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Paradoxical functions on the interval


Author: Víctor Jiménez López
Journal: Proc. Amer. Math. Soc. 120 (1994), 465-473
MSC: Primary 58F13; Secondary 58F11
DOI: https://doi.org/10.1090/S0002-9939-1994-1172952-3
MathSciNet review: 1172952
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Abstract: In this paper it is shown that any expanding with Lipschitz derivative function $ f$ has a contradictory behaviour from the point of view of chaos in the sense of Li and Yorke. On the one hand it cannot generate scrambled sets of positive Lebesgue measure. On the other hand the two-dimensional set $ \operatorname{Ch} (f)$ including the pairs $ (x,y)$ such that $ \{ x,y\} $ is a scrambled set of $ f$ has positive measure. In fact, both the geometric structure (almost everywhere) and measure of $ \operatorname{Ch} (f)$ can be explicitly obtained.


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DOI: https://doi.org/10.1090/S0002-9939-1994-1172952-3
Article copyright: © Copyright 1994 American Mathematical Society

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