ON HIGHER ORDER IRREGULAR SETS

다국어 초록 (Multilingual Abstract)

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.

참고문헌 (Reference)

1 Y. Pesin, "Topological pressure and variational principle for non-compact sets" 18 : 307-318, 1984

2 E. C. Chen, "Topological entropy for divergence points" 25 (25): 1173-1208, 2005

3 S. Albeverio, "Topological and fractal properties of real numbers which are not normal" 129 (129): 615-630, 2005

4 D. Ruelle, "Thermodynamic Formalism: The mathematical structures of classical equi-librium statistical mechanics" Addison Wesley 1978

5 J. J. Li, "The sets of divergence points of self-similar measures are residual" 404 (404): 429-437, 2013

6 D. Thompson, "The irregular set for maps with the specification property has full topo-logical pressure" 25 (25): 25-51, 2010

7 L. Barreira, "Sets of “non-typical” points have full topological entropy and full Hausdorff dimension" 116 : 29-70, 2000

8 A. H. Fan, "Recurrence, dimensions and entropies" 64 (64): 229-244, 2001

9 J. J. Li, "Points with maximal Birkhoff average oscillation" 66(141) (66(141)): 223-241, 2016

10 A. H. Fan, "On the distribution of long-term time averages on symbolic space" 99 (99): 813-856, 2000

11 M. Madritsch, "Non-normal numbers with respect to Markov partitions" 34 (34): 663-676, 2014

12 L. Olsen, "Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages" 82 (82): 1591-1649, 2003

13 M. Pollicott, "Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine ap-proximation" 207 (207): 145-171, 1999

14 J. C. Oxtoby, "Measre and Category" Springer 1996

15 J. Hyde, "Iterated Cesaro averages, frequencies of digits and Baire category" 144 (144): 287-293, 2010

16 L. Barreira, "Irregular sets of two-sided Birkhoff averages and hyperbolic sets" 54 (54): 13-30, 2016

17 L. Barreira, "Irregular sets are residual" 66 (66): 471-489, 2014

18 L. Olsen, "Higher order local dimensions and Baire category" 211 (211): 1-20, 2011

19 T. Jordan, "Higher order Birkhoff averages" 24 (24): 299-313, 2009

20 J. J. Li, "Hausdorff dimensions of the divergence points of self-similar measures with the open set condition" 25 (25): 93-105, 2012

21 J. J. Li, "Generic property of irregular sets in systems satisfying the specification prop-erty" 34 : 635-645, 2014

22 L. Olsen, "Extremely non-normal numbers" 137 (137): 43-53, 2004

23 D.-J. Feng, "Ergodic limits on the conformal repellers" 169 (169): 58-91, 2002

24 M. Denker, "Ergodic Theory on Compact Space" Springer-Verlag 1976

25 C. Bonatti, "Dynamics Beyond Uniform Hyperbolicity" Springer-Verlag 2005

26 J. J. Li, "Divergence points in systems satisfying the specification property" 33 (33): 905-920, 2013

27 I. S. Baek, "Baire category and extremely non-normal points of invariant sets of IFS’s" 27 (27): 935-943, 2010