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

Previous Article
Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case
DCDS Home
This Issue
Next Article
Longtime behavior of the semilinear wave equation with gentle dissipation

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

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

Zheng Yin ^1, and Ercai Chen ^2,

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
Related Papers

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 and Continuous Dynamical Systems, 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), 1419-1443. 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.

[1]	Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118
[2]	Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[3]	Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[4]	Kendry J. Vivas, Víctor F. Sirvent. Metric entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022010
[5]	Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309
[6]	Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421
[7]	Lan Wen. On the preperiodic set. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237
[8]	François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
[9]	James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[10]	Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure and Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
[11]	Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227
[12]	Katrin Gelfert. Lower bounds for the topological entropy. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[13]	Jaume Llibre. Brief survey on the topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[14]	Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[15]	Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . doi: 10.3934/dcds.2011.31.545
[16]	Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[17]	Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
[18]	Sanjit Chatterjee, Chethan Kamath, Vikas Kumar. Private set-intersection with common set-up. Advances in Mathematics of Communications, 2018, 12 (1) : 17-47. doi: 10.3934/amc.2018002
[19]	Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165
[20]	Michel Crouzeix. The annulus as a K-spectral set. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2291-2303. doi: 10.3934/cpaa.2012.11.2291

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Zheng Yin Ercai Chen
on Google Scholar
Zheng Yin Ercai Chen

[Back to Top]