Dynamics of Typical Continuous Functions

Abstract

The set C of continuous functions from the unit interval into itself becomes a Banach space under the supremum norm. A property of a continuous function is termed typical (some authors use ‘generic’) if it is shared by all functions in some residual subset of C (in the sense of Baire category). We begin by studying the iterates of typical continuous functions and the types of orbits which points may have. For example, we show that the typical continuous function can have neither a ‘stable’ (attracting) periodic point nor a point whose orbit is dense in any interval. We then investigate attractive behavior. We find, for example, that typically ‘most’ points in [0,1] are attracted to Cantor sets and that we may prescribe any residual subset of [0,1] to contain the closure of the union of all attractor sets for all typical F.