Typical Continuous Functions Are Not Chaotic in the Sense of Devaney

Abstract

We show that typical continuous functions of the form $f:M \to M$, where $M$ is a compact metric space with the fixed-point property and the absolute-retract property, are not chaotic in the sense of Devaney. Typical continuous functions on the compact interval have been shown to be chaotic in terms of other definitions of chaos. Results are also presented concerning the chain recurrent set for typical continuous functions and concerning functions for which the chain recurrent set is the entire space.