The Ergodic Theory of Lattice Subgroups
The Ergodic Theory of Lattice Subgroups
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格子群的遍历理论
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
A. Nevo
中科院分区：
文献类型：
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作者：
A. Gorodnik;A. Nevo
我们证明了平均和逐点遍历定理一般家庭的平均半单代数（或S-代数）群G，连同一个明确的收敛速度时，行动有一个频谱间隙。给定G中的任意格，我们使用G的遍历定理来解决G中一般域的格点计数问题，并证明了格的任意测度保持作用的平均遍历定理和逐点遍历定理，以及谱间隙存在时的显式收敛速度。
我们还证明了在任意等距作用的格的等分布定理。
为了证明，我们开发了一种通用的方法来推导遍历定理的行动，一个局部紧群G，和一个格子群伽玛，提供一定的自然谱，几何和正则性条件满足的G组，格伽玛，和域的平均值的支持。特别是，我们建立的一般原则，在这些条件下，一个家庭的平均值的一个定量平均遍历定理产生了一个定量的解决方案的格子点计数问题在他们的支持。
我们证明了新的显式误差项，我们得到了各种例子。
We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present.
We also prove an equidistribution theorem in arbitrary isometric actions of the lattice.
For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditions a quantitative mean ergodic theorem for a family of averages gives rise to a quantitative solution of the lattice point counting problem in their supports.
We demonstrate the new explicit error terms that we obtain by a variety of examples.