AI 中文总结
研究环面\(\mathbb{T}^d\)上接近常向量场\(\omega\)的向量场的解析线性化问题，通过提供几何框架，证明相关算术条件等同于Brjuno条件，得到新定量线性化定理及明确估计，控制共轭解析性损失。
AI 中文摘要
我们考虑环面\(\mathbb{T}^d\)上接近常向量场\(\omega\)的向量场的经典解析线性化问题。我们有两个目标。首先，我们提供一个几何框架，其中控制解析线性化的算术条件自然地源于与\(\omega\)相关的幺模格在\(\operatorname{SL}(d,\mathbb{Z})\backslash \operatorname{SL}(d,\mathbb{R})\)上的对角流下的轨道。在这个框架内，一个可和性条件作为收敛的自然标准出现。我们证明它等同于线性形式的Brjuno条件的几种经典表述，包括那些涉及最佳逼近向量和对角流切换时间的表述。作为副产品，我们得到一个具有完全明确估计的新的定量线性化定理。特别地，共轭的解析性损失由一个Brjuno函数控制。
英文摘要
We consider the classical analytic linearization problem for vector fields on the torus $\mathbb{T}^d$ close to a constant vector field $ω$. Our goals are twofold. First, we provide a geometric framework in which the arithmetic condition governing analytic linearization arises naturally from the orbit of a unimodular lattice associated with $ω$ under a diagonal flow on $\operatorname{SL}(d,\mathbb{Z})\backslash \operatorname{SL}(d,\mathbb{R})$. Within this framework, a summability condition emerges as the natural criterion for convergence. We prove that it is equivalent to several classical formulations of the Brjuno condition for linear forms, including those involving best approximation vectors and switching times of the diagonal flow. As a byproduct, we obtain a new quantitative linearization theorem with fully explicit estimates. In particular, the loss of analyticity of the conjugacy is controlled by a Brjuno function.
Comments35 pages
