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Trajectories of chaotic interval maps. (English) Zbl 1361.54023

The author proves that if \((x_n)_{n\in \mathbb{N}}\) is any sequence of numbers in the unit interval \([0,1]\) with the property that there is a map \(x_n\longmapsto x_{n+1}\) which is uniformly continuous on the set \(X=\{ x_n:n\in \mathbb{N} \}\), then, under certain conditions on \(X\), and after extending the map to the closure \(\overline{X}\), there exists a chaotic map of \([0,1]\) for which the given sequence is a trajectory.
The notion of chaos used in the article is in the sense of R. L. Devaney [An introduction to chaotic dynamical systems. 2nd ed. Redwood City, CA etc.: Addison-Wesley Publishing Company, Inc. (1989; Zbl 0695.58002)].
The author also shows sequences which appear as trajectories of maps of \([0,1]\), but not as trajectories of chaotic maps.

MSC:

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

Citations:

Zbl 0695.58002
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