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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.

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