METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

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.