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

Zheng  Yin; Ercai  Chen

doi:10.3934/dcds.2016085

Article Contents

2016, Volume 36, Issue 11: 6581-6597. Doi: 10.3934/dcds.2016085

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

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
2.
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu

Received: November 2015

Revised: June 2016

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 37B40, 37C45; Secondary: 37D25.

Citation:

Related Papers

Cited by

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, arXiv:1110.6091.
[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.