**
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

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Acta Mathematica Universitatis Comenianae

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