A note on metric compactifications and periodic points of maps

In this note, we investigate metric compactifications preserving some properties of periodic points of maps. In particular, we prove that if f : X → X is any map of a locally compact and finite-dimensional separable metric space X, then there exist a metric compactification γX of X and an extension γ f : γ X → γ X of f such that dim γ X = dim X , Cl γ X P i ( f ) = P i ( γ f ) and dim P i ( f ) = dim P i ( γ f ) for each i ∈ N , where P i ( f ) = { x ∈ X | f i ( x ) = x } ( = Fix ( f i ) ) . This is the affirmative answer to Kato (2013) [9, Problem 3.7].