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Two New Types of Chaotic Maps and Minimal Systems

Received: 05 September 2014    Accepted: 16 September 2014    Published: 17 September 2014
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Abstract

In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.

DOI 10.11648/j.pamj.s.2014030601.12
Published in Pure and Applied Mathematics Journal (Volume 3, Issue 6-1, December 2014)

This article belongs to the Special Issue Mathematical Theory and Modeling

Page(s) 7-12
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Topologically θ - Transitive Map, α- Chaotic, Chaotic Amp, α- Transitive, θ- Dense

References
[1] Mohammed Nokhas Murad, Introduction to θ -Type Transitive Maps on Topological spaces. International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:12 No:06 (2012), pp. 104-108 .
[2] Mohammed Nokhas Murad, Topologically α-Transitive Maps and Minimal Systems, Gen. Math. Notes,Vol 10, No. 2, (2012), pp.43-53
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[4] Caldas M., A note on some applications of α-open sets, UMMS, 2(2003). pp. 125-130.
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Author Information
  • University of Sulaimani, Faculty of Science and Science Education, School of Science, Math Department

  • University of Sulaimani, Faculty of physical and Basie Education, School of Basic Education, Department of Computer Science

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  • APA Style

    Mohammed Nokhas Murad Kaki, Sherko Hassan Abdurrahman. (2014). Two New Types of Chaotic Maps and Minimal Systems. Pure and Applied Mathematics Journal, 3(6-1), 7-12. https://doi.org/10.11648/j.pamj.s.2014030601.12

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    Mohammed Nokhas Murad Kaki; Sherko Hassan Abdurrahman. Two New Types of Chaotic Maps and Minimal Systems. Pure Appl. Math. J. 2014, 3(6-1), 7-12. doi: 10.11648/j.pamj.s.2014030601.12

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    AMA Style

    Mohammed Nokhas Murad Kaki, Sherko Hassan Abdurrahman. Two New Types of Chaotic Maps and Minimal Systems. Pure Appl Math J. 2014;3(6-1):7-12. doi: 10.11648/j.pamj.s.2014030601.12

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  • @article{10.11648/j.pamj.s.2014030601.12,
      author = {Mohammed Nokhas Murad Kaki and Sherko Hassan Abdurrahman},
      title = {Two New Types of Chaotic Maps and Minimal Systems},
      journal = {Pure and Applied Mathematics Journal},
      volume = {3},
      number = {6-1},
      pages = {7-12},
      doi = {10.11648/j.pamj.s.2014030601.12},
      url = {https://doi.org/10.11648/j.pamj.s.2014030601.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.s.2014030601.12},
      abstract = {In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.},
     year = {2014}
    }
    

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  • TY  - JOUR
    T1  - Two New Types of Chaotic Maps and Minimal Systems
    AU  - Mohammed Nokhas Murad Kaki
    AU  - Sherko Hassan Abdurrahman
    Y1  - 2014/09/17
    PY  - 2014
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    DO  - 10.11648/j.pamj.s.2014030601.12
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 7
    EP  - 12
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.s.2014030601.12
    AB  - In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.
    VL  - 3
    IS  - 6-1
    ER  - 

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