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A characterization of chaos. (English) Zbl 0577.54041
Summary: 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 \(\neq 2^ n\), \(n=0,1,2,...)\) 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 \(\epsilon >0\). 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.

MSC:
54H20 Topological dynamics (MSC2010)
26A18 Iteration of real functions in one variable
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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