Dynamics of typical continuous functions
Proc. Amer. Math. Soc. 123 (1995), 1703-1707
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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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