Entropy and exact Devaney chaos on totally regular continua (Q379485)

Let \((X,f)\) be a discrete dynamical system, where \(X\) is an infinite compact metric space and \(f : X \to X\) is a continuous map. Recall that \((X,f)\) is said to be Devaney chaotic (resp. totally Devaney chaotic, exactly Devaney chaotic) if \((X,f)\) is transitive (resp. totally transitive, exact) and has a dense set of periodic points.NEWLINENEWLINEThe main goal of the paper is to establish the following result.NEWLINENEWLINETheorem. Every non-degenerate totally regular continuum \(X\) admits an exactly Devaney chaotic map with finite positive entropy. If, in addition, \(X\) contains arbitrarily large generalized stars or \(X\) contains a free arc which does not disconnect \(X\), then \(X\) admitsNEWLINENEWLINE- a Devaney chaotic map which is not totally Devaney chaotic, andNEWLINENEWLINE- an exactly Devaney chaotic map,NEWLINENEWLINEboth with arbitrarily small positive entropy.