Two New Types of Chaotic Maps and Minimal Systems
Pure and Applied Mathematics Journal
Volume 3, Issue 6-1, December 2014, Pages: 7-12
Received: Sep. 5, 2014; Accepted: Sep. 16, 2014; Published: Sep. 17, 2014
Views 4408      Downloads 196
Mohammed Nokhas Murad Kaki, University of Sulaimani, Faculty of Science and Science Education, School of Science, Math Department
Sherko Hassan Abdurrahman, University of Sulaimani, Faculty of physical and Basie Education, School of Basic Education, Department of Computer Science
Article Tools
Follow on us
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.
Topologically θ - Transitive Map, α- Chaotic, Chaotic Amp, α- Transitive, θ- Dense
To cite this article
Mohammed Nokhas Murad Kaki, Sherko Hassan Abdurrahman, Two New Types of Chaotic Maps and Minimal Systems, Pure and Applied Mathematics Journal. Special Issue: Mathematical Theory and Modeling. Vol. 3, No. 6-1, 2014, pp. 7-12. doi: 10.11648/j.pamj.s.2014030601.12
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 .
Mohammed Nokhas Murad, Topologically α-Transitive Maps and Minimal Systems, Gen. Math. Notes,Vol 10, No. 2, (2012), pp.43-53
Mohammed Nokhas Murad, Topologically α- Type Maps and Minimal α-Open Sets, Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering and Medicine Vol. 4 No. 2, (2013) , pp. 177-183
Caldas M., A note on some applications of α-open sets, UMMS, 2(2003). pp. 125-130.
Levine N., Generalized closed sets in topology, Rend. Cire. Math. Paler no. (2) 19 (1970). Pp. 89-96.
Levine N., Semi open sets and semi continuity in topological spaces. Amer. Math. Monthly. 70(1963). 36-41.
Ogata N., On some classes of nearly open sets, Pacific J. Math. 15(1965). 961-970.
Bhattacharya P., and Lahiri K.B., Semi-generalized closed sets in topology. Indian J. Math. 29 (1987). 376-382.
Rosas E., Vielina J., Operator- compact and Operator-connected spaces. Scientific Mathematica 1(2)(1998). 203-208.
Kasahara S., Operation-compact spaces. Mathematica Japonica. 24 (1979).97-105.
Crossley G.S. , Hildebrand, S. K .Semi – topological properties, Fund. Math., 74 (1972), 233-254
Maheshwari N. S., and Thakur S. S., On α-irresolute mappings, Tamkang J. Math. 11 (1980). 209-214.
M. Ganster and I.L. Reilly, A decomposition of continuity, Acta Math. Hungarica 56 (3-4) (1990), 299–301.
M. Ganster and I.L. Reilly, Locally closed sets and LC-continuous functions, Internat. J. Math. Math. Sci. 12 (3) (1989), 417–424.
N. Bourbaki, General Topology Part 1, Addison Wesley, Reading, Mass. 1966.
N. V. Veli•cko, H-closed topological spaces. (Russian) Mat. Sb. (N.S.) 70 (112) (1966), 98-112, English transl. Amer. Math. Soc. Transl. 78(1968), 102-118.
R. F. Dickman, Jr. and J. R. Porter, θ-closed subsets of Hausdorff spaces, Pacific J. Math. 59(1975), 407-415.
R. F. Dickman Jr., J. R. Porter, θ-perfect and θ-absolutely closed functions, Ilinois J. Math. 21(1977), 42-60.
J. Dontchev, H. Maki, Groups of θ-generalized homeomorphisms and the digital line, Topology and its Applications, 20(1998), 1-16.
M. Ganster, T. Noiri, I. L. Reilly, Weak and strong forms of θ-irresolute functions, J. Inst. Math. Comput. Sci. 1(1)(1988), 19-29.
D. S. Jankovic, On some separation axioms and θ-closure, Mat. Vesnik 32 (4)(1980), 439-449.
D. S. Jankovic, θ-regular spaces, Internat. J. Math. & Math. Sci. 8(1986), 615-619.
J. E. Joseph, θ-closure and θ-subclosed graphs, Math., Chronicle 8(1979), 99-117.
S. Fomin, Extensions of topological spaces, Ann. of Math. 44 (1943), 471-480.
S. Iliadis and S. Fomin, The method of centred systems in the theory of topological spaces, Uspekhi Mat. Nauk. 21 (1996), 47-76 (=Russian Math. Surveys, 21 (1966), 37-62). Appl. Math 31(4)(2000) 449-450
Saleh, M., On θ-continuity and strong θ-continuity, Applied Mathematics E-Notes (2003), 42-48.
M. Caldas, S. Jafari and M. M. Kovar, Some properties of θ-open sets, Divulg. Mat 12(2)(2004), 161-169
P. E. Long, L. L. Herrington, The τθ-topology and faintly continuous functions, Kyungpook Math. J. 22(1982), 7-14.
M. Caldas, A note on some applications of α-open sets, UMMS, 2(2003), 125-130
F.H. Khedr and T. Noiri. On θ-irresolute functions. Indian J. Math., 28, 3 (1986), 211-217.
M. Nokhas Murad Kaki 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
M. Nokhas Murad Kaki Topologically α - Transitive Maps and Minimal Systems Gen. Math. Notes, Vol. 10, No. 2, (2012) pp. 43-53 ISSN 2219-7184; Copyright © ICSRS
Engelking, R. Outline of General Topology, North Holland Publishing Company-Amsterdam, 1968.
Andrijevie D., Some properties of the topology of α-sets, Math. Vesnik, (1994). 1 -10
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186