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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Dynamics of typical continuous functions


Author: Hervé Lehning
Journal: Proc. Amer. Math. Soc. 123 (1995), 1703-1707
MSC: Primary 54H20
DOI: https://doi.org/10.1090/S0002-9939-1995-1239798-X
MathSciNet review: 1239798
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Abstract: S. J. Agronsky, A. M. Bruckner, and M. Laczkovic have studied the behaviour of the sequence $ ({f^n}(x))$ where f is the typical continuous function from the closed unit interval I into itself and x the typical point of I. In particular, they have proved that the typical limit set $ \omega (f,x)$ is a Cantor set of Menger-Uryson dimension zero. Using mainly the Tietze extension theorem, we have found a shorter proof of this result which applies to a more general situation. As a matter of fact, we have replaced the closed unit interval by a compact N-dimensional manifold and the Menger-Uryson dimension by the Hausdorff one. We have also proved that, for the typical continuous function f, the function $ x \to \omega (f,x)$ is continuous at the typical point x. It follows that the typical limit set is not a fractal and that, for the typical continuous function f, the sequence $ ({f^n}(x))$ is not chaotic.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1239798-X
Article copyright: © Copyright 1995 American Mathematical Society

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