We discuss the finiteness of the topological entropy of continuous endomorphims for some classes of locally compact groups. Firstly, we focus on the abelian case, imposing the condition of being compactly generated, and note an interesting behaviour of slender groups. Secondly, we remove the condition of being abelian and consider nilpotent periodic locally compact p-groups (p prime), reducing the computations to the case of Sylow p-subgroups. Finally, we investigate locally compact Heisenberg p-groups \(\mathbb{H}_n (\mathbb{Q}_ p )\) on the field \(\mathbb{Q}_ p \) of the p-adic rationals with n arbitrary positive integer.

中文翻译：

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关于小拓扑熵的局部紧群

我们讨论了某些类局部紧群的连续内态的拓扑熵的有限性。首先，我们关注阿贝尔情况，施加紧生成的条件，并注意到细长群的一个有趣的行为。其次，我们去除阿贝尔条件并考虑幂零周期性局部紧p群（p素数），将计算减少到Sylow p子群的情况。最后，我们研究p进有理数域\(\mathbb{Q}_ p \)上的局部紧致海森堡p群\(\mathbb{H}_n (\mathbb{Q}_ p )\) n任意正整数。