Dynamics of typical continuous functions

Author:
Hervé Lehning

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1703-1707

MSC:
Primary 54H20

MathSciNet review:
1239798

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Abstract: S. J. Agronsky, A. M. Bruckner, and M. Laczkovic have studied the behaviour of the sequence 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 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 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 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