




Necessity of the Third Condition from the Definition of Omega Chaos 

PP: 23032307 

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. 




