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通讯网络的弱关系研究.pdf

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Chapter6 ApplicationofInfiniteLabeledGraphs toSymbolicDynamicalSystems KengoMatsumoto AbstractWeapplyatheoryofinfinitelabeledgraphstostudyingpresentations andclassificationsofsymbolicdynamicalsystems,byintroducingaclassofinfinite labeledgraphs,called-graphsystems.Itsmatrixpresentationiscalledasymbolic matrixsystem.Thenotionsofa-graphsystemandsymbolicmatrixsystemare generalizednotionsofafinitelabeledgraphandsymbolicmatrixforsoficsub- shiftstogeneralsubshifts.Strongshiftequivalenceandshiftequivalencebetween symbolicmatrixsystemsareformulatedsothattwosubshiftsaretopologically conjugateifandonlyiftheassociatedcanonicalsymbolicmatrixsystemsare strongshiftequivalent.Weconstructseveralkindsofshiftequivalenceinvariants forsymbolicmatrixsystems.Theyarethedimensiongroups,theK-groups,and theBowen–Franksgroupsthataregeneralizationsofthecorrespondingnotionsfor nonnegativematrices.Theyyieldtopologicalconjugacyinvariantsofsubshifts.The entropicquantitiescalled-entropyandvolumeentropyfor-graphsystemsare alsostudiedrelatedtothetopologicalentropyofsymbolicdynamics. KeywordsSubshiftsSymbolicdynamics-GraphsystemsStrongshift equivalenceBowen–FranksgroupK-theoryTopologicalentropy MSC2000:Primary37B10;Secondary28D20,46L80 6.1Introduction Graphtheoryhasintersectionswithotherbranchesofmathematics.Thetheoryof symbolicdynamicsisoneofthem.Ithassignificantusesforcodingtheoryand formallanguagetheoryininformationsciences.Theclassofsymbolicdynami- calsystemsisabasicpartoftopologicaldynamicalsystems.Graph-theoretical K.Matsumoto() DepartmentofMathematics,JoetsuUniversityofEducation,Joetsu943-8512,Japan e-mail:kengo@juen.ac.jp M.Dehmer(ed.),StructuralAnalysisofComplexNetworks,137 DOI10.1007/978-0-8176-4789-66,cSpringerScience+BusinessMedia,LLC2011 138K.Matsumoto techniquesandlinearalgebraictechniquesareveryusefultostudysymbolic Z dynamicalsystems.QLet˙beafiniteset,calledanalphabet.Let˙betheinfinite 1˙˙˙ productspacesiD1iwhereiD,endowedwiththeproducttopology. Z Thetransformationon˙g