# American Institute of Mathematical Sciences

November  2016, 36(11): 6581-6597. doi: 10.3934/dcds.2016085

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

 1 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, Jiangsu, China 2 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu, China

Received  November 2015 Revised  June 2016 Published  August 2016

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.
Citation: Zheng Yin, Ercai Chen. Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6581-6597. doi: 10.3934/dcds.2016085
##### References:
 [1] J. Barral and M. Mensi, Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum,, Ergodic Theory Dynam. Systems, 27 (2007), 1419. doi: 10.1017/S0143385706001027. Google Scholar [2] L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, University Lecture Series, (2002). doi: 10.1090/ulect/023. Google Scholar [3] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents,, Encyclopedia of Mathematics and its Applications, (2007). doi: 10.1017/CBO9781107326026. Google Scholar [4] L. Barreira and B. Saussol, Variational principles and mixed multifractal spectra,, Trans. Amer. Math. Soc, 353 (2001), 3919. doi: 10.1090/S0002-9947-01-02844-6. Google Scholar [5] L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29. doi: 10.1007/BF02773211. Google Scholar [6] J. Bochi, Genericity of zero Lyapunov exponents,, Ergodic Theory Dynam. Systems, 22 (2002), 1667. doi: 10.1017/S0143385702001165. Google Scholar [7] 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. Google Scholar [8] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar [9] E. Chen, T. Kupper and L. Shu, Topological entropy for divergence points,, Ergodic Theory Dynam. Systems, 25 (2005), 1173. doi: 10.1017/S0143385704000872. Google Scholar [10] V. Climenhaga, Topological pressure of simultaneous level sets,, Nonlinearity, 26 (2013), 241. doi: 10.1088/0951-7715/26/1/241. Google Scholar [11] D. Feng, K. Lau and J. Wu, Ergodic limits on the conformal repellers,, Adv. Math., 169 (2002), 58. doi: 10.1006/aima.2001.2054. Google Scholar [12] M. Hirayama, Periodic probability measures are dense in the set of invariant measures,, Discrete Continuous Dynam. Systems - A, 9 (2003), 1185. doi: 10.3934/dcds.2003.9.1185. Google Scholar [13] T. Jordan and M. Rams, Multifractal analysis for Bedford-McMullen carpets,, Math. Proc. Camb. Phil. Soc., 150 (2011), 147. doi: 10.1017/S0305004110000472. Google Scholar [14] A. Katok, Bernoulli diffeomorphisms on surfaces,, Annals of Math. (2), 110 (1979), 529. doi: 10.2307/1971237. Google Scholar [15] A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphism,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [16] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). doi: 10.1017/CBO9780511809187. Google Scholar [17] C. Liang, G. Liao, W. Sun and X. Tian, Saturated sets for nonuniformly hyperbolic systems,, preprint, (). Google Scholar [18] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages,, J. Math. Pures Appl., 82 (2003), 1591. doi: 10.1016/j.matpur.2003.09.007. Google Scholar [19] L. Olsen, Self-affine multifractal Sierpinski sponges in $\mathbbR^d$,, Pacific J. Math., 183 (1998), 143. doi: 10.2140/pjm.1998.183.143. Google Scholar [20] V. Oseledec, A multiplicative ergodic theorem,, Trans. Mosc. Math. Soc., 19 (1968), 179. Google Scholar [21] Y. Pei and E. Chen, On the variational principle for the topological pressure for certain non-compact sets,, Sci. China Math., 53 (2010), 1117. doi: 10.1007/s11425-009-0109-4. Google Scholar [22] Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, University of Chicago Press, (1997). doi: 10.7208/chicago/9780226662237.001.0001. Google Scholar [23] Y. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets,, Functional Anal. Appl., 18 (1984), 307. doi: 10.1007/BF01083692. Google Scholar [24] C. Pfister and W. Sullivan, On the topological entropy of saturated sets,, Ergodic Theory Dynam. Systems, 27 (2007), 929. doi: 10.1017/S0143385706000824. Google Scholar [25] M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds,, London Mathematical Society Lecture Note Series, (1993). doi: 10.1017/CBO9780511752537. Google Scholar [26] H. Reeve, The packing spectrum for Birkhoff averages on a self-affine repeller,, Ergodic Theory Dynam. Systems, 32 (2012), 1444. doi: 10.1017/S0143385711000368. Google Scholar [27] D. Ruelle, Historical behaviour in smooth dynamical systems,, in Global Analysis of Dynamical Systems, (2001), 63. Google Scholar [28] 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. doi: 10.1017/S0143385702000913. Google Scholar [29] D. Thompson, The irregular set for maps with the specification property has full topological pressure,, Dynamical Systems: An International Journal, 25 (2010), 25. doi: 10.1080/14689360903156237. Google Scholar [30] D. Thompson, Irregular sets, the $\beta$-transformation and the almost specification property,, Trans. Amer. Math. Soc., 364 (2012), 5395. doi: 10.1090/S0002-9947-2012-05540-1. Google Scholar [31] P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures,, J. Stat. Phys., 146 (2012), 330. doi: 10.1007/s10955-011-0392-7. Google Scholar [32] L. Young, Large deviations in dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318. Google Scholar [33] X. Zhou and E. Chen, Multifractal analysis for the historic set in topological dynamical systems,, Nonlinearity, 26 (2013), 1975. doi: 10.1088/0951-7715/26/7/1975. Google Scholar

show all references

##### References:
 [1] J. Barral and M. Mensi, Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum,, Ergodic Theory Dynam. Systems, 27 (2007), 1419. doi: 10.1017/S0143385706001027. Google Scholar [2] L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, University Lecture Series, (2002). doi: 10.1090/ulect/023. Google Scholar [3] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents,, Encyclopedia of Mathematics and its Applications, (2007). doi: 10.1017/CBO9781107326026. Google Scholar [4] L. Barreira and B. Saussol, Variational principles and mixed multifractal spectra,, Trans. Amer. Math. Soc, 353 (2001), 3919. doi: 10.1090/S0002-9947-01-02844-6. Google Scholar [5] L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29. doi: 10.1007/BF02773211. Google Scholar [6] J. Bochi, Genericity of zero Lyapunov exponents,, Ergodic Theory Dynam. Systems, 22 (2002), 1667. doi: 10.1017/S0143385702001165. Google Scholar [7] 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. Google Scholar [8] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar [9] E. Chen, T. Kupper and L. Shu, Topological entropy for divergence points,, Ergodic Theory Dynam. Systems, 25 (2005), 1173. doi: 10.1017/S0143385704000872. Google Scholar [10] V. Climenhaga, Topological pressure of simultaneous level sets,, Nonlinearity, 26 (2013), 241. doi: 10.1088/0951-7715/26/1/241. Google Scholar [11] D. Feng, K. Lau and J. Wu, Ergodic limits on the conformal repellers,, Adv. Math., 169 (2002), 58. doi: 10.1006/aima.2001.2054. Google Scholar [12] M. Hirayama, Periodic probability measures are dense in the set of invariant measures,, Discrete Continuous Dynam. Systems - A, 9 (2003), 1185. doi: 10.3934/dcds.2003.9.1185. Google Scholar [13] T. Jordan and M. Rams, Multifractal analysis for Bedford-McMullen carpets,, Math. Proc. Camb. Phil. Soc., 150 (2011), 147. doi: 10.1017/S0305004110000472. Google Scholar [14] A. Katok, Bernoulli diffeomorphisms on surfaces,, Annals of Math. (2), 110 (1979), 529. doi: 10.2307/1971237. Google Scholar [15] A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphism,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. Google Scholar [16] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). doi: 10.1017/CBO9780511809187. Google Scholar [17] C. Liang, G. Liao, W. Sun and X. Tian, Saturated sets for nonuniformly hyperbolic systems,, preprint, (). Google Scholar [18] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages,, J. Math. Pures Appl., 82 (2003), 1591. doi: 10.1016/j.matpur.2003.09.007. Google Scholar [19] L. Olsen, Self-affine multifractal Sierpinski sponges in $\mathbbR^d$,, Pacific J. Math., 183 (1998), 143. doi: 10.2140/pjm.1998.183.143. Google Scholar [20] V. Oseledec, A multiplicative ergodic theorem,, Trans. Mosc. Math. Soc., 19 (1968), 179. Google Scholar [21] Y. Pei and E. Chen, On the variational principle for the topological pressure for certain non-compact sets,, Sci. China Math., 53 (2010), 1117. doi: 10.1007/s11425-009-0109-4. Google Scholar [22] Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, University of Chicago Press, (1997). doi: 10.7208/chicago/9780226662237.001.0001. Google Scholar [23] Y. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets,, Functional Anal. Appl., 18 (1984), 307. doi: 10.1007/BF01083692. Google Scholar [24] C. Pfister and W. Sullivan, On the topological entropy of saturated sets,, Ergodic Theory Dynam. Systems, 27 (2007), 929. doi: 10.1017/S0143385706000824. Google Scholar [25] M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds,, London Mathematical Society Lecture Note Series, (1993). doi: 10.1017/CBO9780511752537. Google Scholar [26] H. Reeve, The packing spectrum for Birkhoff averages on a self-affine repeller,, Ergodic Theory Dynam. Systems, 32 (2012), 1444. doi: 10.1017/S0143385711000368. Google Scholar [27] D. Ruelle, Historical behaviour in smooth dynamical systems,, in Global Analysis of Dynamical Systems, (2001), 63. Google Scholar [28] 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. doi: 10.1017/S0143385702000913. Google Scholar [29] D. Thompson, The irregular set for maps with the specification property has full topological pressure,, Dynamical Systems: An International Journal, 25 (2010), 25. doi: 10.1080/14689360903156237. Google Scholar [30] D. Thompson, Irregular sets, the $\beta$-transformation and the almost specification property,, Trans. Amer. Math. Soc., 364 (2012), 5395. doi: 10.1090/S0002-9947-2012-05540-1. Google Scholar [31] P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures,, J. Stat. Phys., 146 (2012), 330. doi: 10.1007/s10955-011-0392-7. Google Scholar [32] L. Young, Large deviations in dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318. Google Scholar [33] X. Zhou and E. Chen, Multifractal analysis for the historic set in topological dynamical systems,, Nonlinearity, 26 (2013), 1975. doi: 10.1088/0951-7715/26/7/1975. Google Scholar
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