A note on periodic points of dendrite maps and their induced maps

Let X be a compact metric space and 2X 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 2f:2X→2X. Méndez (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 2F is dense in 2X. This answers in the negative Méndez's conjecture.