复半单李群\(G\)的Anosov子群作用下电流的等分布
Equidistribution of currents under Anosov group actions
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中文总结 AI 辅助
研究复半单李群\(G\)的Anosov子群作用下旗流形上电流行为，通过类似构造对电流求平均，在一定条件下极限电流是吉布斯电流，还证明了光滑形式电流的等分布结果及群作用悬链流的公理\(A\)性质。
中文摘要 AI 辅助
我们研究复半单李群\(G\)的Anosov子群作用下旗流形上电流的行为。给定全旗流形\(F = G/B\)上二维\((k, k)\)的电流\(T\)，通过类似于构造Patterson - Sullivan测度的方法在群作用下对其求平均。我们表明在某些一般性条件下（对于子流形上积分电流的情况），极限电流是吉布斯电流，即由关于吉布斯测度在基于群的极限点的舒伯特簇上适当倍数的积分电流所定义的旗极限集上的积分给出。我们证明了对于\(F\)上由光滑形式定义的电流，同样的等分布结果（但没有任何一般性假设）。我们还证明了群作用在\(F\)上的悬链流的一种公理\(A\)性质形式。
英文摘要
We study the behavior of currents on flag-manifolds under actions of Anosov subgroups of complex semisimple Lie groups G. Given a current T of bidimension (k, k) on the full flag-manifold F = G/B, we average it under the group action via a construction analogous to the construction of Patterson-Sullivan measures. We show that under certain genericity conditions (in the case of currents of integration over subvarieties), the limiting current is a Gibbs current, i.e. is given by integration (with respect to a Gibbs measure) over the flag-limit set of suitable multiples of currents of integration along Schubert varieties based at the limit points of the group. We prove that the same equidistribution result (but without any genericity assumptions) for currents defined by smooth forms on F. We also prove a form of Axiom A property for the suspension flow of the group action on F.
