Semi-uniform sub-additive ergodic theorems for discontinuous skew-product transformations
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- by Meirong Zhang, Zuohuan Zheng and Zhe Zhou PDF
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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
- José F. Alves, Vítor Araújo, and Benoît Saussol, On the uniform hyperbolicity of some nonuniformly hyperbolic systems, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1303–1309. MR 1948124, DOI 10.1090/S0002-9939-02-06857-0
- A. Avila and J. Bochi, On the subadditive ergodic theorem, Preprint, 2009.
- Yongluo Cao, Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity 16 (2003), no. 4, 1473–1479. MR 1986306, DOI 10.1088/0951-7715/16/4/316
- X. Dai, Rotation number of forced set-valued maps of unit circle, Preprint, 2011.
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Michael-R. Herman, Une méthode pour minorer les exposants de Lyapounov 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), no. 3, 453–502 (French). MR 727713, DOI 10.1007/BF02564647
- R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), no. 3, 403–438. MR 667409, DOI 10.1007/BF01208484
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- J. F. C. Kingman, Subadditive ergodic theory, Ann. Probability 1 (1973), 883–909. MR 356192, DOI 10.1214/aop/1176996798
- Hiroaki Niikuni, The rotation number for the generalized Kronig-Penney Hamiltonians, Ann. Henri Poincaré 8 (2007), no. 7, 1279–1301. MR 2360437, DOI 10.1007/s00023-007-0335-7
- John C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 47262, DOI 10.1090/S0002-9904-1952-09580-X
- J. Stark, U. Feudel, P. A. Glendinning, and A. Pikovsky, Rotation numbers for quasi-periodically forced monotone circle maps, Dyn. Syst. 17 (2002), no. 1, 1–28. MR 1888695, DOI 10.1080/14689360110073641
- R. Sturman and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity 13 (2000), no. 1, 113–143. MR 1734626, DOI 10.1088/0951-7715/13/1/306
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
- J. Yan, Lectures on Measure Theory, Science Press, Beijing, 2004 (in Chinese).
- Meirong Zhang and Zhe Zhou, Rotation numbers of linear Schrödinger equations with almost periodic potentials and phase transmissions, Ann. Henri Poincaré 11 (2010), no. 4, 765–780. MR 2677743, DOI 10.1007/s00023-010-0045-4
- Meirong Zhang and Zhe Zhou, Uniform ergodic theorems for discontinuous skew-product flows and applications to Schrödinger equations, Nonlinearity 24 (2011), no. 5, 1539–1564. MR 2785981, DOI 10.1088/0951-7715/24/5/008
- Zuohuan Zheng, Jing Xia, and Zhiming Zheng, Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications, Discrete Contin. Dyn. Syst. 14 (2006), no. 3, 409–417. MR 2171719, DOI 10.3934/dcds.2006.14.409
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
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3068972