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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Dynamics of typical continuous functions
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by Hervé Lehning PDF
Proc. Amer. Math. Soc. 123 (1995), 1703-1707 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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