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 γf:γX→γX of f such that dimγX=dimX, ClγXPi(f)=Pi(γf) and dimPi(f)=dimPi(γf) for each i∈N, where Pi(f)={x∈X|fi(x)=x}(=Fix(fi)). This is the affirmative answer to Kato (2013) [9, Problem 3.7].