February  2014, 34(2): 635-645. doi: 10.3934/dcds.2014.34.635

Generic property of irregular sets in systems satisfying the specification property

1. 

School of Mathematics and statistics, Minnan Normal University, Zhangzhou, 363000, China

2. 

Department of Mathematics, South China University of Technology, Guangzhou, 510641

Received  May 2013 Revised  June 2013 Published  August 2013

Let $f$ be a continuous map on a compact metric space. In this paper, under the hypothesis that $f$ satisfies the specification property, we prove that the set consisting of those points for which the Birkhoff ergodic average does not exist is either residual or empty.
Citation: Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635
References:
[1]

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. doi: 10.1016/j.bulsci.2004.12.004.

[2]

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. doi: 10.3934/dcds.2010.27.935.

[3]

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.

[4]

L. Barreira, J. J. Li and C. Valls, Irregular sets are residual,, preprint., (). 

[5]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.

[6]

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4.

[7]

E. Chen, K. Tassilo and L. Shu, Topological entropy for divergence points, Ergod. Th. Dynam. Sys., 25 (2005), 1173-1208. doi: 10.1017/S0143385704000872.

[8]

M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Space," Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976.

[9]

A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimensions and entropy, J. London Math. Soc. (2), 64 (2001), 229-244. doi: 10.1017/S0024610701002137.

[10]

A.-H. Fan and D.-J. Feng, On the distribution of long-term time averages on symbolic space, J. Stat. Phys., 99 (2000), 813-856. doi: 10.1023/A:1018643512559.

[11]

A.-H. Fan, L. M. Liao and J. Peyrière, Generic points in systems of specification and Banach valued Birkhoff ergodic average, Discrete Contin. Dyn. Syst., 21 (2008), 1103-1128. doi: 10.3934/dcds.2008.21.1103.

[12]

D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91. doi: 10.1006/aima.2001.2054.

[13]

J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz and A. Shaw, Iterated Cesàro averages, frequencies of digits and Baire category, Acta Arith., 144 (2010), 287-293. doi: 10.4064/aa144-3-6.

[14]

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.

[15]

J. J. Li, M. Wu and Y. Xiong, Hausdorff dimensions of the divergence points of self-similar measures with the open set condition, Nonlinearity, 25 (2012), 93-105. doi: 10.1088/0951-7715/25/1/93.

[16]

J. J. Li and M. Wu, Divergence points in systems satisfying the specification property, Discrete Contin. Dyn. Syst., 33 (2013), 905-920.

[17]

J. 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.

[18]

L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43-53. doi: 10.1017/S0305004104007601.

[19]

L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. (9), 82 (2003), 1591-1649. doi: 10.1016/j.matpur.2003.09.007.

[20]

J. C. Oxtoby, "Measure and Category," Springer, New York, 1996.

[21]

Ya. Pesin and B. Pitskel', Topological pressure and variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 307-318.

[22]

M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation, Commun. Math. Phys., 207 (1999), 145-171. doi: 10.1007/s002200050722.

[23]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Ency. Math. and Appl., Vol 5, Addison-Wesley Publishing Co., Reading, Mass., 1978.

[24]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergod. Th. Dynam. Sys., 23 (2003), 317-348. doi: 10.1017/S0143385702000913.

[25]

D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dynamical Systems, 25 (2010), 25-51. doi: 10.1080/14689360903156237.

show all references

References:
[1]

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. doi: 10.1016/j.bulsci.2004.12.004.

[2]

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. doi: 10.3934/dcds.2010.27.935.

[3]

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.

[4]

L. Barreira, J. J. Li and C. Valls, Irregular sets are residual,, preprint., (). 

[5]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.

[6]

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4.

[7]

E. Chen, K. Tassilo and L. Shu, Topological entropy for divergence points, Ergod. Th. Dynam. Sys., 25 (2005), 1173-1208. doi: 10.1017/S0143385704000872.

[8]

M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Space," Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976.

[9]

A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimensions and entropy, J. London Math. Soc. (2), 64 (2001), 229-244. doi: 10.1017/S0024610701002137.

[10]

A.-H. Fan and D.-J. Feng, On the distribution of long-term time averages on symbolic space, J. Stat. Phys., 99 (2000), 813-856. doi: 10.1023/A:1018643512559.

[11]

A.-H. Fan, L. M. Liao and J. Peyrière, Generic points in systems of specification and Banach valued Birkhoff ergodic average, Discrete Contin. Dyn. Syst., 21 (2008), 1103-1128. doi: 10.3934/dcds.2008.21.1103.

[12]

D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math., 169 (2002), 58-91. doi: 10.1006/aima.2001.2054.

[13]

J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz and A. Shaw, Iterated Cesàro averages, frequencies of digits and Baire category, Acta Arith., 144 (2010), 287-293. doi: 10.4064/aa144-3-6.

[14]

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.

[15]

J. J. Li, M. Wu and Y. Xiong, Hausdorff dimensions of the divergence points of self-similar measures with the open set condition, Nonlinearity, 25 (2012), 93-105. doi: 10.1088/0951-7715/25/1/93.

[16]

J. J. Li and M. Wu, Divergence points in systems satisfying the specification property, Discrete Contin. Dyn. Syst., 33 (2013), 905-920.

[17]

J. 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.

[18]

L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43-53. doi: 10.1017/S0305004104007601.

[19]

L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. (9), 82 (2003), 1591-1649. doi: 10.1016/j.matpur.2003.09.007.

[20]

J. C. Oxtoby, "Measure and Category," Springer, New York, 1996.

[21]

Ya. Pesin and B. Pitskel', Topological pressure and variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 307-318.

[22]

M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation, Commun. Math. Phys., 207 (1999), 145-171. doi: 10.1007/s002200050722.

[23]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics," Ency. Math. and Appl., Vol 5, Addison-Wesley Publishing Co., Reading, Mass., 1978.

[24]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergod. Th. Dynam. Sys., 23 (2003), 317-348. doi: 10.1017/S0143385702000913.

[25]

D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dynamical Systems, 25 (2010), 25-51. doi: 10.1080/14689360903156237.

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