Abstract

The infimum, resp. minimum, of the topological entropies of all Devaney chaotic maps in different spaces are studied. After recalling the results implicitly known in the literature for zero and one-dimensional spaces, the paper deals with some higher dimensional spaces. The key role is played by the result saying that a Devaney chaotic map $f$ in a compact metric space $X$ without isolated points can always be extended to a triangular map $F$ in $X\; \times \; [0,1]$ as well as to a triangular map $G$ in $X\times \; S1$ ($S1$ being a circle) in such a way that $F$ and $G$ are also Devaney chaotic and, moreover, $F$ and $G$ have the same topological entropy as $f$. This has several consequences. In particular, the best lower bounds of the topological entropy of Devaney chaotic maps in disks and tori of any dimension are obtained.