# American Institute of Mathematical Sciences

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. Abdenur, C. 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. [2] 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. [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. [4] L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, vol. 272, Birkhäuser, 2008. [5] L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026. [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. [7] A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk. , 38 (1983), p133. doi: 10.1070/RM1983v038n05ABEH003504. [8] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [9] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590. [10] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer, Lecture Notes in Math. , 470,1975. doi: 10.1007/BFb0081279. [11] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. [12] E. Chen, T. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208. [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. [14] M. Dateyama, Invariant Measures for Homeomorphisms with Weak Specification, Tokyo J. of Math., 4 (1981), 389-397. doi: 10.3836/tjm/1270215164. [15] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 1976. doi: 10.1007/BFb0082364. [16] Y. Dong, X. 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. [17] D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers, Advances in Mathematics, 169 (2002), 58-91. [18] J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz and A. Shaw, Iterated Cesaro averages, fre-quencies of digits and Baire category, Acta Arith., 144 (2010), 287-293. [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. [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. [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. [22] C. Liang, W. 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. [23] L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43-53. doi: 10.1017/S0305004104007601. [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. [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. [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. [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. [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. [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. [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. [31] J. Schmeling, Symbolic dynamics for $β$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182. [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. [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. [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. [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. [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. [37] D. Thompson, Irregular sets, the $β$-transformation and the almost specification property, Transactions of the American Mathematical Society, 364 (2012), 5395-5414. [38] M. Todd, Multifractal Analysis of Multimodal Maps, arXiv: 0809.1074v2, 2008. [39] P. Walters, Equilibrium states for $β$-transformations and related transformations, Math. Z., 159 (1978), 65-88. [40] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 2001.

show all references

##### References:
 [1] F. Abdenur, C. 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. [2] 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. [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. [4] L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics, vol. 272, Birkhäuser, 2008. [5] L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026. [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. [7] A. M. Blokh, Decomposition of dynamical systems on an interval, Uspekhi Mat. Nauk. , 38 (1983), p133. doi: 10.1070/RM1983v038n05ABEH003504. [8] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [9] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590. [10] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer, Lecture Notes in Math. , 470,1975. doi: 10.1007/BFb0081279. [11] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. [12] E. Chen, T. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems, 25 (2005), 1173-1208. [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. [14] M. Dateyama, Invariant Measures for Homeomorphisms with Weak Specification, Tokyo J. of Math., 4 (1981), 389-397. doi: 10.3836/tjm/1270215164. [15] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 1976. doi: 10.1007/BFb0082364. [16] Y. Dong, X. 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. [17] D. J. Feng, K. S. Lau and J. Wu, Ergodic limits on the conformal repellers, Advances in Mathematics, 169 (2002), 58-91. [18] J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz and A. Shaw, Iterated Cesaro averages, fre-quencies of digits and Baire category, Acta Arith., 144 (2010), 287-293. [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. [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. [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. [22] C. Liang, W. 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. [23] L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc., 137 (2004), 43-53. doi: 10.1017/S0305004104007601. [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. [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. [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. [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. [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. [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. [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. [31] J. Schmeling, Symbolic dynamics for $β$-shifts and self-normal numbers, Ergodic Theory Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182. [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. [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. [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. [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. [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. [37] D. Thompson, Irregular sets, the $β$-transformation and the almost specification property, Transactions of the American Mathematical Society, 364 (2012), 5395-5414. [38] M. Todd, Multifractal Analysis of Multimodal Maps, arXiv: 0809.1074v2, 2008. [39] P. Walters, Equilibrium states for $β$-transformations and related transformations, Math. Z., 159 (1978), 65-88. [40] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 2001.
 [1] Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545 [2] Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459 [3] Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469 [4] Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103 [5] Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451 [6] Christopher Hoffman. Subshifts of finite type which have completely positive entropy. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1497-1516. doi: 10.3934/dcds.2011.29.1497 [7] Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086 [8] Dou Dou. Minimal subshifts of arbitrary mean topological dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1411-1424. doi: 10.3934/dcds.2017058 [9] Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77 [10] Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010 [11] Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555 [12] Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363 [13] Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 [14] João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465 [15] Nicolás Matte Bon. Topological full groups of minimal subshifts with subgroups of intermediate growth. Journal of Modern Dynamics, 2015, 9: 67-80. doi: 10.3934/jmd.2015.9.67 [16] Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018 [17] Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201 [18] Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827 [19] Bing Li, Tuomas Sahlsten, Tony Samuel. Intermediate $\beta$-shifts of finite type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 323-344. doi: 10.3934/dcds.2016.36.323 [20] Philipp Gohlke, Dan Rust, Timo Spindeler. Shifts of finite type and random substitutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5085-5103. doi: 10.3934/dcds.2019206

2017 Impact Factor: 1.179