{"title":"高维空间上 Solenoid 吸引子维度的二分法","authors":"Haojie Ren","doi":"10.1007/s00220-024-05018-2","DOIUrl":null,"url":null,"abstract":"<p>We study dynamical systems generated by skew products: </p><span>$$T: [0,1)\\times \\mathbb {C}\\rightarrow [0,1)\\times \\mathbb {C} \\quad \\quad T(x,y)=(bx\\mod 1,\\gamma y+\\phi (x))$$</span><p>where integer <span>\\(b\\ge 2\\)</span>, <span>\\(\\gamma \\in \\mathbb {C}\\)</span> are such that <span>\\(0<|\\gamma |<1\\)</span>, and <span>\\(\\phi \\)</span> is a real analytic <span>\\(\\mathbb {Z}\\)</span>-periodic function. Let <span>\\(\\Delta \\in [0,1) \\)</span> be such that <span>\\(\\gamma =|\\gamma |e^{2\\pi i\\Delta }\\)</span>. For the case <span>\\(\\Delta \\notin \\mathbb {Q}\\)</span> we prove the following dichotomy for the solenoidal attractor <span>\\(K^{\\phi }_{b,\\,\\gamma }\\)</span> for <i>T</i>: Either <span>\\(K^{\\phi }_{b,\\,\\gamma }\\)</span> is the graph of a real analytic function, or the Hausdorff dimension of <span>\\(K^{\\phi }_{b,\\,\\gamma }\\)</span> is equal to <span>\\(\\min \\{3,1+\\frac{\\log b}{\\log 1/|\\gamma |}\\}\\)</span>. Furthermore, given <i>b</i> and <span>\\(\\phi \\)</span>, the former alternative only happens for countably many <span>\\(\\gamma \\)</span> unless <span>\\(\\phi \\)</span> is constant.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Dichotomy for the Dimension of Solenoidal Attractors on High Dimensional Space\",\"authors\":\"Haojie Ren\",\"doi\":\"10.1007/s00220-024-05018-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study dynamical systems generated by skew products: </p><span>$$T: [0,1)\\\\times \\\\mathbb {C}\\\\rightarrow [0,1)\\\\times \\\\mathbb {C} \\\\quad \\\\quad T(x,y)=(bx\\\\mod 1,\\\\gamma y+\\\\phi (x))$$</span><p>where integer <span>\\\\(b\\\\ge 2\\\\)</span>, <span>\\\\(\\\\gamma \\\\in \\\\mathbb {C}\\\\)</span> are such that <span>\\\\(0<|\\\\gamma |<1\\\\)</span>, and <span>\\\\(\\\\phi \\\\)</span> is a real analytic <span>\\\\(\\\\mathbb {Z}\\\\)</span>-periodic function. Let <span>\\\\(\\\\Delta \\\\in [0,1) \\\\)</span> be such that <span>\\\\(\\\\gamma =|\\\\gamma |e^{2\\\\pi i\\\\Delta }\\\\)</span>. For the case <span>\\\\(\\\\Delta \\\\notin \\\\mathbb {Q}\\\\)</span> we prove the following dichotomy for the solenoidal attractor <span>\\\\(K^{\\\\phi }_{b,\\\\,\\\\gamma }\\\\)</span> for <i>T</i>: Either <span>\\\\(K^{\\\\phi }_{b,\\\\,\\\\gamma }\\\\)</span> is the graph of a real analytic function, or the Hausdorff dimension of <span>\\\\(K^{\\\\phi }_{b,\\\\,\\\\gamma }\\\\)</span> is equal to <span>\\\\(\\\\min \\\\{3,1+\\\\frac{\\\\log b}{\\\\log 1/|\\\\gamma |}\\\\}\\\\)</span>. Furthermore, given <i>b</i> and <span>\\\\(\\\\phi \\\\)</span>, the former alternative only happens for countably many <span>\\\\(\\\\gamma \\\\)</span> unless <span>\\\\(\\\\phi \\\\)</span> is constant.</p>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-05-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s00220-024-05018-2\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-05018-2","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
where integer \(b\ge 2\), \(\gamma \in \mathbb {C}\) are such that \(0<|\gamma |<1\), and \(\phi \) is a real analytic \(\mathbb {Z}\)-periodic function. Let \(\Delta \in [0,1) \) be such that \(\gamma =|\gamma |e^{2\pi i\Delta }\). For the case \(\Delta \notin \mathbb {Q}\) we prove the following dichotomy for the solenoidal attractor \(K^{\phi }_{b,\,\gamma }\) for T: Either \(K^{\phi }_{b,\,\gamma }\) is the graph of a real analytic function, or the Hausdorff dimension of \(K^{\phi }_{b,\,\gamma }\) is equal to \(\min \{3,1+\frac{\log b}{\log 1/|\gamma |}\}\). Furthermore, given b and \(\phi \), the former alternative only happens for countably many \(\gamma \) unless \(\phi \) is constant.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.