Dense distribution of chaotic maps in continuous map spaces

Abstract

This article is concerned with distribution of several kinds of chaotic maps in a continuous map space, in which the maps are defined in a closed bounded set of a Banach space. It is shown that the map space contains a dense set of maps that are strictly coupled-expanding, have nondegenerate and regular snap-back repellers, have nondegenerate and regular homoclinic orbits to repellers, and consequently that are chaotic in the sense of Devaney as well as in the original sense of Li–Yorke, and have the topological entropy larger than any given positive constant. Further, in the finite-dimensional case, there exists a dense residual set of the map space such that every map f in the set is strictly coupled-expanding in k pairwise disjoint compact sets for any given integer k ≥ 2, is chaotic in the sense of Li–Yorke and has the infinite topological entropy and a nontrivial invariant measure.