Rodrigo Hernández-Gutiérrez

Uniqueness of hyperspaces for Peano continua

For a metric continuum X and a positive integer n, let Cn(X) be the hyperspace of nonempty closed subsets of X with at most n components. We say that X has unique hyperspace Cn(X) provided that, if Y is a continuum and Cn(X) is homeomorphic to Cn(Y ), then X is homeomorphic to Y . In this paper we study which Peano continua X have a unique hyperspace Cn(X). We find some sufficient and also some necessary conditions for a Peano continuum X to have unique hyperspace Cn(X). Our results generalize all the previously known results on this subject. We also give some significant examples.

PERIODIC POINTS AND TRANSITIVITY ON DENDRITES

We study relations between transitivity, mixing and periodic points on dendrites. We prove that when there is a point with dense orbit which is not an endpoint, then periodic points are dense and there is a terminal periodic decomposition (we provide an example of a dynamical system on a dendrite with dense endpoints satisfying this assumption). We also show that it may happen that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It may also happen that dynamical system is transitive but there is a unique periodic point, which in fact is the unique xed point. We also prove that on almost meshed-continua (a class of continua containing topological graphs and dendrites with closed or countable set of endpoints), periodic points are dense if and only if they are dense for the map induced on the hyperspace of all nonempty compact subsets.