Corrections to ‘Invariant measures for higher-rank hyperbolic abelian actions’
Corrections to ‘Invariant measures for higher-rank hyperbolic abelian actions’
复制标题
对“高阶双曲阿贝尔动作的不变测度”的更正
DOI：
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复制
发表时间：
1998
影响因子：
0.9
通讯作者：
R. Spatzier
中科院分区：
文献类型：
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作者：
A. Katok;R. Spatzier
文[2]中定理5.1和7.1的证明存在一个缺口。我们将在下面说明如何在这些定理及其推论中的一些适当的附加假设下封闭它。我们将始终采用[2]的符号。特别地，$\mu$是一个测度不变的，并且在$R^k$-作用$\alpha$下遍历。让我们先解释一下差距。这两个定理证明了建立一个二分法的条件措施$\mu$沿着适当的稳定流形的交集。它们要么是原子的，要么是在适当的平移或幂幺子群U下不变的。原子性最终导致熵为零。不变性的条件措施显示不变性的$\mu$下$U$。然后，我们声称，$\mu$是代数使用，分别，唯一遍历的平移子群的合理subtorus或拉特纳定理（参见。[2，引理5.7]）。然而，这个结论只适用于$\mu$的$U$-遍历分量，它可能不等于$\mu$。事实上，在环面情况下，$R^k$-作用可能有一个零熵因子，使得沿纤维沿着的条件测度是沿有理子环面的叶理沿着的Haar测度。由于零熵不变测度还没有被分类，我们不能得出总测度$\mu$的代数性的结论。在湍流情况下，零熵因子的存在恰恰是我们方法的障碍。外尔室流动的情况有些不同，因为测量的“哈尔”方向可能不可积。在这种情况下，我们需要使用来自环境李群的半单性的额外信息来得到下面给出的定理7.1的版本。
The proofs of Theorems 5.1 and 7.1 of [2] contain a gap. We will show below how to close it under some suitable additional assumptions in these theorems and their corollaries. We will assume the notation of [2] throughout. In particular, $\mu$ is a measure invariant and ergodic under an $R^k$-action $\alpha$. Let us first explain the gap. Both theorems are proved by establishing a dichotomy for the conditional measures of $\mu$ along the intersection of suitable stable manifolds. They were either atomic or invariant under suitable translation or unipotent subgroups $U$. Atomicity eventually led to zero entropy. Invariance of the conditional measures showed invariance of $\mu$ under $U$. We then claimed that $\mu$ was algebraic using, respectively, unique ergodicity of the translation subgroup on a rational subtorus or Ratner's theorem (cf. [2, Lemma 5.7]). This conclusion, however, only holds for the $U$-ergodic components of $\mu$ which may not equal $\mu$. In fact, in the toral case, the $R^k$-action may have a zero-entropy factor such that the conditional measures along the fibers are Haar measures along a foliation by rational subtori. Since invariant measures with zero entropy have not been classified, we cannot conclude algebraicity of the total measure $\mu$ at this time. In the toral case, the existence of zero entropy factors turns out to be precisely the obstruction to our methods. The case of Weyl chamber flows is somewhat different as the ‘Haar’ direction of the measure may not be integrable. In this case, we need to use additional information coming from the semisimplicity of the ambient Lie group to arrive at the versions of Theorem 7.1 presented below.