Liouville测度在凸组合下的不稳定性
Non-stability of Liouville measures under convex combinations
AI总结：
该研究证明了Liouville测度在凸组合下不闭合，并给出了相关群的例子。
摘要：
对于每一个非超FC-中心的可数可换群和每一个k≥2，我们提供了一组对称的、完全支持的概率测度，使得它们的凸组合是非Liouville的（即存在非常数有界的调和函数，等价地，Poisson边界是非平凡的）当且仅当至少k个测度出现在凸组合中。特别地，我们的结果表明，Liouville测度的集合在凸组合下不闭合，这回答了Kaimanovich提出的问题。我们还提供了一个类似的结论，前提是那些非超FC-中心的可数群具有性质：每个对称的、有限支持的概率测度都是Liouville的。这些群是唯一已知的非平凡例子，它们的可数群允许具有有限熵的Liouville测度。例子包括Z上的灯塔群和Z²上的灯塔群，以及无限对称群。
英文摘要：
For every non-hyper-FC-central countable amenable group and every $k\geq 2$, we provide a sequence of symmetric, fully supported probability measures such that their convex combination is non-Liouville (that is it admits a non-constant bounded harmonic function, equivalently, the Poisson boundary is non-trivial) if and only if at least $k$ of them appear in the convex combination. Particularly, our result implies that the set of Liouville measures is not closed under convex combination, which answers a question of Kaimanovich. We also provide a similar result under the additional assumption of finite entropy for those non-hyper-FC-central countable groups with the property that every symmetric, finitely supported probability measure is Liouville. These groups are the only known non-trivial examples of countable groups that admit Liouville measures with finite entropy. Examples include the lamplighter group over $\mathbb{Z}$ and $\mathbb{Z}^2$, and the infinite symmetric group of finite permutations on $\mathbb{Z}$.
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