Orbit Complexity and Entropy for Group Endomorphisms

We consider the pointwise inequality (orbit complexity ≤ topological entropy), known in the case of computable maps and computable metric spaces, for endomorphisms of locally compact groups with an arbitrary upper semicomputable distance. Weaker conditions on the effectiveness of the product and metric neighbourhoods are observed which, in Rn, are transferred to a norm-induced metric and used to prove a version of the inequality on locally compact abelian groups.