  We study topological entropy of exactly Devaney chaotic maps on totally
regular continua, i.e. on (topologically) rectifiable curves. After introducing
the so-called P-Lipschitz maps (where P is a finite invariant set) we give an
upper bound for their topological entropy. We prove that if a non-degenerate
totally regular continuum X contains a free arc which does not disconnect X or
if X contains arbitrarily large generalized stars then X admits an exactly
Devaney chaotic map with arbitrarily small entropy. A possible application for
further study of the best lower bounds of topological entropies of
transitive/Devaney chaotic maps is indicated.
