LAUSR

Abstract Let X be a compact metric space and 2 X be the hyperspace of all nonempty closed subsets of X endowed with the Hausdorff metric. It is well known that for each continuous map f : X → X , the density of periodic points of f implies the density of periodic points of the induced map 2 f : 2 X → 2 X . Mendez (2010) conjectured in [6] that the converse is true when the phase space X is a dendrite. In [8] , Ŝpitalský (2015) constructed a continuous transitive map F : X → X on a dendrite X with only two periodic points. We prove in this note that the set of periodic points of its induced map 2 F is dense in 2 X . This answers in the negative Mendez's conjecture.