Dynamics of typical continuous functions

Abstract

S. J. Agronsky, A. M. Bruckner, and M. Laczkovic have studied the behaviour of the sequence ( f n ( x ) ) ({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 ω ( f , x ) \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 → ω ( f , x ) 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 ) ) ({f^n}(x)) is not chaotic.