Communications in Mathematical Sciences

Volume 22 (2024)

Number 5

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

Pages: 1361 – 1396

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n5.a8

Authors

Zhicheng Tong (College of Mathematics, Jilin University, Changchun, China)

Yong Li (Institute of Mathematics, Jilin University, Changchun, China; School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University,Changchun, Jilin, China)

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.

Keywords

weak-stable manifolds, contractions, differentiable Hartman–Grobman theorem, modulus of continuity, sharp Integrability Condition

2010 Mathematics Subject Classification

35B41, 37L30

Received 20 October 2022

Accepted 22 November 2023

Published 15 July 2024