ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 1(2003)
p. 119 – 127

Two Kinds of Chaos and Relations between them
M. Lampart

Abstract.  In this paper we consider relations between chaos in the sense of Li and Yorke, and $\omega$-chaos. The main aim is to show how important the size of scrambled sets is in definitions of chaos. We provide an example of an $\omega$-chaotic map on a compact metric space which is chaotic in the sense of Li and Yorke, but any scrambled set contains only two points. Chaos in the sense of Li and Yorke cannot be excluded: We show that any continuous map of a compact metric space which is $\omega$-chaotic, must be chaotic in the sense of Li and Yorke. Since it is known that, for continuous maps of the interval, Li and Yorke chaos does not imply $\omega$-chaos, Li and Yorke chaos on compact metric spaces appears to be weaker. We also consider, among others, the relations of the two notions of chaos on countably infinite compact spaces.

AMS subject classification:  37B10, 37D45, secondary 26A18, 54H20
Keywords:  $\omega$-chaos, Li and Yorke chaos, scrambled sets