{"title":"实解析映射的不稳定性问题","authors":"Karim Bekka, Satoshi Koike, Toru Ohmoto, Masahiro Shiota, Masato Tanabe","doi":"10.1112/blms.13124","DOIUrl":null,"url":null,"abstract":"<p>As well known, the <span></span><math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mi>∞</mi>\n </msup>\n <annotation>$C^\\infty$</annotation>\n </semantics></math> stability of proper <span></span><math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mi>∞</mi>\n </msup>\n <annotation>$C^\\infty$</annotation>\n </semantics></math> maps is characterized by the infinitesimal <span></span><math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mi>∞</mi>\n </msup>\n <annotation>$C^\\infty$</annotation>\n </semantics></math> stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal <span></span><math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mi>ω</mi>\n </msup>\n <annotation>$C^\\omega$</annotation>\n </semantics></math> stability does not imply <span></span><math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mi>ω</mi>\n </msup>\n <annotation>$C^\\omega$</annotation>\n </semantics></math> stability; for instance, <i>a Whitney umbrella</i> <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>R</mi>\n <mn>2</mn>\n </msup>\n <mo>→</mo>\n <msup>\n <mi>R</mi>\n <mn>3</mn>\n </msup>\n </mrow>\n <annotation>$\\mathbb {R}^2 \\rightarrow \\mathbb {R}^3$</annotation>\n </semantics></math> <i>is not</i> <span></span><math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mi>ω</mi>\n </msup>\n <annotation>$C^\\omega$</annotation>\n </semantics></math> <i>stable</i>. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3174-3180"},"PeriodicalIF":0.8000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Unstability problem of real analytic maps\",\"authors\":\"Karim Bekka, Satoshi Koike, Toru Ohmoto, Masahiro Shiota, Masato Tanabe\",\"doi\":\"10.1112/blms.13124\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>As well known, the <span></span><math>\\n <semantics>\\n <msup>\\n <mi>C</mi>\\n <mi>∞</mi>\\n </msup>\\n <annotation>$C^\\\\infty$</annotation>\\n </semantics></math> stability of proper <span></span><math>\\n <semantics>\\n <msup>\\n <mi>C</mi>\\n <mi>∞</mi>\\n </msup>\\n <annotation>$C^\\\\infty$</annotation>\\n </semantics></math> maps is characterized by the infinitesimal <span></span><math>\\n <semantics>\\n <msup>\\n <mi>C</mi>\\n <mi>∞</mi>\\n </msup>\\n <annotation>$C^\\\\infty$</annotation>\\n </semantics></math> stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal <span></span><math>\\n <semantics>\\n <msup>\\n <mi>C</mi>\\n <mi>ω</mi>\\n </msup>\\n <annotation>$C^\\\\omega$</annotation>\\n </semantics></math> stability does not imply <span></span><math>\\n <semantics>\\n <msup>\\n <mi>C</mi>\\n <mi>ω</mi>\\n </msup>\\n <annotation>$C^\\\\omega$</annotation>\\n </semantics></math> stability; for instance, <i>a Whitney umbrella</i> <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>R</mi>\\n <mn>2</mn>\\n </msup>\\n <mo>→</mo>\\n <msup>\\n <mi>R</mi>\\n <mn>3</mn>\\n </msup>\\n </mrow>\\n <annotation>$\\\\mathbb {R}^2 \\\\rightarrow \\\\mathbb {R}^3$</annotation>\\n </semantics></math> <i>is not</i> <span></span><math>\\n <semantics>\\n <msup>\\n <mi>C</mi>\\n <mi>ω</mi>\\n </msup>\\n <annotation>$C^\\\\omega$</annotation>\\n </semantics></math> <i>stable</i>. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 10\",\"pages\":\"3174-3180\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-07-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13124\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13124","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
众所周知，适当 C ∞ $C^infty$ 映射的 C ∞ $C^infty$ 稳定性是以无穷小 C ∞ $C^infty$ 稳定性为特征的。在本文中，我们研究了实解析背景下的对应关系。特别是，我们证明了无穷小 C ω $C^\omega$ 稳定性并不意味着 C ω $C^\omega$ 稳定性；例如，惠特尼伞 R 2 → R 3 $\mathbb {R}^2 \rightarrow \mathbb {R}^3$ 不是 C ω $C^\omega$ 稳定性。证明的一个主要工具是惠特尼解析近似定理的一个相对版本，它是通过 H. Cartan 的定理 A 和 B 来证明的。
As well known, the stability of proper maps is characterized by the infinitesimal stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal stability does not imply stability; for instance, a Whitney umbrellais notstable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.
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