12931.一類自仿集的連通性及其自仿測度的譜性研究

1、分類號密級學(xué)號至Q!墨Q21四Z魚墾一類自仿集的連通性及其自仿測度的譜性研究Theconnectednessofselfamnesetandspectralpropertiesofselfaffinemeasure研究生姓名：謝恒文指導(dǎo)教師姓名、職稱：劉競成副教授學(xué)科專業(yè)：基礎(chǔ)數(shù)學(xué)研究方分形幾何湖南師范大學(xué)學(xué)位評定委員會辦公室二零一六年五月ABSTRACTLetM∈^厶(z)is321expandingintegermatrixandD

2、∈Z“isafinitedigitsetThenthefamilyofmaps咖d(z)=M1(zd)d6Varecontractiveunderasuitablenormin碾”anditiswellknownthatthereisauniquenonemptycompactsetT=T(M，口)satisfyingthesetequationT=Ud∈DCd(T)Tiscalledtheinvariantset(orattracto

3、r)oftheiteratedfunctionsystem(IFS)≯d)d∈口Eachinv&riantsetdeftnedaboveisnaturallyequippedwithauniquein—variantprobabilitymeasurep：=PM口，theinv&riancedefinedfromthegivenaffinesystem釓)d∈口by蘆2兩1三川≯‘d∈口Thisisaselfaffineidentity

4、withequal丁(^1，口)Theinvariantmeasure盧MDisweight，andPMDissupportedonalsocalledselfa毋ncmeasureThisarticleconsistsofthreechapters，amongwhichwestudytheconnect—ednessofaclassofplanarselfaffinesetinthefirsthalfofthisarticle，asw

5、ellasthespectralpropertiesofaclassofhigherdimensionalselfaffinemeasurei11thesecondhalfInthefirstChapter，thebackgroundandcurrentsituationoftheconnect—ednessofselfaffinesetsandthespectralpropertiesofselfaffinemeasuresarein

6、troducedAtthesametime，themainresultsofthisarticlearegivenInChapter2、weconsidertheconnectednessoftWOclassofplanarselfaffinesetsT(M，D)：(1)LetthecharacteristicpolynomialoftheexpandingintegermatrixA，bef(x1=z2bxeanddigitsetD=

7、o，1；：m)口，thenecessaryandsufficientconditionsonlydependingontheintegersm，b，C®ivenfortheT(M，口)tobeconnected(2)Letthecharacteristicpolynomialoftheexpandingintegermatrix^彳bef(x)=X2一(Pq)xPqanddigitset口=fo，1，，fPql一2，JPq卜lsv