代数二分法的Hartman-Grobman定理
A Hartman-Grobman theorem for algebraic dichotomies
AI总结：
本文针对代数二分法这一指数二分法的推广概念，提出了相应的Hartman-Grobman线性化定理，推广了Palmer定理，证明了线性化同胚的Hölder连续逆性质，并对比了两类二分法的积分估计差异。
摘要：
代数二分法是指数二分法的一种推广（Lin，JDE2009）。本文在线性系统具有代数二分性的假设下，给出了一个版本的Hartman-Grobman线性化定理，该定理推广了Palmer线性化定理。此外，我们证明了该线性化定理中的同胚映射（且具有Hölder连续的逆）。与指数二分法相比，代数二分法更为复杂。指数二分法会产生收敛的积分估计$\int_{-\infty}^{t}e^{-α(t-s)}ds$和$\int_{t}^{+\infty}e^{-α(s-t)}ds$，而代数二分法会引出$\int_{-\infty}^{t}\left(\frac{μ(t)}{μ(s)}\right)^{-α}ds$或$\int_{t}^{+\infty}\left(\frac{μ(s)}{μ(t)}\right)^{-α}ds$，其黎曼意义下的收敛性尚不明确。
英文摘要：
Algebraic dichotomy is a generalization of an exponential dichotomy (Lin, JDE2009). This paper gives a version of Hartman-Grobman linearization theorem assuming that linear system admits an algebraic dichotomy, which generalizes the Palmer's linearization theorem. Besides, we prove that the homeomorphism in the linearization theorem (and has a Hölder continuous inverse). Comparing with exponential dichotomy, algebraic dichotomy is more complicate. The exponential dichotomy leads to the estimates $\int_{-\infty}^{t}e^{-α(t-s)}ds$ and $\int_{t}^{+\infty}e^{-α(s-t)}ds$ which are convergent. However, the algebraic dichotomy will leads us to $\int_{-\infty}^{t}\left(\frac{μ(t)}{μ(s)}\right)^{-α}ds$ or $\int_{t}^{+\infty}\left(\frac{μ(s)}{μ(t)}\right)^{-α}ds$, whose the convergence is unknown in the sense of Riemann.
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