Certain measures associated with U-flows on compact manifolds
Certain measures associated with U-flows on compact manifolds
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DOI：
10.1007/bf01075620
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发表时间：
1970
影响因子：
0.4
通讯作者：
G. Margulis
中科院分区：
文献类型：
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作者：
G. Margulis
设紧黎曼流形W n上存在DV Anosov意义下的U-流（也称C-流）.正如Anosov所证明的那样，在W n上存在两对叶理，每一对在U流下都是不变的。这些叶理的层次分别称为收缩叶和扩张叶以及收缩球和扩张球（详见§ 1）。本文证明了在所有扩张叶上同时引入a-有限可数可加测度的可能性，使得：1）在U-流的作用下，该测度乘以一个常数; 2）标准同构集（见[7]）的测度重合（注意在[7]中，规范同构只对U-同构定义，但那里给出的定义显然扩展到U-流的情况。类似的命题也证明了其他叶理。
Let a U-flow (also called C-flow) in the sense of DV Anosov be given on a compact Riemann manifold W n. As demonstrated by Anosov, on W n there exist two pairs of foliations, each of which is invariant under the U-flow. The layers of these foliations are called contracting and expanding leaves and contracting and expanding orispheres, respectively (see § 1 for details). In the present paper we prove the theorem that it is possible on all expanding leaves to simultaneously introduce a a-finite countably-additive measure such that: 1) under the action of the U-flow this measure is multiplied by a constant; 2) the measures of the canonically-isomorphic sets (see [7]) coincide (Note that in [7] canonic isomorphism is defined only for U-diffeomorphisms, but the definition given there clearly extends to the case of U-flows.) Analogous propositions are proved for other foliations as well.