关键词: Topological entropy, metric entropy, non-compact metric space, one point compactification

Abstract:

Let T: X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X*=X ∪{x*} the one point compactification of X and T*: X*→X* the homeomorphism on X* satisfying T*|_X=T and T*x*=x*. We show that their topological
entropies satisfy h_d(T, X)≥h(T*, X*) if X is locally compact. We also give a note on Katok's measure theoretic entropy on a compact metric space.

Key words: Topological entropy, metric entropy, non-compact metric space, one point compactification