{"title":"自相似集的熵和Hausdorff维数","authors":"James Evans","doi":"10.1090/PROC/15569","DOIUrl":null,"url":null,"abstract":"Given a $k$-self similar set $X\\subset [0,1]^{d}$ we calculate both its Hausdorff dimension and its entropy, and show that these two quantities are in fact equal. This affirmatively resolves a conjecture of Adamczewski and Bell.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"4 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The entropy and Hausdorff dimension of self-similar sets\",\"authors\":\"James Evans\",\"doi\":\"10.1090/PROC/15569\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given a $k$-self similar set $X\\\\subset [0,1]^{d}$ we calculate both its Hausdorff dimension and its entropy, and show that these two quantities are in fact equal. This affirmatively resolves a conjecture of Adamczewski and Bell.\",\"PeriodicalId\":407889,\"journal\":{\"name\":\"arXiv: Dynamical Systems\",\"volume\":\"4 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-11-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Dynamical Systems\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/PROC/15569\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Dynamical Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/PROC/15569","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The entropy and Hausdorff dimension of self-similar sets
Given a $k$-self similar set $X\subset [0,1]^{d}$ we calculate both its Hausdorff dimension and its entropy, and show that these two quantities are in fact equal. This affirmatively resolves a conjecture of Adamczewski and Bell.
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