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01-Applied Mathematics & Information Sciences
An International Journal
               
 
 
 
 
 
 
 
 
 
 
 

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Volumes > Volume 9 > No. 5

 
   

Necessity of the Third Condition from the Definition of Omega Chaos

PP: 2303-2307
Author(s)
Marek Lampart,
Abstract
The notion of omega chaos was introduced by S. Li in 1993 for continuous maps of compact metric spaces by three conditions: 1. the set difference of omega limit sets is uncountable, 2. intersection of omega limit sets is nonempty and 3. each omega limit set of the point from the omega scrambled set is not contained in the set of all periodic points. It was also pointed that the third condition is superfluous for continuous maps of the compact interval. As a main result of this paper it will be shown that the third condition is essential even in one dimension by construction of two examples of homeomorphisms on one dimensional arcwise connected space having two point set or infinite respectively that satisfies first and second condition but the third condition is not fulfilled from the definition of omega chaos.

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