Dense distribution of chaotic maps in continuous map spaces

摘要

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/ 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.