For convex domains with $C^{1,\epsilon }$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is the almost direct product of a compact group and a non-compact connected simple Lie group with real rank one and finite center. In this case, we also show the limit set is homeomorphic to a sphere and prove a gap theorem: either the domain is biholomorphic to the unit ball (and the limit set is the entire boundary) or the limit set has co-dimension at least two in the boundary.

中文翻译：

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有界域的自同构群和极限集 II：凸情况

对于凸域 $C^{1,\epsilon }$边界我们给出了自同构群的精确描述：如果自同构群的轨道累积在边界的至少两个不同的封闭复面上，那么自同构群的分量是有限的，而恒等式的连通分量是几乎直接的一个紧群和一个实秩为一且中心有限的非紧连通单李群的乘积。在这种情况下，我们还展示了极限集与球体同胚并证明了一个间隙定理：域对单位球是双全纯的（并且极限集是整个边界）或者极限集至少具有共维两个在边界。