Topological pressure for the completely irregular set of birkhoff averages

Xueting Tian

doi:10.3934/dcds.2017118

Article Contents

2017, Volume 37, Issue 5: 2745-2763. Doi: 10.3934/dcds.2017118

This issue Previous Article Diagonal stationary points of the bethe functional Next Article The existence of nontrivial solutions to Chern-Simons-Schrödinger systems

Topological pressure for the completely irregular set of birkhoff averages

Xueting Tian

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received: April 30, 2016

Revised: December 31, 2016

Published: May 2017

The author is supported by National Natural Science Foundation of China (grant no. 11671093, 11301088) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130071120026).

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Abstract

It is well-known that for certain dynamical systems (satisfying specification or its variants), the set of irregular points w.r.t. a continuous function $\phi$ (i.e. points with divergent Birkhoff ergodic averages observed by $\phi$ ) either is empty or carries full topological entropy (or pressure, see [6,17,36,37] etc. for example). In this paper we study the set of irregular points w.r.t. a collection $D$ of finite or infinite continuous functions (that is, points with divergent Birkhoff ergodic averages simultaneously observed by all $\phi∈D$ ) and obtain some generalized results. As consequences, these results are suitable for systems such as mixing shifts of finite type, uniformly hyperbolic diffeomorphisms, repellers and $β-$ shifts.

Keywords:

Topological entropy and topological pressure,
ergodic average,
specification property,
subshifts of finite type and $\beta$-shifts,
hyperbolic systems.

Mathematics Subject Classification: 37C50, 37C45, 37B10, 37D20.

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References

[1]	F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity of C1-generic diffeomorphisms, Israel Journal of Mathematics, 183 (2011), 1-60. doi: 10.1007/s11856-011-0041-5.
[2]	S. Albeverio, M. Pratsiovytyi and G. Torbin, Topological and fractal properties of subsets of real numbers which are not normal, Bull. Sci. Math., 129 (2005), 615-630.
[3]	I.-S. Baek and L. Olsen, Baire category and extremely non-normal points of invariant sets of IFS's, Discrete Contin. Dyn. Syst., 27 (2010), 935-943.
[4]	L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, vol. 272, Birkhäuser, 2008.
[5]	L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.
[6]	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.
[7]	A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk. , 38 (1983), p133. doi: 10.1070/RM1983v038n05ABEH003504.
[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]	R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590.
[10]	R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer, Lecture Notes in Math. , 470,1975. doi: 10.1007/BFb0081279.
[11]	J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754.
[12]	E. Chen, T. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208.
[13]	X. Dai and Y. Jiang, Hausdorff dimensions of zero-entropy sets of dynamical systems with positive entropy, J. Stat. Phys., 120 (2005), 511-519.
[14]	M. Dateyama, Invariant Measures for Homeomorphisms with Weak Specification, Tokyo J. of Math., 4 (1981), 389-397. doi: 10.3836/tjm/1270215164.
[15]	M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 1976. doi: 10.1007/BFb0082364.
[16]	Y. Dong, X. Tian and X. Yuan, Ergodic properties of systems with asymptotic average shadowing property, Journal of Mathematical Analysis and Applications, 432 (2015), 53-73. doi: 10.1016/j.jmaa.2015.06.046.
[17]	D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers, Advances in Mathematics, 169 (2002), 58-91.
[18]	J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz and A. Shaw, Iterated Cesaro averages, fre-quencies of digits and Baire category, Acta Arith., 144 (2010), 287-293.
[19]	M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one dimensional maps, Comm. Math. Phys., 81 (1981), 39-88. doi: 10.1007/BF01941800.
[20]	J. Li and M. Wu, The sets of divergence points of self-similar measures are residual, J. Math. Anal. Appl., 404 (2013), 429-437. doi: 10.1016/j.jmaa.2013.03.043.
[21]	J. Li and M. Wu, Generic property of irregular sets in systems satisfying the specificaiton property, Discrete and Continuous Dynamical Systems, 34 (2014), 635-645.
[22]	C. Liang, W. Sun and X. Tian, Ergodic properties of invariant measures for $C^{1+α}$ non-uniformly hyperbolic systems, Ergodic Theory & Dynam. Systems., 33 (2013), 560-584.
[23]	L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43-53. doi: 10.1017/S0305004104007601.
[24]	L. Olsen and S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages Ⅱ, Bull. Sci. Math., 6 (2007), 518-558.
[25]	Y. Pei and E. Chen, On the varational principle for the topological pressure for certain non-compact sets, Sci China Math, 53 (2010), 1117-1128.
[26]	Ya. Pesin and B. Pitskel', Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 307-318. doi: 10.1007/BF01083692.
[27]	C. Pfister and W. Sullivan, Large deviations estimates for dynamical systems without the specification property. application to the $β$-shifts, Nonlinearity, 18 (2005), 237-261.
[28]	C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824.
[29]	M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation, Comm. Math. Phys., 207 (1999), 145-171. doi: 10.1007/s002200050722.
[30]	D. Ruelle, Historic Behavior in Smooth Dynamical Systems, Global Analysis of Dynamical Systems (H. W. Broer, B. Krauskopf, and G. Vegter, eds. ), Bristol: Institute of Physics Publishing, 2001.
[31]	J. Schmeling, Symbolic dynamics for $β$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182.
[32]	K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X.
[33]	W. Sun and X. Tian, The structure on invariant measures of $C^1$ generic diffeomorphisms, Acta Mathematica Sinica, English Series, 28 (2012), 817-824. doi: 10.1007/s10114-011-9723-5.
[34]	F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity, 21 (2008), T33-T36. doi: 10.1088/0951-7715/21/3/T02.
[35]	F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certainnon-compact sets, Ergodic theory and dynamical systems, 23 (2003), 317-348. doi: 10.1017/S0143385702000913.
[36]	D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst., 25 (2009), 25-51. doi: 10.1080/14689360903156237.
[37]	D. Thompson, Irregular sets, the $β$-transformation and the almost specification property, Transactions of the American Mathematical Society, 364 (2012), 5395-5414.
[38]	M. Todd, Multifractal Analysis of Multimodal Maps, arXiv: 0809.1074v2, 2008.
[39]	P. Walters, Equilibrium states for $β$-transformations and related transformations, Math. Z., 159 (1978), 65-88.
[40]	P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 2001.