发表机构
ETH Zürich; Georgia Institute of Technology（苏黎世联邦理工学院; 佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明共振条件下解析映射稳定不变流形与线性共轭的存在性，引入对数多项式展开并给出正则性结果，推广了Hartman猜想。
AI 中文摘要
我们证明了在存在共振的情况下，解析映射在双曲不动点附近的$C^{r,1-}$-正则稳定不变流形和线性共轭的存在性。正则性指数$r$依赖于最小共振指标，而Hölder指数可以任意接近$1$，即对任意$\varepsilon>0$，可取$1-\varepsilon$。作为推论，我们获得了具有半单线性化的解析系统的Hartman猜想的稳定版本：在完全稳定情形下，对每个$\varepsilon>0$，局部共轭可以选取为$C^{1,1-\varepsilon}$。我们的方法引入了一类基于对数多项式的新函数展开，这使得不变方程能够显式且算法化地求解到任意阶。存在性结果通过在合适的Banach空间中的不动点论证获得。我们进一步给出了几个解析例子，既说明了定理的适用性，也展示了其假设的必要性。
英文摘要
We prove the existence of $C^{r,1-}$-regular stable invariant manifolds and linear conjugacies for analytic maps near a hyperbolic fixed point in the presence of resonances. The regularity exponent $r$ depends on the minimal resonant index, while the Hölder exponent can be chosen arbitrarily close to $1$, i.e., $1-\varepsilon$ for any $\varepsilon>0$. As a consequence, we obtain the stable version of the Hartman conjecture for analytic systems with semisimple linearization: in the fully stable case the local conjugacy can be chosen $C^{1,1-\varepsilon}$ for every $\varepsilon>0$.\\ Our approach introduces a new class of functional expansions based on logarithmic polynomials, which enables the invariance equation to be solved explicitly and algorithmically to arbitrary order. The existence results are obtained via a fixed-point argument in a suitable Banach space. We further present several analytic examples that both illustrate the applicability of the theorem while demonstrating the necessity of its assumptions.
