Topological entropy of Devaney chaotic maps

The infimum respectively minimum of the topological entropies in different spaces are studied for maps which are transitive or chaotic in the sense of Devaney (i.e., transitive with dense periodic points). After a short survey of results explicitly or implicitly known in the literature for zero and one-dimensional spaces the paper deals with chaotic maps in some higher-dimensional spaces. The key role is played by the result saying that a chaotic map f in a compact metric space X without isolated points can always be extended to a triangular (skew product) map F in X x {0}, {1} in such a way that F is also chaotic and has the same topological entropy as f. Moreover, the sets X x (0) and X x {1} are F-invariant which enables to use the factorization and obtain in such a way dynamical systems in the cone and in the suspension over X or in the space X x S-1. This has several consequences. Among others, the best lower bounds for the topological entropy of chaotic maps on disks, tori and spheres of any dimensions are proved to be zero. (C) 2003 Elsevier B.V. All rights reserved.