The transitivity of induced maps

Abstract

For a metric continuum X, we consider the hyperspaces 2X and C (X) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map f : X → X we investigate the transitivity of the induced maps 2f : 2X → 2X and C (f) : C (X) → C (X). Among other results, we show that if X is a dendrite or a continuum of type λ and f : X → X is a map, then C (f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map f : X → X such that 2f and C (f) are transitive. © 2008 Elsevier B.V. All rights reserved.