American Institute of Mathematical Sciences

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

Specification properties and dense distributional chaos

Piotr Oprocha 1,

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.
Keywords: distributional chaos, specification property, generalized specification property..
Mathematics Subject Classification: Primary: 37D45; Secondary: 37B9.
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]

Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto. Regular maps with the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2991-3009. doi: 10.3934/dcds.2013.33.2991
[2]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231
[3]

Jinjun Li, Min Wu. Divergence points in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 905-920. doi: 10.3934/dcds.2013.33.905
[4]

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
[5]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635
[6]

Manfred G. Madritsch, Izabela Petrykiewicz. Non-normal numbers in dynamical systems fulfilling the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4751-4764. doi: 10.3934/dcds.2014.34.4751
[7]

Shair Ahmad, Alan C. Lazer. On a property of a generalized Kolmogorov population model. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 1-6. doi: 10.3934/dcds.2013.33.1
[8]

Welington Cordeiro, Manfred Denker, Xuan Zhang. On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1941-1957. doi: 10.3934/dcds.2017082
[9]

Welington Cordeiro, Manfred Denker, Xuan Zhang. Corrigendum to: On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3705-3706. doi: 10.3934/dcds.2018160
[10]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675
[11]

Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475
[12]

Piotr Oprocha, Pawel Wilczynski. Distributional chaos via isolating segments. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 347-356. doi: 10.3934/dcdsb.2007.8.347
[13]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082
[14]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803
[15]

V. M. Gundlach, Yu. Kifer. Expansiveness, specification, and equilibrium states for random bundle transformations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 89-120. doi: 10.3934/dcds.2000.6.89
[16]

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069
[17]

Harry L. Johnson, David Russell. Transfer function approach to output specification in certain linear distributed parameter systems. Conference Publications, 2003, 2003 (Special) : 449-458. doi: 10.3934/proc.2003.2003.449
[18]

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
[19]

Kazuhiro Sakai. The oe-property of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 581-591. doi: 10.3934/dcds.1998.4.581
[20]

Pablo Sánchez, Jaume Sempere. Conflict, private and communal property. Journal of Dynamics & Games, 2016, 3 (4) : 355-369. doi: 10.3934/jdg.2016019

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences