May  2017, 37(5): 2745-2763. doi: 10.3934/dcds.2017118

Topological pressure for the completely irregular set of birkhoff averages

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  May 2016 Revised  January 2017 Published  February 2017

Fund Project: The author is supported by National Natural Science Foundation of China (grant no. 11671093, 11301088) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130071120026)

It is well-known that for certain dynamical systems (satisfying specification or its variants), the set of irregular points w.r.t. a continuous function $\phi$ (i.e. points with divergent Birkhoff ergodic averages observed by $\phi$) either is empty or carries full topological entropy (or pressure, see [6,17,36,37] etc. for example). In this paper we study the set of irregular points w.r.t. a collection $D$ of finite or infinite continuous functions (that is, points with divergent Birkhoff ergodic averages simultaneously observed by all $\phi∈D$) and obtain some generalized results. As consequences, these results are suitable for systems such as mixing shifts of finite type, uniformly hyperbolic diffeomorphisms, repellers and $β-$shifts.

Citation: Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118
References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Nonuniform hyperbolicity of C1-generic diffeomorphisms, Israel Journal of Mathematics, 183 (2011), 1-60. doi: 10.1007/s11856-011-0041-5. Google Scholar

[2]

S. AlbeverioM. Pratsiovytyi and G. Torbin, Topological and fractal properties of subsets of real numbers which are not normal, Bull. Sci. Math., 129 (2005), 615-630. Google Scholar

[3]

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. Google Scholar

[4]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, vol. 272, Birkhäuser, 2008.Google Scholar

[5]

L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026. Google Scholar

[6]

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. Google Scholar

[7]

A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk. , 38 (1983), p133. doi: 10.1070/RM1983v038n05ABEH003504. Google Scholar

[8]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar

[9]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590. Google Scholar

[10]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer, Lecture Notes in Math. , 470,1975. doi: 10.1007/BFb0081279. Google Scholar

[11]

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. Google Scholar

[12]

E. ChenT. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208. Google Scholar

[13]

X. Dai and Y. Jiang, Hausdorff dimensions of zero-entropy sets of dynamical systems with positive entropy, J. Stat. Phys., 120 (2005), 511-519. Google Scholar

[14]

M. Dateyama, Invariant Measures for Homeomorphisms with Weak Specification, Tokyo J. of Math., 4 (1981), 389-397. doi: 10.3836/tjm/1270215164. Google Scholar

[15]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 1976. doi: 10.1007/BFb0082364. Google Scholar

[16]

Y. DongX. Tian and X. Yuan, Ergodic properties of systems with asymptotic average shadowing property, Journal of Mathematical Analysis and Applications, 432 (2015), 53-73. doi: 10.1016/j.jmaa.2015.06.046. Google Scholar

[17]

D. J. FengK. S. Lau and J. Wu, Ergodic limits on the conformal repellers, Advances in Mathematics, 169 (2002), 58-91. Google Scholar

[18]

J. HydeV. LaschosL. OlsenI. Petrykiewicz and A. Shaw, Iterated Cesaro averages, fre-quencies of digits and Baire category, Acta Arith., 144 (2010), 287-293. Google Scholar

[19]

M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one dimensional maps, Comm. Math. Phys., 81 (1981), 39-88. doi: 10.1007/BF01941800. Google Scholar

[20]

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. Google Scholar

[21]

J. Li and M. Wu, Generic property of irregular sets in systems satisfying the specificaiton property, Discrete and Continuous Dynamical Systems, 34 (2014), 635-645. Google Scholar

[22]

C. LiangW. Sun and X. Tian, Ergodic properties of invariant measures for $C^{1+α}$ non-uniformly hyperbolic systems, Ergodic Theory & Dynam. Systems., 33 (2013), 560-584. Google Scholar

[23]

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

[24]

L. Olsen and S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages Ⅱ, Bull. Sci. Math., 6 (2007), 518-558. Google Scholar

[25]

Y. Pei and E. Chen, On the varational principle for the topological pressure for certain non-compact sets, Sci China Math, 53 (2010), 1117-1128. Google Scholar

[26]

Ya. Pesin and B. Pitskel', Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 307-318. doi: 10.1007/BF01083692. Google Scholar

[27]

C. Pfister and W. Sullivan, Large deviations estimates for dynamical systems without the specification property. application to the $β$-shifts, Nonlinearity, 18 (2005), 237-261. Google Scholar

[28]

C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824. Google Scholar

[29]

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

[30]

D. Ruelle, Historic Behavior in Smooth Dynamical Systems, Global Analysis of Dynamical Systems (H. W. Broer, B. Krauskopf, and G. Vegter, eds. ), Bristol: Institute of Physics Publishing, 2001.Google Scholar

[31]

J. Schmeling, Symbolic dynamics for $β$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182. Google Scholar

[32]

K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X. Google Scholar

[33]

W. Sun and X. Tian, The structure on invariant measures of $C^1$ generic diffeomorphisms, Acta Mathematica Sinica, English Series, 28 (2012), 817-824. doi: 10.1007/s10114-011-9723-5. Google Scholar

[34]

F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity, 21 (2008), T33-T36. doi: 10.1088/0951-7715/21/3/T02. Google Scholar

[35]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certainnon-compact sets, Ergodic theory and dynamical systems, 23 (2003), 317-348. doi: 10.1017/S0143385702000913. Google Scholar

[36]

D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst., 25 (2009), 25-51. doi: 10.1080/14689360903156237. Google Scholar

[37]

D. Thompson, Irregular sets, the $β$-transformation and the almost specification property, Transactions of the American Mathematical Society, 364 (2012), 5395-5414. Google Scholar

[38]

M. Todd, Multifractal Analysis of Multimodal Maps, arXiv: 0809.1074v2, 2008.Google Scholar

[39]

P. Walters, Equilibrium states for $β$-transformations and related transformations, Math. Z., 159 (1978), 65-88. Google Scholar

[40]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 2001.Google Scholar

show all references

References:
[1]

F. AbdenurC. Bonatti and S. Crovisier, Nonuniform hyperbolicity of C1-generic diffeomorphisms, Israel Journal of Mathematics, 183 (2011), 1-60. doi: 10.1007/s11856-011-0041-5. Google Scholar

[2]

S. AlbeverioM. Pratsiovytyi and G. Torbin, Topological and fractal properties of subsets of real numbers which are not normal, Bull. Sci. Math., 129 (2005), 615-630. Google Scholar

[3]

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. Google Scholar

[4]

L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, vol. 272, Birkhäuser, 2008.Google Scholar

[5]

L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026. Google Scholar

[6]

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. Google Scholar

[7]

A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk. , 38 (1983), p133. doi: 10.1070/RM1983v038n05ABEH003504. Google Scholar

[8]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar

[9]

R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590. Google Scholar

[10]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer, Lecture Notes in Math. , 470,1975. doi: 10.1007/BFb0081279. Google Scholar

[11]

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. Google Scholar

[12]

E. ChenT. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208. Google Scholar

[13]

X. Dai and Y. Jiang, Hausdorff dimensions of zero-entropy sets of dynamical systems with positive entropy, J. Stat. Phys., 120 (2005), 511-519. Google Scholar

[14]

M. Dateyama, Invariant Measures for Homeomorphisms with Weak Specification, Tokyo J. of Math., 4 (1981), 389-397. doi: 10.3836/tjm/1270215164. Google Scholar

[15]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 1976. doi: 10.1007/BFb0082364. Google Scholar

[16]

Y. DongX. Tian and X. Yuan, Ergodic properties of systems with asymptotic average shadowing property, Journal of Mathematical Analysis and Applications, 432 (2015), 53-73. doi: 10.1016/j.jmaa.2015.06.046. Google Scholar

[17]

D. J. FengK. S. Lau and J. Wu, Ergodic limits on the conformal repellers, Advances in Mathematics, 169 (2002), 58-91. Google Scholar

[18]

J. HydeV. LaschosL. OlsenI. Petrykiewicz and A. Shaw, Iterated Cesaro averages, fre-quencies of digits and Baire category, Acta Arith., 144 (2010), 287-293. Google Scholar

[19]

M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one dimensional maps, Comm. Math. Phys., 81 (1981), 39-88. doi: 10.1007/BF01941800. Google Scholar

[20]

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. Google Scholar

[21]

J. Li and M. Wu, Generic property of irregular sets in systems satisfying the specificaiton property, Discrete and Continuous Dynamical Systems, 34 (2014), 635-645. Google Scholar

[22]

C. LiangW. Sun and X. Tian, Ergodic properties of invariant measures for $C^{1+α}$ non-uniformly hyperbolic systems, Ergodic Theory & Dynam. Systems., 33 (2013), 560-584. Google Scholar

[23]

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

[24]

L. Olsen and S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages Ⅱ, Bull. Sci. Math., 6 (2007), 518-558. Google Scholar

[25]

Y. Pei and E. Chen, On the varational principle for the topological pressure for certain non-compact sets, Sci China Math, 53 (2010), 1117-1128. Google Scholar

[26]

Ya. Pesin and B. Pitskel', Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 307-318. doi: 10.1007/BF01083692. Google Scholar

[27]

C. Pfister and W. Sullivan, Large deviations estimates for dynamical systems without the specification property. application to the $β$-shifts, Nonlinearity, 18 (2005), 237-261. Google Scholar

[28]

C. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynam. Sys., 27 (2007), 929-956. doi: 10.1017/S0143385706000824. Google Scholar

[29]

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

[30]

D. Ruelle, Historic Behavior in Smooth Dynamical Systems, Global Analysis of Dynamical Systems (H. W. Broer, B. Krauskopf, and G. Vegter, eds. ), Bristol: Institute of Physics Publishing, 2001.Google Scholar

[31]

J. Schmeling, Symbolic dynamics for $β$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182. Google Scholar

[32]

K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X. Google Scholar

[33]

W. Sun and X. Tian, The structure on invariant measures of $C^1$ generic diffeomorphisms, Acta Mathematica Sinica, English Series, 28 (2012), 817-824. doi: 10.1007/s10114-011-9723-5. Google Scholar

[34]

F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity, 21 (2008), T33-T36. doi: 10.1088/0951-7715/21/3/T02. Google Scholar

[35]

F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certainnon-compact sets, Ergodic theory and dynamical systems, 23 (2003), 317-348. doi: 10.1017/S0143385702000913. Google Scholar

[36]

D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst., 25 (2009), 25-51. doi: 10.1080/14689360903156237. Google Scholar

[37]

D. Thompson, Irregular sets, the $β$-transformation and the almost specification property, Transactions of the American Mathematical Society, 364 (2012), 5395-5414. Google Scholar

[38]

M. Todd, Multifractal Analysis of Multimodal Maps, arXiv: 0809.1074v2, 2008.Google Scholar

[39]

P. Walters, Equilibrium states for $β$-transformations and related transformations, Math. Z., 159 (1978), 65-88. Google Scholar

[40]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 2001.Google Scholar

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