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首页> 外文期刊>Journal d'analyse mathematique >FINITE ENTROPY ACTIONS OF FREE GROUPS, RIGIDITY OF STABILIZERS, AND A HOWE-MOORE TYPE PHENOMENON
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FINITE ENTROPY ACTIONS OF FREE GROUPS, RIGIDITY OF STABILIZERS, AND A HOWE-MOORE TYPE PHENOMENON

机译：自由群的有限熵作用，稳定器的刚度和霍夫-莫尔型现象

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摘要

We study a notion of entropy, called f-invariant entropy, introduced by Lewis Bowen for probability measure preserving actions of finitely generated free groups. In the degenerate case, the f-invariant entropy is -infinity. In this paper, we investigate the qualitative consequences of an action having finite f-invariant entropy. We find three main properties of such actions. First, the stabilizers occurring in factors of such actions are highly restricted. Specifically, the stabilizer of almost every point must be either trivial or of finite index. Second, such actions are very chaotic in the sense that when the space is not essentially countable, every non-identity group element acts with infinite Kolmogorov-Sinai entropy. Finally, we show that such actions display behavior reminiscent of the Howe-Moore property. Specifically, if the action is ergodic, there exists an integer n such that for every non-trivial normal subgroup K, the number of K-ergodic components is at most n. Our results are based on a new formula for f-invariant entropy.
机译：我们研究了一种由刘易斯·鲍文（Lewis Bowen）引入的称为f不变熵的熵概念，用于概率度量保持有限生成的自由组的作用。在简并的情况下，f不变熵为-无限大。在本文中，我们研究了具有有限f不变熵的行为的定性结果。我们发现此类动作的三个主要属性。首先，高度限制了由于这种作用而产生的稳定剂。具体而言，几乎每个点的稳定器都必须是微不足道的或有限指数的。其次，从空间上不可数的角度来看，每个动作都具有无限的Kolmogorov-Sinai熵，因此这种动作非常混乱。最后，我们证明了这些动作显示的行为让人联想到Howe-Moore属性。具体地，如果动作是遍历的，则存在整数n，使得对于每个非平凡的正常子组K，遍历的K分量的数量最多为n。我们的结果基于f不变熵的新公式。

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