可积系统上的非交换共边方程

IF 0.7 1区 数学 Q2 MATHEMATICS
Rafael de la Llave, Maria Saprykina
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引用次数: 0

摘要

我们证明了具有Banach代数$ {{\mathscr G}} $或李群的可积系统上的实解析族环的Livshits定理的一个类似。也就是说,我们考虑一个可积动力系统$ f:{{\mathscr M}} \equiv{{\mathbb T}}^d \times [-1, 1]^d\to {{\mathscr M}} $, $ f(\theta, I) = (\theta + I, I) $和一个由复参数$ {\varepsilon} $索引的实解析族共环$ \eta_{\varepsilon} : {{\mathscr M}} \to {{\mathscr G}} $在一个开放的球$ {{\mathscr E}}_\rho \subset {{\mathbb C}} $中。我们证明了如果$ \eta_{\varepsilon} $是接近恒等的,并且具有一般周期数据,即对于每个周期点$ p = f^n p $和每个$ {\varepsilon} \in {{\mathscr E}}_{\rho} $,那么存在一个实解析族的映射$ \phi_{\varepsilon}: {{\mathscr M}} \to {{\mathscr G}} $满足所有$ (\theta, I)\in {{\mathscr M}} $和$ {\varepsilon} \in {{\mathscr E}}_{\rho/2} $的协边方程$   \eta_{\varepsilon}(\theta, I) = (\phi_{\varepsilon}\circ f(\theta, I))^{-1} \cdot \phi_{\varepsilon} (\theta, I)    $。我们还证明了如果上面的协边方程$ \eta_{\varepsilon} $在$ {\varepsilon} $上具有形式幂级数意义上的解,然后它有一个解析解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Noncommutative coboundary equations over integrable systems
We prove an analog of the Livshits theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra$ {{\mathscr G}} $or a Lie group. Namely, we consider an integrable dynamical system$ f:{{\mathscr M}} \equiv{{\mathbb T}}^d \times [-1, 1]^d\to {{\mathscr M}} $,$ f(\theta, I) = (\theta + I, I) $, and a real-analytic family of cocycles$ \eta_{\varepsilon} : {{\mathscr M}} \to {{\mathscr G}} $indexed by a complex parameter$ {\varepsilon} $in an open ball$ {{\mathscr E}}_\rho \subset {{\mathbb C}} $. We show that if$ \eta_{\varepsilon} $is close to identity and has trivial periodic data, i.e.,for each periodic point$ p = f^n p $and each$ {\varepsilon} \in {{\mathscr E}}_{\rho} $, then there exists a real-analytic family of maps$ \phi_{\varepsilon}: {{\mathscr M}} \to {{\mathscr G}} $satisfying the coboundary equation$   \eta_{\varepsilon}(\theta, I) = (\phi_{\varepsilon}\circ f(\theta, I))^{-1} \cdot \phi_{\varepsilon} (\theta, I)    $for all$ (\theta, I)\in {{\mathscr M}} $and$ {\varepsilon} \in {{\mathscr E}}_{\rho/2} $.We also show that if the coboundary equation above with an analytic left-hand side$ \eta_{\varepsilon} $has a solution in the sense of formal power series in$ {\varepsilon} $, then it has an analytic solution.
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
11
审稿时长
>12 weeks
期刊介绍: The Journal of Modern Dynamics (JMD) is dedicated to publishing research articles in active and promising areas in the theory of dynamical systems with particular emphasis on the mutual interaction between dynamics and other major areas of mathematical research, including: Number theory Symplectic geometry Differential geometry Rigidity Quantum chaos Teichmüller theory Geometric group theory Harmonic analysis on manifolds. The journal is published by the American Institute of Mathematical Sciences (AIMS) with the support of the Anatole Katok Center for Dynamical Systems and Geometry at the Pennsylvania State University.
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