关于法双曲不变流形定理中扰动的$C^1$范数阈值的尖锐性——以玩具模型为视角
On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective
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中文总结 AI 辅助
本文以玩具模型为视角，针对类标准耗散扭转映射，证明法双曲不变流形定理中扰动的$C^1$范数阈值$(1-\text{sqrt}(\text{lambda}))^2$是尖锐的，明确了该阈值上下不变图的保持特性，揭示了临界阈值现象。
中文摘要 AI 辅助
经典法双曲不变流形定理断言，$C^1$法双曲不变流形在$C^1$小扰动下保持不变。对于一类类标准耗散扭转映射，本文证明扰动的$C^1$范数阈值$(1-\text{sqrt}(\text{lambda}))^2$是尖锐的：存在$C^\text{infty}$扰动$\text{phi}$，其$\text{||phi||}_{C^1}=(1-\text{sqrt}(\text{lambda}))^2$，使得该映射保持唯一的不变图，但该图存在不可微点；而当$\text{||phi||}_{C^1}<(1-\text{sqrt}(\text{lambda}))^2$时，$C^1$法双曲不变流形保持不变，其中$\text{lambda}$为映射的雅可比行列式，这为耗散扭转映射中不变图的保持提供了临界阈值现象。
英文摘要
The classical normally hyperbolic invariant manifold theorem asserts that a \(C^1\) normally hyperbolic invariant manifold persists under \(C^1\) small perturbations. For a family of standard-like dissipative twist maps, we show that the threshold \((1-\sqrtλ)^2\) for the \(C^1\)-norm of the perturbation is sharp: there exists a $C^\infty$ perturbation \(ϕ\) with \(\|ϕ\|_{C^1} = (1-\sqrtλ)^2\) such that the map preserves a unique invariant graph, but this graph possesses non-differentiable points. On the other hand, whenever \(\|ϕ\|_{C^1} < (1-\sqrtλ)^2\), the \(C^1\) normally hyperbolic invariant manifold persists, where \(λ\) denotes the Jacobian determinant of the map. This provides a critical threshold phenomenon for the persistence of invariant graphs in dissipative twist maps.
