  We study the locally compact abelian groups in the class $\mathfrak
E_{<\infty}$, that is, having only continuous endomorphisms of finite
topological entropy, and in its subclass $\mathfrak E_0$, that is, having all
continuous endomorphisms with vanishing topological entropy. We discuss the
reduction of the problem to the case of periodic locally compact abelian
groups, and then to locally compact abelian $p$-groups.
  We show that locally compact abelian $p$-groups of finite rank belong to
$\mathfrak E_{<\infty}$, and that those of them that belong to $\mathfrak E_0$
are precisely the ones with discrete maximal divisible subgroup. Furthermore,
the topological entropy of endomorphisms of locally compact abelian $p$-groups
of finite rank coincides with the logarithm of their scale.
  The backbone of the paper is the Addition Theorem for continuous
endomorphisms of locally compact abelian groups. Various versions of the
Addition Theorem are established in the paper and used in the proofs of the
main results, but its validity in the general case remains an open problem.
