ON DENSITY OF PERIODIC POINTS FOR INDUCED HYPERSPACE MAPS

Abstract

Let X be a continuum and 2(X) be the hyperspace of all nonempty closed subsets of X endowed with the Hausdorff metric. It is known that for each continuous map f : X -> X the density of periodic points of the induced map 2(f) : 2(X) -> 2(X) implies the density of periodic points of the base map f provided that X is a graph. In this note we describe a continuum X and a continuous map f : X -> X where the density of periodic points of the induced map 2f does not imply the density of periodic points of the base map f. Also we study a condition of f equivalent to the density of periodic points of 2(f)