论文标题
稳定固定点和周期性轨道的Koopman本征函数的通用特性
Generic Properties of Koopman Eigenfunctions for Stable Fixed Points and Periodic Orbits
论文作者
论文摘要
我们最近的工作确定了$ \ Mathcal {C}^{k，α} _ {\ text {loc}} $全球定义的线性化的半jugacies的$ \ Mathcal {c}^{k，α} _ {c}^{k，α} _ {c}^{k，α} _ {c} $ $ \ mathcal {c}^1 $全球吸引了超级固定点或周期性的kvalheim（kvalheim）。应用程序包括（i）改进，例如唯一性语句，用于sternberg线性化和floquet正常形式定理； （ii）关于“应用的Koopmanism”文献中出现的各种数量的存在，独特性，分类和收敛的结果，例如主要特征性，等值函数和拉普拉斯平均值。 在这项工作中，我们介绍了其中一些结果，重点是Koopmanism的应用，并考虑了它们的广泛适用性。特别是，我们表明，对于“几乎所有” $ \ mathcal {c}^\ infty $流动，具有全球吸引的双曲线固定点或周期性轨道，$ \ Mathcal {c}^\ infty $ koopman eigenfunctions可以完全分类，从而将结果分类为已知的分析系统。对于此类系统，每个$ \ MATHCAL {C}^\ infty $ eigenFunction都由其特征值模量标量乘法唯一决定。
Our recent work established existence and uniqueness results for $\mathcal{C}^{k,α}_{\text{loc}}$ globally defined linearizing semiconjugacies for $\mathcal{C}^1$ flows having a globally attracting hyperbolic fixed point or periodic orbit (Kvalheim and Revzen, 2019). Applications include (i) improvements, such as uniqueness statements, for the Sternberg linearization and Floquet normal form theorems; (ii) results concerning the existence, uniqueness, classification, and convergence of various quantities appearing in the "applied Koopmanism" literature, such as principal eigenfunctions, isostables, and Laplace averages. In this work we give an exposition of some of these results, with an emphasis on the Koopmanism applications, and consider their broadness of applicability. In particular we show that, for "almost all" $\mathcal{C}^\infty$ flows having a globally attracting hyperbolic fixed point or periodic orbit, the $\mathcal{C}^\infty$ Koopman eigenfunctions can be completely classified, generalizing a result known for analytic systems. For such systems, every $\mathcal{C}^\infty$ eigenfunction is uniquely determined by its eigenvalue modulo scalar multiplication.
