Mather $\beta$-函数的射流刚性与解析标准映射中关于势的全纯复化KAM曲线
Jet rigidity of Mather's $β$-function and complexified KAM curves holomorphic in potential for analytic standard maps
- Institute of Science and Technology Austria (ISTA)（奥地利科学技术学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明解析标准映射在KAM域中，单个代数丢番图旋转数处Mather $\beta$-函数的射流局部决定势，且保持射流的有限傅里叶支撑实解析形变必为常数，结果非微扰。
AI中文摘要：
Mather的$\beta$-函数将每个旋转数关联到扭转映射轨道所携带的最小平均作用量。对于KAM机制中的解析标准映射，我们证明以下结论。对于满足自然对称条件的任意$d \ge 1$个扰动方向族，在单个代数丢番图旋转数处$\beta$-函数的射流局部地决定KAM域中一个开且普遍（从而稠密）的基势集合的势。特别地，我们得到：对于KAM域中一个剩余且普遍的偶势集合，任何具有有限傅里叶支撑且保持固定代数丢番图旋转数处$\beta$-函数完整射流的实解析形变都是常数。这些结果在KAM域内是非微扰的。作为一个具有独立意义的中间结果，我们建立了一个KAM定理，给出不变曲线（从而$\beta$-函数）对复全纯势以及边界包含实丢番图数的复化旋转数域上的联合$C^\infty$--全纯依赖性。
英文摘要：
Mather's $β$-function associates with each rotation number the least average action carried by an orbit of a twist map. For analytic standard maps in the KAM regime, we prove the following. For any family of $d \ge 1$ directions of perturbation satisfying a natural symmetry condition, the jet of the $β$-function at a single algebraic Diophantine rotation number locally determines the potential for an open and prevalent, hence dense, set of base potentials in the KAM domain. In particular, we obtain that for a residual and prevalent set of even potentials in the KAM domain, every real analytic deformation with finite Fourier support that preserves the full jet of the $β$-function at a fixed algebraic Diophantine rotation number is constant. These results are non perturbative within the KAM domain. As an intermediate result of independent interest, we establish a KAM theorem giving joint $C^\infty$--holomorphic dependence of the invariant curves, and hence of the $β$-function, on complex holomorphic potentials and on a domain of complexified rotation numbers whose boundary contains real Diophantine numbers.
