{"title":"自相似性和光谱动力学","authors":"Bryan Goldberg, Rongwei Yang","doi":"10.7900/jot.2020sep27.2329","DOIUrl":null,"url":null,"abstract":"This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group D∞. The main theorem states that the Julia set of the induced rational map F on P2 for D∞ is the union of the projective spectrum with F's extended indeterminacy set. Moreover, the limit function of the iteration sequence {F∘n} on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G's projective spectrum is contained in the Julia set of G.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2020-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":"{\"title\":\"Self-similarity and spectral dynamics\",\"authors\":\"Bryan Goldberg, Rongwei Yang\",\"doi\":\"10.7900/jot.2020sep27.2329\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group D∞. The main theorem states that the Julia set of the induced rational map F on P2 for D∞ is the union of the projective spectrum with F's extended indeterminacy set. Moreover, the limit function of the iteration sequence {F∘n} on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G's projective spectrum is contained in the Julia set of G.\",\"PeriodicalId\":50104,\"journal\":{\"name\":\"Journal of Operator Theory\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2020-02-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Operator Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7900/jot.2020sep27.2329\",\"RegionNum\":4,\"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":"Journal of Operator Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7900/jot.2020sep27.2329","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group D∞. The main theorem states that the Julia set of the induced rational map F on P2 for D∞ is the union of the projective spectrum with F's extended indeterminacy set. Moreover, the limit function of the iteration sequence {F∘n} on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G's projective spectrum is contained in the Julia set of G.
期刊介绍:
The Journal of Operator Theory is rigorously peer reviewed and endevours to publish significant articles in all areas of operator theory, operator algebras and closely related domains.