We study the locally compact abelian groups in the class ${\mathfrak E_{ \lt \infty }}$ , 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 ${\mathfrak E_{ \lt \infty }}$ , 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.

中文翻译：

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局部紧致阿贝尔群的拓扑熵的有限性

我们研究班级中的局部紧致阿贝尔群${\mathfrak E_{ \lt \infty }}$，即仅具有有限拓扑熵的连续自同态，并且在其子类中$\mathfrak E_0$，即具有所有拓扑熵消失的连续自同态。我们讨论将问题简化为周期性局部紧致阿贝尔群的情况，然后到局部紧致阿贝尔群p-团体。我们证明了局部紧致阿贝尔p- 有限秩的群属于${\mathfrak E_{ \lt \infty }}$，并且那些属于$\mathfrak E_0$正是具有离散最大可分子群的那些。此外，局部紧致阿贝尔自同态的拓扑熵p- 有限秩组与其尺度的对数一致。论文的主干是局部紧致阿贝尔群的连续自同态的加法定理。论文中建立了各种版本的加法定理并用于主要结果的证明，但其在一般情况下的有效性仍然是一个悬而未决的问题。