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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.bulsci.2004.12.004.  Google Scholar

[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.  doi: 10.3934/dcds.2010.27.935.  Google Scholar

[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.  doi: 10.1007/BF02773211.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Space,", Lecture Notes in Mathematics, 527 (1976).   Google Scholar

[9]

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

[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.  doi: 10.1023/A:1018643512559.  Google Scholar

[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.  doi: 10.3934/dcds.2008.21.1103.  Google Scholar

[12]

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

[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.  doi: 10.4064/aa144-3-6.  Google Scholar

[14]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar

[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.  doi: 10.1088/0951-7715/25/1/93.  Google Scholar

[16]

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

[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.  doi: 10.1016/j.jmaa.2013.03.043.  Google Scholar

[18]

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

[19]

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

[20]

J. C. Oxtoby, "Measure and Category,", Springer, (1996).   Google Scholar

[21]

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

[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.  doi: 10.1007/s002200050722.  Google Scholar

[23]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Ency. Math. and Appl., (1978).   Google Scholar

[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.  doi: 10.1017/S0143385702000913.  Google Scholar

[25]

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

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.  doi: 10.1016/j.bulsci.2004.12.004.  Google Scholar

[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.  doi: 10.3934/dcds.2010.27.935.  Google Scholar

[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.  doi: 10.1007/BF02773211.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Space,", Lecture Notes in Mathematics, 527 (1976).   Google Scholar

[9]

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

[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.  doi: 10.1023/A:1018643512559.  Google Scholar

[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.  doi: 10.3934/dcds.2008.21.1103.  Google Scholar

[12]

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

[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.  doi: 10.4064/aa144-3-6.  Google Scholar

[14]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar

[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.  doi: 10.1088/0951-7715/25/1/93.  Google Scholar

[16]

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

[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.  doi: 10.1016/j.jmaa.2013.03.043.  Google Scholar

[18]

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

[19]

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

[20]

J. C. Oxtoby, "Measure and Category,", Springer, (1996).   Google Scholar

[21]

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

[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.  doi: 10.1007/s002200050722.  Google Scholar

[23]

D. Ruelle, "Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics,", Ency. Math. and Appl., (1978).   Google Scholar

[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.  doi: 10.1017/S0143385702000913.  Google Scholar

[25]

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

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