自相似集的Hausdorff测度与上凸密度的估计与计算

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1、AbstractABSTRACT.Inthisthesis,wemainlystudyhowtocalculatetheexactHausdorffmeasureofaclassofself-similarsetsandhowtoestimatetheupperboundofupperconvexdensityofsomeconcretepointsinsomeself-similarsets.Inaddition,wepresentanecessaryandsufficientconditionforaclassofself-similarsetsonreallinet

2、ohavebestcoveringsandsomesufficientconditionsforanalmosteverywherebestcoveringtobeabestonefortheself-similarsetssatisfyingtheopensetcondition.Thisthesisisdividedintofourchapters.Inthefirstchapterweintroducetheresearchbackgroundaboutfractal,anddescribeadetailedbasicdefinitionsandlemmasabou

3、tfractal.InthesecondchapterwestudytheproblemofcalculationofHausdorffmeasureofaclassofself-similarsetsinaregularhexahedron.Whenthesimilarratiossatisfiescertainconditions,weprovethatthenaturalcoveristhebestshapeforupperconvexdensity.Itmeansthatthenaturalcoverisabestone,thentheexactvalueofHa

4、usdorffsmeasureofthisclassofself-similarsetsis3,wheresisHausdorffmeasure.InthethirdchapterwefirstprovethatthefamilyofbestshapsofthevertexsofaclassofSierpinskicarpetsisacoveringoftheSierpinskicarpets,thenprovetheminimumoftheupperconvexdensityisachievedattheirvertexs.Thisresultextendssome

5、recentresults.Intheforthchapter,wefirsttakeadvantagedofanimportantcharacteristicthatthebestshapofself-similarsetsonreallinemustbeclosedinterval,thenpresentanecessaryandsufficientconditionforaclassofself-similarsetsonreallinetohavebestcoverings,andsomesufficientconditionsforanalmosteverywh

6、erebestcoveringtobeabestonefortheself-similarsetssatisfyingtheopensetcondition.KeyWords：selfsimilarset;Hausdorffmeasure;Hausdorffdimension;Upperconvexdensity;naturalcovering;bestcoveringIII万方数据目录目录第一章绪论.......................................................................................

7、......................................11.1分形研究背景及现状...........................................................................................11.2分形基本概念.........................................................................................................3第二章一类正六面