Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space

IF 1.2 4区 数学 Q2 MATHEMATICS, APPLIED
Zhicheng Tong, Yong Li
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引用次数: 0

Abstract

As is well known the classical Hartman–Grobman theorem states that a $C^1$ mapping can be $C^0$ linearized near its hyperbolic fixed point in $\mathbb{R}^n$. However, it is quite nontrivial to guarantee the local homeomorphism to be differentiable. Recently, the regularity assumption on derivative of the mapping has been weakened to Hölder’s type, significantly improving the work of $C^\infty$, but still unknown for only differentiable case. We will try to touch this question in this paper. Without Hölder’s type, we first consider the existence and regularity of weak-stable manifolds for homeomorphisms with contraction in a Banach space, and further study linearization of mappings near hyperbolic fixed points. More precisely, we propose an Integrability Condition for regularity on linearization which is proved to be sharp, and establish a differentiable Hartman–Grobman theorem via modulus of continuity in a general Banach space. Thus we provide an almost complete answer to the question mentioned above.
通过连续性模数的可微哈特曼-格罗布曼定理一般巴拿赫空间线性化的尖锐结果
众所周知,经典的哈特曼-格罗布曼定理指出,$C^1$映射可以在$\mathbb{R}^n$的双曲定点附近被$C^0$线性化。然而,要保证局部同构是可微分的并不容易。最近,关于映射导数的正则性假设被弱化为荷尔德类型,大大改进了 $C^\infty$ 的工作,但对于仅可微分的情况仍是未知数。我们将在本文中尝试探讨这个问题。在不考虑荷尔德类型的情况下,我们首先考虑巴拿赫空间中同构收缩的弱稳定流形的存在性和正则性,并进一步研究双曲定点附近映射的线性化。更确切地说,我们提出了线性化正则性的积分性条件(Integrability Condition),并证明该条件是尖锐的,同时通过一般巴拿赫空间中的连续性模数建立了可微哈特曼-格罗布曼定理。因此,我们几乎完整地回答了上述问题。
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来源期刊
CiteScore
1.70
自引率
10.00%
发文量
59
审稿时长
6 months
期刊介绍: Covers modern applied mathematics in the fields of modeling, applied and stochastic analyses and numerical computations—on problems that arise in physical, biological, engineering, and financial applications. The journal publishes high-quality, original research articles, reviews, and expository papers.
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