# American Institute of Mathematical Sciences

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.

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##### 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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