AI 中文总结
本文通过复相切几何刻画了平面上 Siegel 奇点的解析线性化，提出了圆球上的强准则及多项式与对数边界判据，并研究了相切环面的横截动力学与刚性。
AI 中文摘要
我们利用与收缩的严格伪凸超曲面的复相切，给出了 $\C^2$ 中解析可线性化的非退化 Siegel 叶层的几何刻画。我们的主要结果处理一个固定的有界光滑严格伪凸 Reinhardt 域：在与线性部分的特征方向相容的坐标下，沿收缩序列的二维相切轨迹刻画了解析 Siegel 线性化。对于圆球，我们证明了更强的结论：不需要预先将欧氏坐标轴与特征方向对齐；相切假设本身迫使线性部分成为具有负实特征值比率的酉对角化矩阵。我们还建立了圆球上的多项式 Shilov 边界准则，以及适用于任意光滑严格伪凸域的、嵌入闭双圆盘位于伪凸侧时的对数单边界准则。最后，我们研究了光滑相切环面的横截全纯动力学，包括周期情形下的刚性定理。
英文摘要
We give geometric characterizations of analytically linearizable nondegenerate Siegel foliations in $\C^2$ using complex tangencies with shrinking strictly pseudoconvex hypersurfaces. Our principal result treats a fixed bounded smoothly bounded strictly pseudoconvex Reinhardt domain: in coordinates compatible with the eigendirections of the linear part, two-dimensional tangency loci along a shrinking sequence characterize analytic Siegel linearizability. For round spheres we prove a stronger statement: no a priori alignment of the Euclidean coordinate axes with the eigendirections is required; the tangency hypothesis itself forces the linear part to be unitarily diagonalizable with negative real eigenvalue ratio. We also establish a polynomial Shilov-boundary criterion on a round sphere and a logarithmic one-boundary criterion valid for arbitrary smooth strictly pseudoconvex domains, provided the embedded closed bidisc lies on the pseudoconvex side. Finally, we study the transverse-holomorphic dynamics of smooth tangency tori, including a rigidity theorem in the periodic case.
