{"title":"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","authors":"Zhicheng Tong, Yong Li","doi":"10.4310/cms.2024.v22.n5.a8","DOIUrl":null,"url":null,"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.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"28 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space\",\"authors\":\"Zhicheng Tong, Yong Li\",\"doi\":\"10.4310/cms.2024.v22.n5.a8\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":50659,\"journal\":{\"name\":\"Communications in Mathematical Sciences\",\"volume\":\"28 1\",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Sciences\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/cms.2024.v22.n5.a8\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cms.2024.v22.n5.a8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space
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.
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.