- Volume 54 Issue 1
To indicate the statistical complexity of dynamical systems, we introduce the notions of higher order irregular set and higher order maximal Birkhoff average oscillation in this paper. We prove that, in the setting of topologically mixing Markov chain, the set consisting of those points having maximal k-order Birkhoff average oscillation for all positive integers k is as large as the whole space from the topological point of view. As applications, we discuss the corresponding results on a repeller.
연구 과제 주관 기관 : National Natural Science Foundation of China
- S. Albeverio, M. Pratsiovytyi, and G. Torbin, Topological and fractal properties of real numbers which are not normal, Bull. Sci. Math. 129 (2005), no. 8, 615-630. https://doi.org/10.1016/j.bulsci.2004.12.004
- 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), no. 3, 935-943. https://doi.org/10.3934/dcds.2010.27.935
- L. Barreira, J. J. Li, and C. Valls, Irregular sets are residual, Tohoku Math. J. 66 (2014), no. 4, 471-489. https://doi.org/10.2748/tmj/1432229192
- L. Barreira, J. J. Li, and C. Valls, Irregular sets of two-sided Birkhoff averages and hyperbolic sets, Ark. Mat. 54 (2016), no. 1, 13-30. https://doi.org/10.1007/s11512-015-0214-2
- 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. https://doi.org/10.1007/BF02773211
- C. Bonatti, L. Diaz, and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Springer-Verlag, 2005.
- E. C. Chen, K. Tassilo, and L. Shu, Topological entropy for divergence points, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1173-1208. https://doi.org/10.1017/S0143385704000872
- M. Denker, C. Grillenberger, and K. Sigmund, Ergodic Theory on Compact Space, volume 527 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1976.
- A. H. Fan and D.-J. Feng, On the distribution of long-term time averages on symbolic space, J. Stat. Phys. 99 (2000), no. 3-4, 813-856. https://doi.org/10.1023/A:1018643512559
- A. H. Fan, D.-J. Feng, and J. Wu, Recurrence, dimensions and entropies, J. London Math. Soc. 64 (2001), no. 1, 229-244. https://doi.org/10.1017/S0024610701002137
- D.-J. Feng, K.-S. Lau, and J. Wu, Ergodic limits on the conformal repellers, Adv. Math. 169 (2002), no. 1, 58-91. https://doi.org/10.1006/aima.2001.2054
- J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz, and A. Shaw, Iterated Cesaro averages, frequencies of digits and Baire category, Acta Arith. 144 (2010), no. 3, 287-293. https://doi.org/10.4064/aa144-3-6
- T. Jordan, V. Naudot, and T. Young, Higher order Birkhoff averages, Dyn. Syst. 24 (2009), no. 3, 299-313. https://doi.org/10.1080/14689360802676269
- J. J. Li and M. Wu, Divergence points in systems satisfying the specification property, Discrete Contin. Dyn. Syst. 33 (2013), no. 2, 905-920. https://doi.org/10.3934/dcds.2013.33.905
- J. J. Li and M. Wu, The sets of divergence points of self-similar measures are residual, J. Math. Anal. Appl. 404 (2013), no. 2, 429-437. https://doi.org/10.1016/j.jmaa.2013.03.043
- J. J. Li and M. Wu, Generic property of irregular sets in systems satisfying the specification property, Discrete Contin. Dyn. Syst. 34 (2014), 635-645.
- J. J. Li and M. Wu, Points with maximal Birkhoff average oscillation, Czechoslovak Math. J. 66(141) (2016), no. 1, 223-241.
- 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), no. 1, 93-105. https://doi.org/10.1088/0951-7715/25/1/93
- M. Madritsch, Non-normal numbers with respect to Markov partitions, Discrete Contin. Dyn. Syst. 34 (2014), no. 2, 663-676. https://doi.org/10.3934/dcds.2014.34.663
- L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. 82 (2003), no. 12, 1591-1649. https://doi.org/10.1016/j.matpur.2003.09.007
- L. Olsen, Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 43-53. https://doi.org/10.1017/S0305004104007601
- L. Olsen, Higher order local dimensions and Baire category, Studia Math. 211 (2011), no. 1, 1-20. https://doi.org/10.4064/sm211-1-1
- J. C. Oxtoby, Measre and Category, Springer, New York, 1996.
- Y. Pesin and B. S. Pitskel, Topological pressure and variational principle for non-compact sets, Functional Anal. Appl. 18 (1984), 307-318. https://doi.org/10.1007/BF01083692
- 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), no. 1, 145-171. https://doi.org/10.1007/s002200050722
- D. Ruelle, Thermodynamic Formalism: The mathematical structures of classical equilibrium statistical mechanics, Ency. Math. and Appl. Vol 5, Addison Wesley, 1978.
- D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst. 25 (2010), no. 1, 25-51. https://doi.org/10.1080/14689360903156237