发表机构
School of Mathematics, Jilin University; Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University（吉林大学数学学院; 东北师范大学数学与交叉科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完全刻画了平面微分同胚的局部$C^1$线性化条件，给出三个最优Dini型阈值，并确定双曲情形下线性化共轭的最优正则性。
AI 中文摘要
我们建立了平面微分同胚局部$C^1$线性化的完全刻画。对于每个给定的双曲实若尔当标准形，我们获得了导数连续模对局部$C^1$线性化的必要且充分的可积性条件。这些判据归结为三个典型阈值：经典的、多项式加权的和平方对数加权的Dini条件。每个阈值都是最优的：每当相应条件不满足时，我们构造一个具有相同线性部分的微分同胚，它不是局部$C^1$可线性化的。对于每个非双曲线性部分，显式的多项式反例表明，仅光滑性甚至不能保证局部拓扑线性化。在$C^1$存在性之外，我们确定了每个双曲情形下线性化共轭导数的最优正则性，并通过反例建立了最优性。
英文摘要
We establish a complete characterization of local $C^1$ linearizability for planar diffeomorphisms. For every prescribed hyperbolic real Jordan form, we obtain necessary and sufficient integrability conditions on the modulus of continuity of the derivative for local $C^1$ linearizability. These criteria reduce to three canonical thresholds: the classical, polynomially weighted, and squared-logarithmically weighted Dini conditions. Each threshold is optimal: whenever the corresponding condition fails, we construct a diffeomorphism with the identical linear part that is not locally $C^1$ linearizable. For every non-hyperbolic linear part, explicit polynomial counterexamples show that smoothness alone cannot guarantee even local topological linearization. Beyond $C^1$ existence, we determine the optimal regularity for the derivative of the linearizing conjugacy in every hyperbolic case, with the optimality established by counterexamples.
Comments57 pages, 5 tables, and 12 figures. Comments are welcome!
