Advanced Search
Article Contents
Article Contents

Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems

Abstract Related Papers Cited by
  • This article is devoted to the study of the irregular sets of Birkhoff averages in some nonuniformly hyperbolic systems via Pesin theory. Particularly, we give a conditional variational principle for the topological entropy of the irregular sets. Our result can be applied (i) to the diffeomorphisms on surfaces, (ii) to the nonuniformly hyperbolic diffeomorphisms described by Katok.
    Mathematics Subject Classification: Primary: 37B40, 37C45; Secondary: 37D25.


    \begin{equation} \\ \end{equation}
  • [1]

    J. Barral and M. Mensi, Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum, Ergodic Theory Dynam. Systems, 27 (2007), 1419-1443.doi: 10.1017/S0143385706001027.


    L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series, 23, American Mathematical Society, Providence, 2002.doi: 10.1090/ulect/023.


    L. Barreira and Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007.doi: 10.1017/CBO9781107326026.


    L. Barreira and B. Saussol, Variational principles and mixed multifractal spectra, Trans. Amer. Math. Soc, 353 (2001), 3919-3944.doi: 10.1090/S0002-9947-01-02844-6.


    L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.doi: 10.1007/BF02773211.


    J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.doi: 10.1017/S0143385702001165.


    T. Bomfim and P. Varandas, Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets, Ergodic Theory Dynam. Systems. doi: 10.1017/etds.2015.46.


    R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.doi: 10.1090/S0002-9947-1973-0338317-X.


    E. Chen, T. Kupper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208.doi: 10.1017/S0143385704000872.


    V. Climenhaga, Topological pressure of simultaneous level sets, Nonlinearity, 26 (2013), 241-268.doi: 10.1088/0951-7715/26/1/241.


    D. Feng, K. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91.doi: 10.1006/aima.2001.2054.


    M. Hirayama, Periodic probability measures are dense in the set of invariant measures, Discrete Continuous Dynam. Systems - A, 9 (2003), 1185-1192.doi: 10.3934/dcds.2003.9.1185.


    T. Jordan and M. Rams, Multifractal analysis for Bedford-McMullen carpets, Math. Proc. Camb. Phil. Soc., 150 (2011), 147-156.doi: 10.1017/S0305004110000472.


    A. Katok, Bernoulli diffeomorphisms on surfaces, Annals of Math. (2), 110 (1979), 529-547.doi: 10.2307/1971237.


    A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphism, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.


    A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511809187.


    C. Liang, G. Liao, W. Sun and X. Tian, Saturated sets for nonuniformly hyperbolic systems, preprint, arXiv:1110.6091.


    L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649.doi: 10.1016/j.matpur.2003.09.007.


    L. Olsen, Self-affine multifractal Sierpinski sponges in $\mathbbR^d$, Pacific J. Math., 183 (1998), 143-199.doi: 10.2140/pjm.1998.183.143.


    V. Oseledec, A multiplicative ergodic theorem, Trans. Mosc. Math. Soc., 19 (1968), 179-210.


    Y. Pei and E. Chen, On the variational principle for the topological pressure for certain non-compact sets, Sci. China Math., 53 (2010), 1117-1128.doi: 10.1007/s11425-009-0109-4.


    Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.doi: 10.7208/chicago/9780226662237.001.0001.


    Y. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 307-318.doi: 10.1007/BF01083692.


    C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956.doi: 10.1017/S0143385706000824.


    M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, London Mathematical Society Lecture Note Series, 180, Cambridge University Press, Cambridge, 1993.doi: 10.1017/CBO9780511752537.


    H. Reeve, The packing spectrum for Birkhoff averages on a self-affine repeller, Ergodic Theory Dynam. Systems, 32 (2012), 1444-1470.doi: 10.1017/S0143385711000368.


    D. Ruelle, Historical behaviour in smooth dynamical systems, in Global Analysis of Dynamical Systems, Inst. Phys., Bristol, (2001), 63-66.


    F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems, 23 (2003), 317-348.doi: 10.1017/S0143385702000913.


    D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dynamical Systems: An International Journal, 25 (2010), 25-51.doi: 10.1080/14689360903156237.


    D. Thompson, Irregular sets, the $\beta$-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414.doi: 10.1090/S0002-9947-2012-05540-1.


    P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures, J. Stat. Phys., 146 (2012), 330-358.doi: 10.1007/s10955-011-0392-7.


    L. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.doi: 10.2307/2001318.


    X. Zhou and E. Chen, Multifractal analysis for the historic set in topological dynamical systems, Nonlinearity, 26 (2013), 1975-1997.doi: 10.1088/0951-7715/26/7/1975.

  • 加载中

Article Metrics

HTML views() PDF downloads(153) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint