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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 |
References:
[1] |
J. Barral and M. Mensi, Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum, Ergodic Theory Dynam. Systems, 27 (2007), 1419-1443.
doi: 10.1017/S0143385706001027. |
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.
doi: 10.1007/BF02773211. |
[6] |
J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
[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. |
[8] |
R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.
doi: 10.1090/S0002-9947-1973-0338317-X. |
[9] |
E. Chen, T. Kupper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208.
doi: 10.1017/S0143385704000872. |
[10] |
V. Climenhaga, Topological pressure of simultaneous level sets, Nonlinearity, 26 (2013), 241-268.
doi: 10.1088/0951-7715/26/1/241. |
[11] |
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. |
[12] |
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. |
[13] |
T. Jordan and M. Rams, Multifractal analysis for Bedford-McMullen carpets, Math. Proc. Camb. Phil. Soc., 150 (2011), 147-156.
doi: 10.1017/S0305004110000472. |
[14] |
A. Katok, Bernoulli diffeomorphisms on surfaces, Annals of Math. (2), 110 (1979), 529-547.
doi: 10.2307/1971237. |
[15] |
A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphism, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. |
[16] |
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. |
[17] |
C. Liang, G. Liao, W. Sun and X. Tian, Saturated sets for nonuniformly hyperbolic systems,, preprint, ().
|
[18] |
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. |
[19] |
L. Olsen, Self-affine multifractal Sierpinski sponges in $\mathbbR^d$, Pacific J. Math., 183 (1998), 143-199.
doi: 10.2140/pjm.1998.183.143. |
[20] |
V. Oseledec, A multiplicative ergodic theorem, Trans. Mosc. Math. Soc., 19 (1968), 179-210. |
[21] |
Y. Pei and E. Chen, On the variational principle for the topological pressure for certain non-compact sets, Sci. China Math., 53 (2010), 1117-1128.
doi: 10.1007/s11425-009-0109-4. |
[22] |
Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.
doi: 10.7208/chicago/9780226662237.001.0001. |
[23] |
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. |
[24] |
C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956.
doi: 10.1017/S0143385706000824. |
[25] |
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. |
[26] |
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. |
[27] |
D. Ruelle, Historical behaviour in smooth dynamical systems, in Global Analysis of Dynamical Systems, Inst. Phys., Bristol, (2001), 63-66. |
[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), 317-348.
doi: 10.1017/S0143385702000913. |
[29] |
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. |
[30] |
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. |
[31] |
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. |
[32] |
L. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.
doi: 10.2307/2001318. |
[33] |
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. |
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), 1419-1443.
doi: 10.1017/S0143385706001027. |
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.
doi: 10.1007/BF02773211. |
[6] |
J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696.
doi: 10.1017/S0143385702001165. |
[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. |
[8] |
R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.
doi: 10.1090/S0002-9947-1973-0338317-X. |
[9] |
E. Chen, T. Kupper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208.
doi: 10.1017/S0143385704000872. |
[10] |
V. Climenhaga, Topological pressure of simultaneous level sets, Nonlinearity, 26 (2013), 241-268.
doi: 10.1088/0951-7715/26/1/241. |
[11] |
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. |
[12] |
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. |
[13] |
T. Jordan and M. Rams, Multifractal analysis for Bedford-McMullen carpets, Math. Proc. Camb. Phil. Soc., 150 (2011), 147-156.
doi: 10.1017/S0305004110000472. |
[14] |
A. Katok, Bernoulli diffeomorphisms on surfaces, Annals of Math. (2), 110 (1979), 529-547.
doi: 10.2307/1971237. |
[15] |
A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphism, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. |
[16] |
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. |
[17] |
C. Liang, G. Liao, W. Sun and X. Tian, Saturated sets for nonuniformly hyperbolic systems,, preprint, ().
|
[18] |
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. |
[19] |
L. Olsen, Self-affine multifractal Sierpinski sponges in $\mathbbR^d$, Pacific J. Math., 183 (1998), 143-199.
doi: 10.2140/pjm.1998.183.143. |
[20] |
V. Oseledec, A multiplicative ergodic theorem, Trans. Mosc. Math. Soc., 19 (1968), 179-210. |
[21] |
Y. Pei and E. Chen, On the variational principle for the topological pressure for certain non-compact sets, Sci. China Math., 53 (2010), 1117-1128.
doi: 10.1007/s11425-009-0109-4. |
[22] |
Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.
doi: 10.7208/chicago/9780226662237.001.0001. |
[23] |
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. |
[24] |
C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956.
doi: 10.1017/S0143385706000824. |
[25] |
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. |
[26] |
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. |
[27] |
D. Ruelle, Historical behaviour in smooth dynamical systems, in Global Analysis of Dynamical Systems, Inst. Phys., Bristol, (2001), 63-66. |
[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), 317-348.
doi: 10.1017/S0143385702000913. |
[29] |
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. |
[30] |
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. |
[31] |
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. |
[32] |
L. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.
doi: 10.2307/2001318. |
[33] |
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. |
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