Entropy and exact Devaney chaos on totally regular continua

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.