Skip to main content Accessibility help
×
Home

The Hausdorff dimension of multiply Xiong chaotic sets

  • JIAN LI (a1), JIE LÜ (a2) and YUANFEN XIAO (a3)

Abstract

We construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere that is contained in some multiply proximal cell for the full shift over finite symbols and the Gauss system, respectively.

Copyright

References

Hide All
[1] Akin, E. and Kolyada, S.. Li–Yorke sensitivity. Nonlinearity 16(4) (2003), 14211433.
[2] Auslander, J.. On the proximal relation in topological dynamics. Proc. Amer. Math. Soc. 11 (1960), 890895.
[3] Balibrea, F. and López, V. J.. The measure of scrambled sets: a survey. Acta Univ. M. Belii Ser. Math. 7 (1999), 311.
[4] Blanchard, F. and Huang, W.. Entropy sets, weakly mixing sets and entropy capacity. Discrete Contin. Dyn. Syst. 20(2) (2008), 275311.
[5] Blanchard, F., Huang, W. and Snoha, L.. Topological size of scrambled sets. Colloq. Math. 110(2) (2008), 293361.
[6] Bruin, H. and López, V. J.. On the Lebesgue measure of Li–Yorke pairs for interval maps. Comm. Math. Phys. 299(2) (2010), 523560.
[7] Falconer, K.. Fractal Geometry (Mathematical Foundations and Applications) , 3rd edn. John Wiley & Sons, Chichester, 2014.
[8] Fang, C., Huang, W., Yi, Y. and Zhang, P.. Dimensions of stable sets and scrambled sets in positive finite entropy systems. Ergod. Th. & Dynam. Sys. 32(2) (2012), 599628.
[9] Hu, H. and Yu, Y.. On Schmidt’s game and the set of points with non-dense orbits under a class of expanding maps. J. Math. Anal. Appl. 418(2) (2014), 906920.
[10] Huang, W., Li, J., Ye, X. and Zhou, X.. Positive topological entropy and 𝛥-weakly mixing sets. Adv. Math. 306 (2017), 653683.
[11] Huang, W., Shao, S. and Ye, X.. Mixing and proximal cells along sequences. Nonlinearity 17(4) (2004), 12451260.
[12] Iosifescu, M. and Kraaikamp, C.. Metrical Theory of Continued Fractions (Mathematics and its Applications, 547) . Kluwer Academic Publishers, Dordrecht, 2002.
[13] Jarník, V.. Zur metrischen Theorie der diophantischen Approximationen. Prace Matematyczno-Fizyczne 36(1) (1928), 91106.
[14] Liu, K.. 𝛥-weakly mixing subset in positive entropy actions of a nilpotent group. J. Differential Equations 267 (2019), 525546.
[15] Liu, W. and Li, B.. Chaotic and topological properties of continued fractions. J. Number Theory 174 (2017), 369383.
[16] Misiurewicz, M.. On Bowen’s definition of topological entropy. Discrete Contin. Dyn. Syst. Ser. A 10 (2004), 827833.
[17] Rohlin, V. A.. Exact endomorphisms of a Lebesgue space. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499530 (in Russian).
[18] Wu, C. L. and Tan, F.. Hausdorff dimension of a chaotic set of shift in a countable symbolic space. J. South China Norm. Univ. Natur. Sci. Ed. 4 (2007), 1116 (in Chinese, with English and Chinese summaries).
[19] Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.
[20] Wu, J.. A remark on the growth of the denominators of convergents. Monatsh. Math. 147(3) (2006), 259264.
[21] Xiong, J. C. and Yang, Z. G.. Chaos caused by a topologically mixing map. Dynamical Systems and Related Topics (Nagoya, 1990) (Advanced Series in Dynamical Systems, 9) . World Scientific, River Edge, NJ, 1991, pp. 550572.
[22] Xiong, J.. Erratic time dependence of orbits for a topologically mixing map. J. China Univ. Sci. Tech. 21(4) (1991), 387396.
[23] Xiong, J. C.. Hausdorff dimension of a chaotic set of shift of a symbolic space. Sci. China Ser. A 38(6) (1995), 696708.

Keywords

MSC classification

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed