October  2007, 17(4): 821-833. doi: 10.3934/dcds.2007.17.821

Specification properties and dense distributional chaos

1. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland

Received  July 2006 Revised  September 2006 Published  January 2007

The notion of distributional chaos was introduced by Schweizer and Smítal in [Trans. Amer. Math. Soc., 344 (1994) 737] for continuous maps of a compact interval. Further, this notion was generalized to three versions $d_1C$--$d_3C$ for maps acting on general compact metric spaces (see e.g. [Chaos Solitons Fractals, 23 (2005) 1581]). The main result of [ J. Math. Anal. Appl. , 241 (2000) 181] says that a weakened version of the specification property implies existence of a two points scrambled set which exhibits a $d_1 C$ version of distributional chaos. In this article we show that much more complicated behavior is present in that case. Strictly speaking, there exists an uncountable and dense scrambled set consisting of recurrent but not almost periodic points which exhibit uniform $d_1 C$ versions of distributional chaos.
Citation: Piotr Oprocha. Specification properties and dense distributional chaos. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 821-833. doi: 10.3934/dcds.2007.17.821
[1]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[2]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[3]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[4]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[5]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[6]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]