. Avila and J. Bochi, On the subadditive ergodic theorem, Preprint, 2009.
  • 3. Y. Cao, Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity 16 (2003), 1473-1479. MR 1986306 (2005g:37061)
  • 4. X. Dai, Rotation number of forced set-valued maps of unit circle, Preprint, 2011.
  • 5. N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience Publishers, New York, 1958. MR 0117523 (22:8302)
  • 6. M. R. Herman, Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58 (1983), 453-502. MR 727713 (85g:58057)
  • 7. R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), 403-438. Erratum, Comm. Math. Phys. 90 (1983), 317-318. MR 667409 (83h:34018); MR 0714441 (85a:34032)
  • 8. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge, 1995. MR 1326374 (96c:58055)
  • 9. J. F. C. Kingman, Subadditive ergodic theory, Ann. Prob. 1 (1973), 883-909. MR 0356192 (50:8663)
  • 10. H. Niikuni, The rotation number for the generalized Kronig-Penney Hamiltonians, Ann. Henri Poincaré 8 (2007), 1279-1301. MR 2360437 (2008m:81049)
  • 11. J. C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116-136. MR 0047262 (13:850e)
  • 12. J. Stark, U. Feudel, P. A. Glendinning and A. Pikovsky, Rotation numbers for quasi-periodically forced monotone circle maps, Dyn. Syst. 17 (2002), 1-28. MR 1888695 (2003h:37058)
  • 13. R. Sturman and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity 13 (2000), 113-143. MR 1734626 (2000m:37041)
  • 14. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York/Berlin, 1982. MR 648108 (84e:28017)
  • 15. J. Yan, Lectures on Measure Theory, Science Press, Beijing, 2004 (in Chinese).
  • 16. M. Zhang and Z. Zhou, Rotation numbers of linear Schrödinger equations with almost periodic potentials and phase transmissions, Ann. Henri Poincaré 11 (2010), 765-780. MR 2677743 (2011j:47139)
  • 17. M. Zhang and Z. Zhou, Uniform ergodic theorems for discontinuous skew-product flows and applications to Schrödinger equations, Nonlinearity 24 (2011), 1539-1564. MR 2785981
  • 18. Z.-H. Zheng, J. Xia and Z. Zheng, Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications, Discrete Contin. Dyn. Syst. 14 (2006), 409-417. MR 2171719 (2006g:37041)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37A20, 28A35, 58D05

Retrieve articles in all journals with MSC (2010): 37A20, 28A35, 58D05


Additional Information

Meirong Zhang
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: mzhang@math.tsinghua.edu.cn

Zuohuan Zheng
Affiliation: Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: zhzheng@amt.ac.cn

Zhe Zhou
Affiliation: Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: zzhou@amss.ac.cn

DOI: https://doi.org/10.1090/S0002-9939-2013-11580-7
Keywords: Skew-product transformation, invariant Borel probability measure, semi-uniform convergence, ergodic theorem, sub-additive process
Received by editor(s): January 26, 2011
Received by editor(s) in revised form: November 29, 2011
Published electronically: May 29, 2013
Communicated by: Yingfei Yi
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society