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February  2013, 33(2): 905-920. doi: 10.3934/dcds.2013.33.905

## Divergence points in systems satisfying the specification property

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

Received  July 2011 Revised  February 2012 Published  September 2012

Let $f$ be a continuous transformation of a compact metric space $(X,d)$ and $\varphi$ any continuous function on $X$. In this paper, under the hypothesis that $f$ satisfies the specification property, we determine the topological entropy of the following sets: $$K_{I}=\Big\{x\in X: A\big(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^{i}(x))\big)=I\Big\},$$ where $I$ is a closed subinterval of $\mathbb{R}$ and $A(a_{n})$ denotes the set of accumulation points of the sequence $\{a_{n}\}_{n}$. Our result generalizes the classical result of Takens and Verbitskiy ( Ergod. Th. Dynam. Sys., 23 (2003), 317-348 ). As an application, we present another concise proof of the fact that the irregular set has full topological entropy if $f$ satisfies the specification property.
Citation: Jinjun Li, Min Wu. Divergence points in systems satisfying the specification property. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 905-920. doi: 10.3934/dcds.2013.33.905
##### References:
 [1] 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. [2] L. Barreira, "Dimension and Recurrence in Hyperbolic Dynamical," Progress in Mathematics 272, Birkhäuser, 2008. [3] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [4] R. Bowen, Periodic points and measures for axiom-A-diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. doi: 10.1090/S0002-9947-1971-0282372-0. [5] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4. [6] Chen Ercai, Tassilo Küpper and Shu Lin, Topological entropy for divergence points, Ergod. Th. Dynam. Sys., 25 (2005), 1173-1208. doi: 10.1017/S0143385704000872. [7] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Space," Of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 527, 1976, iv+360 pp. [8] A. H. Fan, D. J. Feng and J. Wu, Recurrence, dimensions and entropies, J. London Math. Soc., 64 (2001), 229-244. doi: 10.1017/S0024610701002137. [9] 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. [10] 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. [11] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, 1995. [12] A. Katok, Fifty years of entropy in dynamics: 1958-2007, J. Mod. Dyn.,1 (2007), 545-596. [13] K. S. Lau and L. Shu, The spectrum of Poincare recurrence, Ergod. Th. Dynam. Sys., 28 (2008), 1917-1943. doi: 10.1017/S0143385707001095. [14] 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. [15] E. Olivier, Analyse multifractale de fonctions continues, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1171-1174. doi: 10.1016/S0764-4442(98)80221-8. [16] L. Olsen and S. Winter, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. London Math. Soc., 67 (2003), 103-122. doi: 10.1112/S0024610702003630. [17] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649. [18] Y. B. Pesin, "Dimension Theory in Dynamical System: Contemporary Views and Applications," University of Chicago Press, Chicago, 1997. [19] C. E. Pfister and W. G. Sullivan, On the topological entropy of saturated set, Ergod. Th. Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824. [20] 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. [21] D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dynamical Systems : An International Journal, 25(1) (2010), 25-51. [22] P. Water, "An Introduction to Ergodic Theory," Springer-Verlage, New York-Berlin, 1982.

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##### References:
 [1] 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. [2] L. Barreira, "Dimension and Recurrence in Hyperbolic Dynamical," Progress in Mathematics 272, Birkhäuser, 2008. [3] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [4] R. Bowen, Periodic points and measures for axiom-A-diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. doi: 10.1090/S0002-9947-1971-0282372-0. [5] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4. [6] Chen Ercai, Tassilo Küpper and Shu Lin, Topological entropy for divergence points, Ergod. Th. Dynam. Sys., 25 (2005), 1173-1208. doi: 10.1017/S0143385704000872. [7] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Space," Of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 527, 1976, iv+360 pp. [8] A. H. Fan, D. J. Feng and J. Wu, Recurrence, dimensions and entropies, J. London Math. Soc., 64 (2001), 229-244. doi: 10.1017/S0024610701002137. [9] 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. [10] 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. [11] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, 1995. [12] A. Katok, Fifty years of entropy in dynamics: 1958-2007, J. Mod. Dyn.,1 (2007), 545-596. [13] K. S. Lau and L. Shu, The spectrum of Poincare recurrence, Ergod. Th. Dynam. Sys., 28 (2008), 1917-1943. doi: 10.1017/S0143385707001095. [14] 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. [15] E. Olivier, Analyse multifractale de fonctions continues, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 1171-1174. doi: 10.1016/S0764-4442(98)80221-8. [16] L. Olsen and S. Winter, Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. London Math. Soc., 67 (2003), 103-122. doi: 10.1112/S0024610702003630. [17] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649. [18] Y. B. Pesin, "Dimension Theory in Dynamical System: Contemporary Views and Applications," University of Chicago Press, Chicago, 1997. [19] C. E. Pfister and W. G. Sullivan, On the topological entropy of saturated set, Ergod. Th. Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824. [20] 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. [21] D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dynamical Systems : An International Journal, 25(1) (2010), 25-51. [22] P. Water, "An Introduction to Ergodic Theory," Springer-Verlage, New York-Berlin, 1982.
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