For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and connected Lie group locally isomorphic to ${ \rm Aut}(\mathbb{B}_k)$. Further, the limit set is a smooth submanifold diffeomorphic to the sphere of dimension $2k-1$. As applications we prove a new finite jet determination theorem and a Tits alternative theorem. The geometry of the Kobayashi metric plays an important role in the paper.

中文翻译：

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有界域的自同构群和极限集 I：有限类型情况

对于有限类型的有界伪凸域，我们给出了自同构群的精确描述：如果自同构群的轨道在边界的至少两个不同点上累积，那么自同构群的分量是有限的，并且几乎是紧群和连通李群​​局部同构于 ${ \rm Aut}(\mathbb{B}_k)$。此外，极限集是一个光滑的子流形，对维度为 $2k-1$ 的球体微分。作为应用，我们证明了一个新的有限射流确定定理和一个 Tits 替代定理。Kobayashi 度量的几何形状在本文中起着重要作用。