Results on Scrambled Sets of Full Measurable Functions
Keywords:
Scrambled sets, Fσ-set, Gδσ-subset, regular space
Abstract
In this paper we constructed a function f possessing a scrambled set of full measure, then the function g is strictly increasing on [0, 1/2] and strictly decreasing on [1/2, 1] and further we extend our result that lim sup n→∞ |fn(x) − fn(y)| = 1 and lim inf n→∞ |fn(x) − fn(y)| = 0.These claims to extend results such a kind obtained by Brukner A.M and [3]. Simtal .J [10] and Jiehu Mai [8].
References
N.Aoki and Hiraide, Topological theory of Dynamical systems Recent Advances, North-Holland, (1994).
L.S.Block and Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, Springer-Verlag, (1513).
A.M.Brukner and Thakyin Hu, On Scrambled sets for chaotic functions, Trans. Amer. Mathe. Soc., 301(1987), 289-297.
Devaney, Introduction to Chaotic dynamical systems, Addison–Wesely 2nd Edition, (1989).
W.Gorman, The homeomorphic transformation of c-sets to d-sets, Proc. Amer. Math. Soc., 17(1956), 825-830.
I.Kan, A Chaotic fuction possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc., 92(1984), 45-49.
T.Li and J.Yorke, A Period three implies chaos, American Mathematical Monthly.
Jiehua Mai, Scrambled sets of continuous maps of one-dimensional polyhedral, Trans. Amer. Math. Soc., 351(1999), 353-362.
A.Yu.Shashkin, Fixed Points, Mathematical World, Universities Press.
J.A.Simtal, Chaotic function with some external properties, Proc. Amer. Maths. Soc., 87(1983), 54-56.
L.S.Block and Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, Springer-Verlag, (1513).
A.M.Brukner and Thakyin Hu, On Scrambled sets for chaotic functions, Trans. Amer. Mathe. Soc., 301(1987), 289-297.
Devaney, Introduction to Chaotic dynamical systems, Addison–Wesely 2nd Edition, (1989).
W.Gorman, The homeomorphic transformation of c-sets to d-sets, Proc. Amer. Math. Soc., 17(1956), 825-830.
I.Kan, A Chaotic fuction possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc., 92(1984), 45-49.
T.Li and J.Yorke, A Period three implies chaos, American Mathematical Monthly.
Jiehua Mai, Scrambled sets of continuous maps of one-dimensional polyhedral, Trans. Amer. Math. Soc., 351(1999), 353-362.
A.Yu.Shashkin, Fixed Points, Mathematical World, Universities Press.
J.A.Simtal, Chaotic function with some external properties, Proc. Amer. Maths. Soc., 87(1983), 54-56.
How to Cite
Dr.Pralahad Mahagaonkar. (2016). Results on Scrambled Sets of Full Measurable Functions. International Journal of Current Research in Science and Technology, 2(6), 1-4. Retrieved from https://crst.gfer.org/index.php/crst/article/view/69
Issue
Section
Articles
