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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.

show all references

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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