Bulletin of the Australian Mathematical Society

Research Article

A characterization of chaos

K. Jankováa1 and J. Smítala1

a1 Department of Mathematics, Komensky University, 842 15 Bratislava, Czechoslovakia.


Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level xs025B > 0. Recent results of the second author then imply that any continuous map has exactly one of the following properties: It is either strongly chaotic or every trajectory is approximable by cycles. Also some other conditions characterizing chaos are given.

(Received November 25 1985)