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

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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

Abstract
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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.

Keywords: Pesin set, Irregular set, topological entropy..

Mathematics Subject Classification: Primary: 37B40, 37C45; Secondary: 37D2.

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.
[2]	L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, University Lecture Series, (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, (2007). doi: 10.1017/CBO9781107326026.
[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.
[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.
[6]	J. Bochi, Genericity of zero Lyapunov exponents,, Ergodic Theory Dynam. Systems, 22 (2002), 1667. 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. 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. doi: 10.1017/S0143385704000872.
[10]	V. Climenhaga, Topological pressure of simultaneous level sets,, Nonlinearity, 26 (2013), 241. 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. 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. 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. doi: 10.1017/S0305004110000472.
[14]	A. Katok, Bernoulli diffeomorphisms on surfaces,, Annals of Math. (2), 110 (1979), 529. doi: 10.2307/1971237.
[15]	A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphism,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.
[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.
[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. 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. doi: 10.2140/pjm.1998.183.143.
[20]	V. Oseledec, A multiplicative ergodic theorem,, Trans. Mosc. Math. Soc., 19 (1968), 179.
[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.
[22]	Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, University of Chicago Press, (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. doi: 10.1007/BF01083692.
[24]	C. Pfister and W. Sullivan, On the topological entropy of saturated sets,, Ergodic Theory Dynam. Systems, 27 (2007), 929. doi: 10.1017/S0143385706000824.
[25]	M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds,, London Mathematical Society Lecture Note Series, (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. doi: 10.1017/S0143385711000368.
[27]	D. Ruelle, Historical behaviour in smooth dynamical systems,, in Global Analysis of Dynamical Systems, (2001), 63.
[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.
[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.
[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.
[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.
[32]	L. Young, Large deviations in dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318.
[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.

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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.
[2]	L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, University Lecture Series, (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, (2007). doi: 10.1017/CBO9781107326026.
[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.
[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.
[6]	J. Bochi, Genericity of zero Lyapunov exponents,, Ergodic Theory Dynam. Systems, 22 (2002), 1667. 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. 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. doi: 10.1017/S0143385704000872.
[10]	V. Climenhaga, Topological pressure of simultaneous level sets,, Nonlinearity, 26 (2013), 241. 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. 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. 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. doi: 10.1017/S0305004110000472.
[14]	A. Katok, Bernoulli diffeomorphisms on surfaces,, Annals of Math. (2), 110 (1979), 529. doi: 10.2307/1971237.
[15]	A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphism,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.
[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.
[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. 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. doi: 10.2140/pjm.1998.183.143.
[20]	V. Oseledec, A multiplicative ergodic theorem,, Trans. Mosc. Math. Soc., 19 (1968), 179.
[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.
[22]	Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, University of Chicago Press, (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. doi: 10.1007/BF01083692.
[24]	C. Pfister and W. Sullivan, On the topological entropy of saturated sets,, Ergodic Theory Dynam. Systems, 27 (2007), 929. doi: 10.1017/S0143385706000824.
[25]	M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds,, London Mathematical Society Lecture Note Series, (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. doi: 10.1017/S0143385711000368.
[27]	D. Ruelle, Historical behaviour in smooth dynamical systems,, in Global Analysis of Dynamical Systems, (2001), 63.
[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.
[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.
[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.
[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.
[32]	L. Young, Large deviations in dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 525. doi: 10.2307/2001318.
[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.

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