Let $f: X\longrightarrow X$ be a map of a continuum. In this paper we examine the following dynamical conditions on $f$: (1) $f$ is continuum-wise fully expansive; (2) $f$ is weakly continuum-wise fully expansive; (3) $f$ is mixing; (4) $f$ is weakly mixing. We first show that (1) implies (2), (2) implies (3) and (3) implies (4). Then we investigate what topological conditions will force the reverse implications to hold and give examples of when the reverse conditions do not hold. In particular, a map of the universal dendrite is given that is weakly mixing but not mixing.
Abstract. It is shown that if X is a chainable continuum and h : X −→ X is a homeomorphism such that the topological entropy of h is greater than 0, then X must contain an indecomposable subcontinuum. This answers a question of Barge.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.