A characterization of nonchaotic continuous maps of the interval stable under small perturbations

Preiss, D.; Smítal, J.

doi:10.1090/S0002-9947-1989-0997677-9

A characterization of nonchaotic continuous maps of the interval stable under small perturbations
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by D. Preiss and J. Smítal PDF

Trans. Amer. Math. Soc. 313 (1989), 687-696 Request permission

Abstract:

Recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles (sometimes this is possible even in the case when the trajectory is not asymptotically periodic) or is $\varepsilon$-chaotic for some $\varepsilon > 0$. In certain cases, the first property is stable under small perturbations. This means that a perturbed map can be chaotic but the chaos must be small whenever the perturbation is small. In other words, there are nonchaotic maps without "chaos explosions". In the paper we give a characterization of these maps along with some consequences. Using the main result it is possible to prove that generically the nonchaotic maps are stable.

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