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Topological Efficiency of Stirring with Obstacles

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Title: Topological Efficiency of Stirring with Obstacles
Physical Description: 1 online resource (64 p.)
Language: english
Creator: Harrington, Jason
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: choas, efficient, fluid, hypotrochoid, mixing, stirring, topology
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We investigate the problem of stirring by a single stirrer moving around a finite number of obstacles; the goal is to find the stirring protocol which is the most efficient, denoted by E(n) where n is the number of obstacles. Efficiency is defined as the ratio of mixing by the number of stirring motions for a given stirring protocol. In other words, we want to increase the amount the stirrer mixes while decreasing the number of motions needed. Through numerical experimentation, we find that the maximum efficiency seems to be realized by a specific protocol which is, essentially, ?looping around each obstacle once.? This stirring protocol is used as a lower bound for E(n). The upper bound is computed by overestimating the amount of mixing. Together with the upper bound and the lower bound we can show that as the number of obstacles goes to infinity, then E(n) goes to log(3).
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jason Harrington.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Boyland, Philip L.

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Classification: lcc - LD1780 2010
System ID: UFE0042383:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042383/00001

Material Information

Title: Topological Efficiency of Stirring with Obstacles
Physical Description: 1 online resource (64 p.)
Language: english
Creator: Harrington, Jason
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: choas, efficient, fluid, hypotrochoid, mixing, stirring, topology
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We investigate the problem of stirring by a single stirrer moving around a finite number of obstacles; the goal is to find the stirring protocol which is the most efficient, denoted by E(n) where n is the number of obstacles. Efficiency is defined as the ratio of mixing by the number of stirring motions for a given stirring protocol. In other words, we want to increase the amount the stirrer mixes while decreasing the number of motions needed. Through numerical experimentation, we find that the maximum efficiency seems to be realized by a specific protocol which is, essentially, ?looping around each obstacle once.? This stirring protocol is used as a lower bound for E(n). The upper bound is computed by overestimating the amount of mixing. Together with the upper bound and the lower bound we can show that as the number of obstacles goes to infinity, then E(n) goes to log(3).
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jason Harrington.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Boyland, Philip L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042383:00001


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TOPOLOGICALEFFICIENCYOFSTIRRINGWITHOBSTACLESByJASONC.HARRINGTONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

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c2010JasonC.Harrington 2

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Tomywifeandmyfamilywhosupportedmethroughoutthewholeprocess 3

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ACKNOWLEDGMENTS IamindebtedtomyPhDadvisor,ProfessorBoyland,forhispatienceandguidancethroughoutthewholePhDprocess.Withouthishelp,thisdissertationwouldnotbepossible.Iwouldliketoexpressmygratitudetothecommitteemembers.IwouldliketothankProfessorKeeslingandProfessorBlockforintroducingmetodynamicsandfortheirexcellentteaching.IwouldalsoliketoexpressmygratitudetoProfessorRobinsonforhisinsightfulcommentsonthedissertationandhisgeneralguidancewhileIwasastudentingraduateschool.Lastly,IwouldliketothankProfessorPetersforservingastheexternalcommitteemember. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 10 1.1Introduction ................................... 10 1.2StirringProtocol ................................ 12 1.3StirringasaBraid ............................... 13 1.4MappingClassGroups ............................. 15 1.5TopologicalEntropy .............................. 16 1.6Thurston'sAnalysisofMappingClasses ................... 16 1.7FluidMechanics ................................ 17 2BURAUREPRESENTATION ............................ 19 2.1BurauRepresentation ............................. 19 2.1.1BurauastheActiononHomology ................... 19 2.1.2RightAutomorphism .......................... 20 2.2ReducedBurauRepresentation ........................ 23 2.3BurauRepresentationandEntropy ...................... 24 2.4ConventionandNotation ........................... 25 3FUNDAMENTALSTIRRINGPROTOCOLS .................... 26 3.1Denition .................................... 26 3.2CongurationSpaces ............................. 26 4NUMERICALEXPERIMENTS ........................... 29 4.1GroupActions ................................. 29 4.2BurauRepresentation ............................. 30 4.3MousirAlgorithm ............................... 30 4.4ResultsofExperimentation .......................... 30 5PROPERTIESOFHSP ............................... 33 5.1EntropyLowerBoundforHSP ......................... 33 5.2GearabilityofHSP ............................... 43 5

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6UPPERBOUNDS .................................. 45 6.1ConstructionofUpperBound ......................... 45 6.2SpectralRadiusofincidenceMatrices .................... 51 APPENDIX:SOMEGENERALMATRIXLEMMAS ................... 59 A.1GeneralMatrixTheory ............................. 59 A.2SpectralRadiusProperties .......................... 60 REFERENCES ....................................... 61 BIOGRAPHICALSKETCH ................................ 64 6

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LISTOFFIGURES Figure page 1-1ParticleDance .................................... 13 1-2GeometricBraid ................................... 14 1-3SigmaGenerators .................................. 15 2-11Generators ..................................... 20 2-2iActingon1Generators ............................. 20 2-3Zcover ........................................ 21 2-4LiftofapathinM2 .................................. 21 2-51-SkeletonofM2anditslift ............................. 22 2-6Inducedactionof^1onZ-cover ........................... 22 3-14Example ...................................... 28 4-1TwoversionsofHSP6 ................................ 31 4-2Comparisonbetweentwostirringprotocols .................... 31 5-1EfciencyofHSP ................................... 33 5-2TrainTrack ...................................... 38 5-3ConstructingHSP(8) ................................. 44 6-1NewGeneratorsfor1(M2) ............................. 47 6-2NewAlphaGenerators ................................ 47 7

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTOPOLOGICALEFFICIENCYOFSTIRRINGWITHOBSTACLESByJasonC.HarringtonDecember2010Chair:PhillipBoylandMajor:MathematicsWeinvestigatethetwo-dimensionalproblemofstirringbyasinglestirrermovingaroundanitenumberofobstacles.Thegoalistondthestirringprotocolwhichyieldsthemaximumtopologicalefciency.Topologicalefciencyisdenedtobetheentropyassociatedwiththestirringprotocoldividedbythenumberofstirrermotionsaroundeachobstacle,andwedenotethemaximumefciencywithmobstaclesasE(m).Ourinvestigationbeginswithnumericalsimulationsofthetopologicalefciencyforavarietyofpathsandnumberofobstacles.Sinceweseekthemaximumefciency,thetopologicalefciencyofanyspecicstirringprotocolyieldsalowerboundforE(m).Tothedegreepossiblecomputationallywendthatthemaximumefciencyseemstoberealizedbyaspecicprotocolwhichisessentiallyloopingaroundeachobstacleonce,andwedenoteitHSPm.UsingtechniquesfromThurston-NielsentheorywecanrigorouslycomputetheefciencyofHSPm,useitasalowerboundforE(m),andshowitconvergestolog(3)asm!1.Foranupperbound,wewilluseBowen'sTheoremthatentropyisboundedbelowbythegrowthrateoftheactionofthemaponthefundamentalgroup.Weusenon-standardgeneratorsforourspace,thediskwithm+1holes.Thestirringpathinducesanactiononthesegenerators.Thegrowthrateon1isoverestimatedbyignoringcancellations.Thisallowsustousetheincidencematricesoftheactionandthusgetanentropyupperboundusingthegeneralizedspectralradiusoftheincidence 8

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matricesofthefundamentalmotionsofthestirrer.WeprovethatthisgeneralizedspectralradiusisactuallyachievedbyaniteproductofincidencematriceswhicharecloselyrelatedtotheprotocolHSPm.Finally,weshowthattheupperboundforthetopologicalefciencyconvergestolog(3)asm!1,andso,E(m)islog(3)intheasymptoticlimit. 9

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CHAPTER1INTRODUCTION 1.1IntroductionWeinvestigatethetwo-dimensionalproblemofstirringbyasinglestirrermovingaroundanitenumberofobstacles.Wecallsuchastirringmotionasafundamentalstirringprotocol.Thegoalistondthefundamentalstirringprotocolwhichyieldsthemaximumtopologicalefciency.Topologicalefciencyisdenedtobetheentropyassociatedwiththestirringprotocoldividedbythenumberofstirrermotionsaroundeachobstacle,andwedenotethemaximumefciencywithmobstaclesasE(m).Inotherwords,E(m)=supfE():isafundamentalstirringprotocolwithn)]TJ /F8 11.955 Tf 11.95 0 Td[(1xedholesgwhereEdenotesthetopologicalefciencyofthefundamentalstirringprotocol. Theorem1.1(MainTheorem). Consideradiskwithm)]TJ /F8 11.955 Tf 12.45 0 Td[(1obstacles,andonefunda-mentalstirrer.Thenlog((3m)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 11.95 0 Td[(3m+2)=(m)]TJ /F8 11.955 Tf 11.95 0 Td[(1)) m)]TJ /F8 11.955 Tf 11.96 0 Td[(1E(m)log(3).Webeginbylookingattwo-dimensionalstirringfromdifferentpointsofview.First,wewillconsiderastirringprotocolasaparticledance,wherethestirrersmovearoundaspointsintwo-dimensions.Thiswilldescribethepossiblepathsofthestirrers.Next,weconsiderastirringprotocolasabraid.Sincebraidshavegenerators,thisenablesustohaveagroupstructureonstirringprotocols.Wewillusej'stodenotetheelementsofafundamentalstirringprotocol.Next,wewillconsiderastirringprotocolasanelementofthemappingclassgroup,whichenablesustouseThurston-Nielsentheory.Whenastirrermovesaroundanobstacle,itwillpullandstretchtheuidaround.However,givenaspecicstirringprotocol,differentuidsmaybehavedifferently.Forexample,stirringcoffeewithcream 10

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israthereffortlessandcanbedonewithaickofaspoon,whereasthismotionwillnotmixthickeruidslikecement.Ifweconsiderauidasasurfacehomeomorphism,thenThurston-Nielsentheorygivesustheasurfacehomeomorphismcalled'thathasthesmallestentropyassociatedwiththestirringprotocol.WewillusetheBuraurepresentationforbraidswhichgivesalowerboundfortheentropyof'.SinceanystirringprotocolgivesalowerboundforE(m),weusedseveralnumericaltechniquesforndingonethatappearedtobemostefcientandthenrigorouslycomputeditsefciency.Threedifferentdifferentnumericaltechniquesforapproximatingtheentropyofafundamentalstirringprotocolwithn)]TJ /F8 11.955 Tf 12.23 0 Td[(1xedholeswereused.Severalthousanddifferentstirringprotocolsweregeneratedandtestedwiththesetechniques. 1. ThersttechniqueusestheBuraurepresentationofthefundamentalstirringprotocol.Thematrixgeneratedatt=)]TJ /F8 11.955 Tf 9.29 0 Td[(1givesalowerboundfortheentropyofthebraid. 2. Thenexttechniqueconsiderstheactionofthefundamentalstirringprotocolonthefundamentalgroupofthediskwithnholesbyallowingthefundamentalstirrertobeonetheholes.Thestirringprotocolwastheniteratedupto10times.Thiswasusedtoapproximatetheexponentialgrowthon1andthustheentropy. 3. Thelasttechniqueusesaprogramcalledtrains.execreatedbyTobyHallthatusestraintrackstocomputetheentropyofabraid.ArandomfundamentalstirringprotocolwasgeneratedandthenconvertedintothegeneratorsfortheArtinbraidgroupandinputedthatwordintotheprogram.Afterconsideringseveralthousanddifferentstirringprotocolsthatwereproducedrandomly,theonethatwasmostefcientoutoftestedprotocolswasfoundtobeaprotocolwhichloopsexactlyoncearoundeachobstacleinsuccession.Thisstirringprotocolwasusedtocomputealowerbound.Inordertocomputetheupperbound,weusedtheactionthatisinducedonthefundamentalgroupbythegeneratorsforthefundamentalstirringprotocols.Bynotconsideringcancellation,wewereabletondanupperboundfortheexponentialgrowthon1andthustheentropy. 11

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Puttingbothoftheseideastogether,weareabletoshowthatlog((3m)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 11.95 0 Td[(3m+2)=(m)]TJ /F8 11.955 Tf 11.95 0 Td[(1)) m)]TJ /F8 11.955 Tf 11.96 0 Td[(1log((HSPm)) m)]TJ /F8 11.955 Tf 11.96 0 Td[(1E(m)log(3).Thismeansthatlimm!1E(m)=log(3)andthatlimm!1log((HSPm))=(m)]TJ /F8 11.955 Tf 11.96 0 Td[(1)=log(3).Theprocessofcomputingtheupperboundproducedamatrix,namedtheincidencematrix,foreachgeneratorofthefundamentalstirringprotocol.LetAjdenotetheincidencematrixforj.Thegeneralizedspectralradiusofthesetofincidencematricesiscomputed.ThegeneralizedspectralradiusofthesetofmatricesisrealizedbythematrixA2...Anwhichisexactlythesameorderasinthelowerbound.ThissupportstheconjecturethatE(n)isactuallyachievedbytheHSP. 1.2StirringProtocolWenowdiscusstheset-uptostudyuidmixingfromatopologicalpointofview.ThisbranchofmathematicsiscalledTopologicalFluidMechanics.Firstwewilldenewhatismeantbyageneralstirringprotocol.LetD2betheunitdiskinR2andletnbeapositiveinteger. Denition1(StirringProtocol). Considerndistinctpointsx1,x2,...,xnintheinteriorofD2.WedeneastirringprotocolSPbySP=(f1(t),...,fn(t))wherefi:[0,1]!D2suchthat: 1. fiiscontinuousforeachi, 2. fi(t)6=fj(t)ifi6=j,forallt2[0,1], 3. fi(0)=xiforeachi, 4. fi(1)2fx1,x2,...,xngforeachi.Eachficorrespondstothemotionofastirrer.Somepeoplepreferthatthepathofthestirrerbedifferentiable,however,differentiabilityisnotneededforthisdiscussion.Condition2impliesthatthestirrersdonothiteachotherateachtimet,i.e.ff1(t),...,fn(t)garendistinctpoints.Condition3merelytellsusthestartingpositionofthestirringprotocol.Condition4impliesthestirrersmustendbackonthesetfx1,x2,...,xng.Thisdenitionofastirringprotocolissometimesreferredtoasaparticledanceas 12

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discussedbyRolfsen[ 1 ].Figure 1-1 givesanexampleofastirringprotocol,wherethereareinitiallyfourstirrersandthearrowshowsthepathofeachstirrerbacktothebaseposition.Notethatitispossibleforfi(1)6=fi(0)forsomei.Ifwereplace Figure1-1. Thestirrersactasparticlesmovingintheuid. condition 4 withfi(1)=fi(0)forallithenwehaveaspecialcasecalledapurestirringprotocol. 1.3StirringasaBraidWewillnowshowthattheconstructionofastirringprotocolcanbedescribedintermsoftheArtinBraidgroup.TherigorousproofsoftheclaimsinthesectioncanbefoundintheclassicBirmanbook[ 2 ].LetSPbeastirringprotocolasdenedabove.LetI=[0,1]andAi:I!ID2bedenedbyAi(t)=(t,fi(t))foreachi.Byconstruction,eachAiisacontinuousarc.WewillcalleachAiabraidstring.LetA=fA1,A2,...,Angbeafamilyofarcsthatstartatthepointsf(0,x1),(0,x2),...,(0,xn)gandendupatf(1,f1(1)),(1,f2(1)),...(1,fn(1))g.Byconstruction,eachAiintersectseachintermediateparallelplaneexactlyonce,andAintersectseachintermediateparallelplaneatexactlyndistinctpoints.WecallAageometricn-braid. 13

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Denition2(PureBraid). Ifallthearcsofageometricn-braid,A,havethepropertythatfk(1)=xkforallk,wherefisastirringprotocolandxkisthestartingpointasdenedpreviously,thenwecallthisbraidapurebraid. Figure1-2. AstirringprotocolasaGeometricBraid. Next,wewilldiscusstheequivalenceoftwogeometricn-braids.Theideaisthatweshouldbeabletodeformabraidwithoutchangingthestructure,i.e.abraidstringcannotpassthroughotherbraidstringsbutcanwigglearound.Wemakethisconceptmoreprecisebythefollowingdenition. Denition3(EquivalenceofGeometricBraids). LetX=ID2.Wesaythattwogeometricbraidsand0areequivalentifthereisanisotopicdeformationHs:XI!Xwheretheimagesetsisageometricbraidforalls2Iand0=and1=0.LetB(n)denotethesetofallequivalenceclassesofthegeometricn-braids.Let^1,^22B(n),wedenetheproduct(composition),denotedas^1^2asfollows.Let^1=[B1]where[B1]denotestheequivalenceclassforthegeometricbraidB1,similarly,^2=[B2].SoB2isafamilyofarcsthatstartatf(0,x1),(0,x2),...,(0,xn)gandendatf(1,x1),(1,x2),...,(1,xn)g.ShiftB1upsothatB1isafamilyofarcsthatstartatf(1,x1),(1,x2),...,(1,xn)gandendatf(2,x1),(2,x2),...,(2,xn)g.WeconcatenatethebraidstogetherwithB1ontopandB2below.Together,wehaveabraidin[0,2]D2.Wemayrescaletime,sothattheconcatenatedbraidtsinID2.Theequivalenceclassofthisnewlyformedbraidis^1^2.Thereisalsoanidentityelement,namely,thebraidwhereeachstringissimplyastraightlineanddoesnotcrossotherstrings.Given 14

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abraid1,thereexistsabraid3thatwillundo1inthesensethat13istheidentitybraid.TheArtinBraidgroupisthegroupformedfrom(B(n),).Artinidentiedasetofgeneratorsandrelations,i.e.apresentationfortheArtinBraidgroup.Letidenotethebraidwhoseithandi+1thbraidstringchangeplacesasshowninFigure 1-3 .InArtin's Figure1-3. iand)]TJ /F5 7.97 Tf 6.59 0 Td[(1i rstpaperonbraidgroupsfrom1925[ 3 ],hegaveapresentationofthebraidgroupintermsofthei'sandtheirrelations,namely: Theorem1.2(Artin). ThegroupB(n)ofgeometricbraidsonn-stringsadmitsapresen-tationwithgenerators1,2,...,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1withthedeningrelations; 1. ij=jiforji)]TJ /F3 11.955 Tf 11.96 0 Td[(jj>2, 2. ii+1i=i+1ii+1for1in)]TJ /F8 11.955 Tf 11.95 0 Td[(2.Byconvention,wewillhavetimegoingdownandwillthinkofthe'sascomposition,sowewillreadthemfromrighttoleft.Forexample,fromthebraidinthegure 1-2 thebraidpresentationis2)]TJ /F5 7.97 Tf 6.58 0 Td[(11.WewilldescribethestirringprotocolintermsofthegeneratorsoftheArtinbraidgroup. 1.4MappingClassGroupsWehaveanotherwaywemaydescribeastirringprotocol.Thedescriptionwillhelpusunderstandthedynamicsgivenbyastirringprotocol.LetM2beanorientable,compactsurfacewithnegativeEulercharacteristic,possiblywithboundary.Wewillallowanitenumberofpointsthatactlikeobstacles. 15

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Denition4. ThemappingclassgrouponM2isMCG(M2)=f[f]wheref:M2!M2isahomeomorphismandfj@M2=idand[f]istheisotopyclassforfgwhereisotopiesaretakentoxtheboundary,@M2,pointwise.MCGhasanaturalgroupoperation,namelycomposition.LetM2=D2)-261(fn)]TJ /F1 11.955 Tf 12.43 0 Td[(ptsg.Imaginethatthepointsremovedarestirrersinauidandthepointspulltheirsurroundingswiththemastheymoveabout.Topologically,themotionofthepointsextends(notuniquely)toacontinuousfamilyofhomeomorphismsofM2.ThisisotopyclassdescribesanequivalencebetweenB(n)andMCG(M2). 1.5TopologicalEntropyLet:M2!M2whereisahomeomorphismand:1(M2)!1(M2)bethegroupactioninducedby.Inotherwords,asmovesthepointsofthespacearound,thenwillalsomovetheelementsofthefundamentalgroupinM2inthesamemanner.Thereisanicegeometricaldescriptioningure 2-2 when=i.LetLdenotethewordlength.ByBowen[ 4 ]withatheoremin[ 5 ]thetopologicalentropyofapseudo-Anosovmapisgivenbyh()=supg21(M2)limm!1log(L(mg)) m.Entropycanbeusedtodeterminetheasymptoticbehaviorofiteratinganinducedaction.Forexample,ifh()>0,thenweknowthatthewordlengthofmgisgrowingexponentiallyforallg21(M2)asmincreases.Furthermore,ifh(1)>h(2)forsomeinducedactionson1(M2),then(1)mgisgrowingexponentiallyfasterthan(2)mgforallgasmincreasesThisgivesameasureofthecomplexityoftheaction. 1.6Thurston'sAnalysisofMappingClassesWewouldliketodiscussthedynamicsofthestirringprotocol.Wewillmakeuseofthefollowingtheorem: 16

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Theorem1.3(Thurston-NielsenClassication). Iffisahomeomorphismofacompactsurface,S,withperhapsanitenumberofpunctures,thenfisisotopictoahomeomor-phism,',ofoneofthefollowingtypes: 1. Finiteorder:'m=idforsomeintegerm1; 2. Pseudo-Anosov:'preservesapairoftransverse,measuredfoliations,FuandFsandthereisa>1suchthat'stretchesFubyafactorofandcontractsFsby1=; 3. Reducible:'xesafamilyofreducingcurves,andonthecomplementarysurface'satises(1)or(2).ThistheorywasdevelopedbyThurston[ 6 ].Lettingf2MCG(M2),wehavethreecasesfortheThurstonrepresentative'f.Theniteordercasestatesthatafteranitenumberofiterationsoff,wewillendupbackwherewestarted.Thepseudo-Anosovcase,abbreviatedpA,tellsusthatthetopologicalentropyof'fislog()whereisthevaluestatedinthetheorem[ 5 ].Thethirdcase,thereduciblecase,impliesthatthespacecanbedividedintoinvariantpieceswhere'fhaseitherniteorderorispAonthepieces. 1.7FluidMechanicsInthissection,wewillexplainhowastirringprotocolrelatestothephysicalapplicationofuidmixing.BythefollowingtheoremofHandel[ 7 ],weseethatapAmapgivesusalowerboundforthedynamicsinducedbyastirringprotocol. Theorem1.4(Handel'sIstopyStability). If'isapAandfisisotopicto',thenthereisacompact,f-invariantset,Y,andacontinuous,ontomapping:Y!M2,sothatf='.Thus,wehavethecommutativediagram,Y f// Y M2'// M2 17

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Thistheoremstatesthatthedynamicsoffareatleastascomplicatedasthedynamicsof'.Inthecasewithuidmixing,theuidmovescontinuouslyonthesurfaceD2)-178(fn)]TJ /F1 11.955 Tf 11.43 0 Td[(ptsg.Letfdenotetheuidow.SoifwehaveapAmapcalled,'f,associatedtof,thenthetopologicalentropylog()=h('f)h(f)whereisthevaluefromThurston'stheorem.SothepAgivesthelowerboundforthetopologicalentropyofanyuid,regardlessofitsphysicalproperties.Sowitheachbraid,thereisaThurstonrepresentative.Theserepresentativesgiveusalowerboundfortheentropyoftheactualuidow.Theideaistomaximizetheamountofentropyassociatedwithstirring,whileminimizingthenumberofmotionsneededforthestirrertomove.Historically,uidmixingwassolvedonacase-by-casebasis.Theuidowwouldbeasolutiontoaspecicsetofpartialdifferentialequations,andacomputerexperimentwasrunwithvariousstirringprotocols.Thentodeterminethebeststirringprotocol,onewoulduseaspecicmixingmeasureandcompare.Finn,CoxandBurns[ 8 ],listafewvariousmeasuresthatarestillusedtoday.Also,Ottino[ 9 ]andChorin[ 10 ]aregoodreferencesfortraditionaluidmechanics. 18

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CHAPTER2BURAUREPRESENTATION 2.1BurauRepresentationArepresentationofagroupGonamoduleMoveraringRisagrouphomomorphismfromGtoGL(M).Wedeneahomomorphismb:B(n)!GLn(Z[t,t)]TJ /F5 7.97 Tf 6.58 0 Td[(1])calledtheBuraurepresentation,bytheimagesofthegeneratorsofB(n).b(i)=266666664Ii)]TJ /F5 7.97 Tf 6.59 0 Td[(1 00 0 0 1)]TJ /F3 11.955 Tf 11.96 0 Td[(tt 00 10 0 0 00 In)]TJ /F6 7.97 Tf 6.59 0 Td[(i)]TJ /F5 7.97 Tf 6.58 0 Td[(1377777775ithrowwhereIkdenotesthekkidentitymatrix.ThisrepresentationisnamedafteritsdiscovererBurauin1936[ 11 ].Weseebydirectcomputationthat 1. b(i)b(j)=b(j)b(i)forji)]TJ /F3 11.955 Tf 11.95 0 Td[(jj>2, 2. b(i)b(i+1)b(i)=b(i+1)b(i)b(i+1)for1in)]TJ /F8 11.955 Tf 11.95 0 Td[(2.TheBuraurepresentationforn=2,3isfaithfulandtheproofcanbefoundinBirman'sbook[ 2 ].In1999,Bigelow[ 12 ]showedthattherepresentationisnotfaithfulforn=5.Forn6,LongandPatonshowedin1993thattheBuraurepresentationisnotfaithful[ 13 ].However,itisstillunknownwhetherornotitisfaithfulwhenn=4. 2.1.1BurauastheActiononHomologyWewillshowthatthisrepresentationcanbeinterpretedastheinducedactiononhomologyinZ-coversbyabraid.Referencesforthissectionare[ 14 ]and[ 15 ].Firstwemustintroduceanotheraction. 19

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2.1.2RightAutomorphismWeneedtocomputetheactionofthebraidgeneratorsonthefundamentalgroupofourdiskminusnpoints,M2.LetFndenoteafreegrouponnsymbols.Recallthenotationforthebraidgeneratorsfrom 1.2 .WewilldenearightautomorphismofFn=.Leti2B(n)andwewilldene^2Aut(Fn)by xi^i=xixi+1x)]TJ /F5 7.97 Tf 6.59 0 Td[(1i (2) xi+1^i=xi (2) xj^i=xjwhenj6=iandj6=i+1. (2) Recallthatiswitchesthebraidithhole.Thereisagraphicalinterpretationforthisaction.LetxibeageneratorforM2=D2)-271(fn)]TJ /F1 11.955 Tf 12.55 0 Td[(ptsg,i.e.Wedeterminehowthei Figure2-1. 1Generators generatorsacton1(M2,p).Wethinkof^iastwistingtheloopsxiandxi+1.Noticethattheotherloops,xjdonotmove,i.e.xj^i=xjforallj6=iandj6=i+1. Figure2-2. iActingon1Generators 20

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Now,wecanconstructacoveringspace~M2ofM2bymakingcutsfromeachpuncturetotheboundaryandconnectingcopiesofM2.Forexample,whenn=3wehavethefollowinggure.Thus,thedecktransformationsmappingfromonecopyto Figure2-3. Coverofadiskwith3holes thenextaregivenbythegroup,thecyclicgroupofinniteorder.ThisprocessofpassingfromhomotopytohomologyisknownasAbelianization.Forexample,whenn=3wehavethefollowinggure. Figure2-4. LoopinM2liftstoapathfrom~ptot~p 21

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Bythehomotopyliftingproperty,weknowafterxingabasepointinthecoverthataloopliftsuniquelytoapathinthecover.Wewillnowconsidertheactionon~M2bylookingata1-skeletonofM2anditslift,asweseebelow: Figure2-5. 1-SkeletonofM2anditslift Next,weconsideraspecicexample,with1wehavethata7!aba)]TJ /F5 7.97 Tf 6.58 0 Td[(1from 2 wherea21(M2)from 2-5 and^aisintheliftofa.Let~1denotetheliftoftheaction^1inthe1-skeleton.Thus,a~1=(1)]TJ /F3 11.955 Tf 12.58 0 Td[(t)a+bandsimilarlya~1=aandc~1=c.We Figure2-6. InducedactiononZ-cover canrepeattheprocesstocomputetheactionoftheothergenerators.Wewillwritethe 22

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action~ibyamatrix.Thisistheactionof~1,~x17!~x1)]TJ /F3 11.955 Tf 11.96 0 Td[(t~x1+t~x2,~x27!~x1,~x37!~x3.Wecanwrite~1asmatrixbyobservingthat~1:266664~x1~x2~x33777757!2666641)]TJ /F3 11.955 Tf 11.96 0 Td[(tt0100001377775266664~x1~x2~x3377775.WehavenowarrivedattheBuraurepresentationfor1,andsimilarlywecanconstructmatrixrepresentationfortheothergenerators.Fromourexampleinwhichn=3and~1wemaychooseadifferentbasis,(~x1+t~x2+t2~x3,~x2)]TJ /F8 11.955 Tf 14.02 0 Td[(~x1,~x2)]TJ /F8 11.955 Tf 14.03 0 Td[(~x3)Thisyields^b(i)266664~x1+t~x2+t2~x3~x2)]TJ /F8 11.955 Tf 14.02 0 Td[(~x1~x2)]TJ /F8 11.955 Tf 14.02 0 Td[(~x3377775=2666641 00 0 )]TJ /F3 11.955 Tf 9.3 0 Td[(t00 )]TJ /F8 11.955 Tf 9.3 0 Td[(11377775266664x1+tx2+t2x3x2)]TJ /F3 11.955 Tf 11.95 0 Td[(x1x2)]TJ /F3 11.955 Tf 11.95 0 Td[(x3377775ThematrixintheblockiscalledthereducedBuraurepresentation.ForabraidgroupB(n),weonlyneedtoconsiderthereducedBuraurepresentationsinGLn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(Z[t,t)]TJ /F5 7.97 Tf 6.59 0 Td[(1])tocomputethelowerboundforentropy. 2.2ReducedBurauRepresentationWedeneahomomorphismrb:B(n)!GLn)]TJ /F5 7.97 Tf 6.58 0 Td[(1(Z[t,t)]TJ /F5 7.97 Tf 6.59 0 Td[(1]) 23

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calledthereducedBuraurepresentation,bytheimagesofthegeneratorsofB(n).rb(1)=266664)]TJ /F3 11.955 Tf 9.3 0 Td[(t1...00I377775rb(j)=26666666666666666641...100t)]TJ /F3 11.955 Tf 9.3 0 Td[(t1001...13777777777777777775where(0,...,0,t,)]TJ /F3 11.955 Tf 9.29 0 Td[(t,1,0,...,0)occursonthejthrowforr(j)and1
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Theorem2.2(BandandBoyland). ForapAbraid,theBurauestimateissharpfortheentropyatsomerootofunityifandonlyifitissharpat)]TJ /F8 11.955 Tf 9.3 0 Td[(1. 2.4ConventionandNotationTheBraidgroupon(n+1)-stringsf1,...,ngistraditionallythoughtofasacompositionoffunctions,i.e.thebraid12isreadfromrighttoleftsowewouldperform2rst.However,matrixmultiplicationisperformedfromlefttotheright.Foreaseofnotation,wewouldliketheorderoftheBraidgroupandtheBuraurepresentationtobereadinthesameway.WeareonlyinterestedinthespectralradiusoftheBuraumatricesandamatrixanditstransposehavethesamespectralradius.(rb(12))=(rb(2)rb(1))=((rb(2)r(1))T)=(rb(1)Trb(2)T)Noticethattheorderonthelefthandsidematchestheorderontherighthandside.Itisbecauseofthisthatwewilluser:B(n)!GLn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(Z[t,t)]TJ /F5 7.97 Tf 6.59 0 Td[(1])wherer(i)=rb(i)T. 25

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CHAPTER3FUNDAMENTALSTIRRINGPROTOCOLS 3.1DenitionFortherestofthispaper,weonlyconsidertheveryspecialcasewherewehavenobstaclesandasinglestirrer.Inthisspeciccase,wecallstirringprotocolsthefundamentalstirringprotocols.Inotherwords,afundamentalstirringprotocolcanbethoughtofasapurebraid,whereonlyonebraidstringisallowedtomovearoundtheotherstationarybraidstrings. Denition5(FundamentalStirringProtocol). Considerndistinctpointsx1,x2,...,xnintheinteriorofD2.WewilldeneafundamentalstirringprotocolFSP,byFSP=(f1(t),...,fn(t))wherefi:[0,1]!D2suchthat: 1. f1iscontinuous, 2. fiisconstantforeachi>1, 3. fi(t)6=fj(t)ifi6=j,forallt2[0,1], 4. fi(0)=xiforeachi, 5. f1(1)=f1(0).Therststirrerassociatedwithf1willbecalleda)]TJ /F1 11.955 Tf 9.3 0 Td[(stirrer,whiletheotherstirrerswillbereferredtoasobstacles. 3.2CongurationSpacesThefundamentalgroupofthecongurationspacewillgiveanalternatedenitionofthestirringprotocol.Thisnewwayatlookingatastirringprotocolwillgiveinsightonthegeneratorsneededtodescribethefundamentalstirringprotocol.LetMbeamanifoldofdimensiongreaterthanorequalto2,letQni=1Mdenotethen-foldproductspace,andletF0,ndenotethespace F0,n(M)=f(z1,...,zn)2nYi=1Mjzi6=zjifi6=jg(3)LetQm=fq1,...qmgbeasetofxeddistinguishedpointsofM.NowdeneFm,n(M)=F0,n(M)]TJ /F3 11.955 Tf 11.95 0 Td[(Qm). 26

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LetM=D2bethediskwiththeusualEuclideanmetric.FromBirman,[ 19 ]weseethatthefundamentalgroup1(F0,n(D2))=PnwherePnisthepurebraidgroup.Thepointsfz1,...,znginequation 3 areorderedandwecanpermutethembyactingwiththesymmetricgroup,Pn.Therightaction:F0,n(M)Pn!F0,nisdenedby((x0,...,xn),a)=(xa(0),,xa(n)).PnactsfreelyonF0,n.Ingeneral,wehave1(F0,n(D2))=Pn=Bn,wherePnisthesymmetricgroupandBnisthefullbraidgroup.ConsiderthebraidgroupCn=1(Fn)]TJ /F5 7.97 Tf 6.58 0 Td[(1,1),whichisasubgroupofPn(andthusasubgroupofBn)describingthecasewheretherearen)]TJ /F8 11.955 Tf 12.38 0 Td[(1obstaclesandonestir.Wehavethefollowingtheorem. Theorem3.1. Thegroupformedbythebraidsgeneratedbyasinglestirrerwithn)]TJ /F8 11.955 Tf 12.39 0 Td[(1obstaclesinthediskD2isisomorphictoFn)]TJ /F5 7.97 Tf 6.59 0 Td[(1,thefreegrouponn)]TJ /F8 11.955 Tf 11.96 0 Td[(1letters. Proof. Cn(D2)=1(Fn,1(D2))=1(D2)]TJ /F3 11.955 Tf 11.96 0 Td[(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=Fn)]TJ /F5 7.97 Tf 6.59 0 Td[(1whereFmisthefreegroupgeneratedbymelements. Wecanrepresentbraidsbyanitenumberofj's.Inourspeciccase,therstbraidstring,(thestringtofar-left),isallowedtomovearoundtheothern)]TJ /F8 11.955 Tf 12.28 0 Td[(1obstacles.InFigure 3-1 ,weseeanexample.Letkdenotethebraid,wheretherstbraidstringloopsclockwisearoundthejthstringpassinginfrontoftheotherstrings.Wecanrepresentkintermsof's,as:k=)]TJ /F5 7.97 Tf 6.59 0 Td[(11)]TJ /F5 7.97 Tf 6.59 0 Td[(12)]TJ /F5 7.97 Tf 6.59 0 Td[(13...)]TJ /F5 7.97 Tf 6.58 0 Td[(1k)]TJ /F5 7.97 Tf 6.59 0 Td[(2)]TJ /F5 7.97 Tf 6.59 0 Td[(1k)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F5 7.97 Tf 6.59 0 Td[(1k)]TJ /F5 7.97 Tf 6.59 0 Td[(1k)]TJ /F5 7.97 Tf 6.59 0 Td[(2...21where,1
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Figure3-1. Thetheleftstirrermovesaroundthe4thobstacle arenorelationsamongthegenerators.TheyformthefreegroupFnasmentionedinTheorem 3.1 Example1. Considerthecasewith4obstaclesandafundamentalstirrer.Letbethestirringprotocolwhere=223)]TJ /F5 7.97 Tf 6.58 0 Td[(145.Let#denotethenumberof'sinthestirringprotocol.Soforourpreviousexample,weseethat#=5.Wedenewhatismeantbytopologicalefciency. Denition6(TopologicalEfciency). Letbeafundamentalstirringprotocol.LetbethemapfromtheThurston'sClassicationintheisotopyclassoftheprotocol.Lethdenotethetopologicalentropyof,thenthetopologicalefciencyisE()=h() #Awaytothinkaboutthisdenitionistheentropypergenerator.Theentropywillusuallyincreasewithmore's,however,theefciencymaylower. 28

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CHAPTER4NUMERICALEXPERIMENTSTohelpwiththeinvestigation,therewerethreetypesofnumericalexperimentsthatweredone.Theresultswillbediscussedlast.Fromeachexperiment,wewerelookingforpatternsandtrendswiththeoverallefciencyofthefundamentalmixingprotocols.InapaperbyBoyland,Aref,andStremler[ 20 ]statedthatatleast3stirrersareneededtoinduceamixingprotocol.Sowewillassumethatnisgreaterthanorequaltothree. 4.1GroupActionsWewillassumethatwehavean)]TJ /F8 11.955 Tf 10.68 0 Td[(1obstaclesandafundamentalstirrer.Recallthatthegroupformedbyasinglestirrerwithn)]TJ /F8 11.955 Tf 12.3 0 Td[(1obstaclesis1(D2)]TJ /F3 11.955 Tf 12.3 0 Td[(Qn)]TJ /F5 7.97 Tf 6.59 0 Td[(1)=Fn)]TJ /F5 7.97 Tf 6.58 0 Td[(1.Thestirrerinducesanactiononthefundamentalgroupofthediskminusnpoints,namely,k:Fn)]TJ /F5 7.97 Tf 6.59 0 Td[(1!Aut(Fn).Adirectcalculationshowsthat k(xj)=8>>>>>>><>>>>>>>:xjifj
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Bowen[ 4 ]withatheoremin[ 5 ]tellsushowtocomputetheentropyofapseudo-Anosovmapusingwordgrowth.Namely,h(SP)=supxj21(M2)limm!1log(L(SPmxj))=m.Fortheexperiment,variousxj'swerepicked.Iftheprotocolistrulymixing,i.e.haspositiveentropy,thenthewordlengthwillgrowexponentially.Ascriptingcomputerlanguage,Perl,wasusedtotestthetopologicalefciencyofdifferentstirringprotocols.However,theprogramwouldrunoutofmemory(2gigs)after10iterationsforanyprotocolthathadpositiveentropy. 4.2BurauRepresentationThesecondexperimentwastocomputetheBuraurepresentationforthej'sgeneratorswiththesubstitutiont=)]TJ /F8 11.955 Tf 9.3 0 Td[(1asstatedabove.Thenrandomlygeneratedstirringprotocolsofdifferentlengths.Theentropywasapproximatedbythelogarithmofthespectralradius.ThisprogramwaswritteninMathematica. 4.3MousirAlgorithmTheprevioustwomethodsonlygaveapproximations.Thenextonegivestheexactentropy.ThetwoprogramswerewrittenbyTobyHall[ 21 ].Therstprogramtrains.execomputestheentropyofabraidbyusingtheBestvina-Handelalgorithm.Thesecondprogramdynn.execomputestheentropyofabraidbyusingtheMousirAlgorithm.RandomfundamentalstirringprotocolswerecreatedandwereturnedintobraidsasdescribedabovebyaC++programandtheninputedthatwordintoTobyHall'sprograms.TheoutputwasanalyzedbyExcelwheretopologicalefciencywascomputed. 4.4ResultsofExperimentationEveryexperimentyieldedthesameresult,namelytheprotocoln)]TJ /F5 7.97 Tf 6.59 0 Td[(1n)]TJ /F5 7.97 Tf 6.59 0 Td[(22hasthelargestefciencyforn)]TJ /F8 11.955 Tf 12.52 0 Td[(1obstacleswithasinglestirrer.Wewillcallthisprotocol 30

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HSPn.Rememberthatthefundamentalstirrercanalsobeconsideredasabraidstring,soHSPnhasatotalofnbraidstringsorinotherwords,then)]TJ /F8 11.955 Tf 12.78 0 Td[(1obstaclesandthefundamentalstirrer.Ifwechangethepositionsoftheopenholes,weseethatHSPhasalotsymmetry.InFigure 4-1 weseethecaseforHSP6. Figure4-1. TwoversionsofHSP6 Anotherinterestingfactthatcameout,isthattheefciencydoesnotdependonessentialcrossings,theminimalnumberoftimesthefundamentalstirrercrossesitsownpath,ofthestirringprotocol.LetuscomparethetwofollowingstirringprotocolsinFigure 4-2 .ThestirringprotocolontheleftisHSP6,whilethestirringprotocolontherightusesthesamealpha'sastheleftbutinadifferentorder.Theprotocolonthelefthasthreetimestheamountoftopologicalentropyastheoneontheright. Figure4-2. Theleftstirringprotocolismoreefcient. InapaperbyFinn,CoxandBryne[ 8 ],anumericalexperimentwasrunonrandomstirringprotocols,withoutobstacles.Specically,HSP8'spathwasstudiedbutminusany 31

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obstacles.TheirexperimentfocusedonspecicuidswhereastheHSPwasderivedwithoutspecicuidpropertiesinmind.Theauthorsmadeanoteintheirpaper,thatHSP8protocoldidnottraversethroughthecenteroftheuid,whiletheirotherprotocolsdid,andyetHSP8stillperformedremarkablywell.Theyalsouseddifferentmeasurestodeterminemixing,likediffusion,whichisnotconsideredhere. 32

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CHAPTER5PROPERTIESOFHSPFromtheexperimentsweconjecturethefollowing: Conjecture1. Themostefcientfundamentalstirringprotocolwithn)]TJ /F8 11.955 Tf 12.42 0 Td[(1obstaclesisHSPn=2...n. 5.1EntropyLowerBoundforHSPInthissectionwegivesomeofthepropertiesofHSPn.InFigure 5-1 weseethebehavioroftheefciencyofHSPovernwherenisthenumberofobstacles. Figure5-1. EfciencyofHSP Wewilldenote[HSPnastheBuraurepresentationofHSPnevaluatedatt=)]TJ /F8 11.955 Tf 9.3 0 Td[(1. Theorem5.1. LetHSPnbethestirringprotocoldenedinthepreviouschapter.Thenthetraceof[HSPnis)]TJ /F8 11.955 Tf 9.3 0 Td[(3n)]TJ /F5 7.97 Tf 6.58 0 Td[(1+3n)]TJ /F8 11.955 Tf 11.95 0 Td[(2. Proof. Firstwewillintroducenotation.RecalltheinotationfromtheArtinBraidinsection 1.2 .Wewillnowkeeptrackofthetotalnumberofstrings.Soforiinabraidgroupwithatotalofn-strings,wewilldenoteasi,n.Also,forsimplicity,wewilluseahatoverabraidtodenotethereducedBuraurepresentationevaluatedatt=)]TJ /F8 11.955 Tf 9.3 0 Td[(1.ConsiderthereducedBuraurepresentationforthebraidgeneratorsfornbraidsf^1,n,^2,n,...,^n)]TJ /F5 7.97 Tf 6.58 0 Td[(1,ngandcomparethattothegeneratorsforn+1braidsf^1,n+1,^2,n+1,...,^n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n+1,^n,n+1g. 33

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BythedenitionofthereducedBuraurepresentationand1in)]TJ /F8 11.955 Tf 11.96 0 Td[(1wehave,^i,n+1=2666666666664 0^i,n ... 0 0...0 13777777777775Now,wewillalsodenei,ninasimilarmannerasthei,n,wheretheadditionalktellsusthetotalnumberofstrings.For2knwehavethefollowing:k,n=)]TJ /F5 7.97 Tf 6.59 0 Td[(11,n)]TJ /F5 7.97 Tf 6.58 0 Td[(12,n)]TJ /F5 7.97 Tf 6.58 0 Td[(13,n...)]TJ /F5 7.97 Tf 6.58 0 Td[(1k)]TJ /F5 7.97 Tf 6.59 0 Td[(2,n)]TJ /F5 7.97 Tf 6.59 0 Td[(1k)]TJ /F5 7.97 Tf 6.58 0 Td[(1,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1k)]TJ /F5 7.97 Tf 6.58 0 Td[(1,nk)]TJ /F5 7.97 Tf 6.59 0 Td[(2,n...2,n1,nFor2jn)]TJ /F8 11.955 Tf 11.95 0 Td[(1wehave^j,n+1=2666666666664 0^j,n ... 0 0...0 13777777777775 34

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Now,wejusthavetond^n,n+1and^n+1,n+1.Thesecasesaredifferentbecausethelastrowof^n,n+1and^n+1,n+1havemorenonzeroentries.Firstwewillcompute,^n,n+1.^n,n+1=(^1,n+1^2,n+1...^n)]TJ /F5 7.97 Tf 6.59 0 Td[(2,n+1)^)]TJ /F5 7.97 Tf 6.58 0 Td[(2n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,n+1(^1,n+1^2,n+1...^n)]TJ /F5 7.97 Tf 6.59 0 Td[(2,n+1))]TJ /F5 7.97 Tf 6.59 0 Td[(1=(^1,n+1^2,n+1...^n)]TJ /F5 7.97 Tf 6.59 0 Td[(2,n+1)2666666666640In)]TJ /F5 7.97 Tf 6.59 0 Td[(1...00...021)]TJ /F8 11.955 Tf 9.3 0 Td[(20...01377777777775(^1,n+1^2,n+1...^n)]TJ /F5 7.97 Tf 6.58 0 Td[(2,n+1))]TJ /F5 7.97 Tf 6.59 0 Td[(1=2666666666666666664000...011100...011010...011001...011...00...0111000...00137777777777777777752666666666666666664100...000010...000001...000...000...10000...021)]TJ /F8 11.955 Tf 9.3 0 Td[(2000...00137777777777777777752666666666666666664000...011100...011010...011001...011...00...0111000...0013777777777777777775)]TJ /F5 7.97 Tf 6.58 0 Td[(1=2666666666664 )]TJ /F8 11.955 Tf 9.29 0 Td[(2^n,n ... )]TJ /F8 11.955 Tf 9.29 0 Td[(2 0...0 13777777777775 35

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Sincethe[HSPn=^n,n^n)]TJ /F5 7.97 Tf 6.59 0 Td[(1,nn)]TJ /F5 7.97 Tf 6.58 0 Td[(2,n^2,n,wehavethat[HSPn+1=^n+1,n+1266666664)]TJ /F8 11.955 Tf 9.3 0 Td[(2[HSPn...)]TJ /F8 11.955 Tf 9.3 0 Td[(20...01377777775where^n+1,n+1=2666666666643 00 )]TJ /F8 11.955 Tf 9.29 0 Td[(22 )]TJ /F8 11.955 Tf 9.29 0 Td[(2... In)]TJ /F5 7.97 Tf 6.59 0 Td[(2 ...2 )]TJ /F8 11.955 Tf 9.29 0 Td[(22 00 )]TJ /F8 11.955 Tf 9.29 0 Td[(1377777777775andIn)]TJ /F5 7.97 Tf 6.59 0 Td[(2denotesthen)]TJ /F8 11.955 Tf 11.96 0 Td[(2byn)]TJ /F8 11.955 Tf 11.96 0 Td[(2identitymatrix.Thenbyconstruction,[HSPn+1=^n+1266666664)]TJ /F8 11.955 Tf 9.3 0 Td[(2[HSPn...)]TJ /F8 11.955 Tf 9.3 0 Td[(20...01377777775[HSPn(i,j)=8>>>>>>>>>><>>>>>>>>>>:3[HSPn)]TJ /F5 7.97 Tf 6.58 0 Td[(1(1,j)i=1and1jn)]TJ /F8 11.955 Tf 11.96 0 Td[(2)]TJ /F8 11.955 Tf 9.3 0 Td[(8i=n)]TJ /F8 11.955 Tf 11.96 0 Td[(1and1jn)]TJ /F8 11.955 Tf 11.96 0 Td[(22[HSPn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1,j)+[HSPn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(i,j)2in)]TJ /F8 11.955 Tf 11.95 0 Td[(2and2jn)]TJ /F8 11.955 Tf 11.96 0 Td[(22[HSPn)]TJ /F5 7.97 Tf 6.58 0 Td[(1(1,j)1in)]TJ /F8 11.955 Tf 11.95 0 Td[(2andj=n)]TJ /F8 11.955 Tf 11.96 0 Td[(1)]TJ /F8 11.955 Tf 9.3 0 Td[(5i=n)]TJ /F8 11.955 Tf 11.96 0 Td[(1andj=n)]TJ /F8 11.955 Tf 11.95 0 Td[(1Fromthiscomplicateddescription,wecanseeaformulaforthetraceTr([HSPn).Forconvenience,letAn=[HSPnandletTn=Tr(An). 36

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Tn=An(1,1)+An(n,n)=3An)]TJ /F5 7.97 Tf 6.58 0 Td[(1(1,1)+2n)]TJ /F5 7.97 Tf 6.59 0 Td[(2Xk=2An)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1,k)+n)]TJ /F5 7.97 Tf 6.59 0 Td[(2Xk=2An)]TJ /F5 7.97 Tf 6.59 0 Td[(1(k,k))]TJ /F8 11.955 Tf 11.95 0 Td[(5=2An)]TJ /F5 7.97 Tf 6.58 0 Td[(1(1,1)+2n)]TJ /F5 7.97 Tf 6.59 0 Td[(2Xk=2An)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1,k)+n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xk=1An)]TJ /F5 7.97 Tf 6.59 0 Td[(1(k,k)=2An)]TJ /F5 7.97 Tf 6.58 0 Td[(1(1,1)+2n)]TJ /F5 7.97 Tf 6.59 0 Td[(2Xk=2An)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1,k)+Tn)]TJ /F5 7.97 Tf 6.59 0 Td[(1Thus,wehavethefollowingequationforthetrace, Tn=2An)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1,1)+2n)]TJ /F5 7.97 Tf 6.59 0 Td[(2Xk=2An)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1,k)+Tn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(5)LetDn)]TJ /F5 7.97 Tf 6.59 0 Td[(1=Pn)]TJ /F5 7.97 Tf 6.59 0 Td[(2k=2An)]TJ /F5 7.97 Tf 6.58 0 Td[(1(1,k).NotethatthesubscriptisusedtoindicatethesumofthetoprowforAn)]TJ /F5 7.97 Tf 6.58 0 Td[(1.Bytheinductiveconstruction,weseethat D3=)]TJ /F8 11.955 Tf 9.3 0 Td[(8 (5) D4=)]TJ /F8 11.955 Tf 9.3 0 Td[(8)]TJ /F8 11.955 Tf 11.96 0 Td[(38 (5) D5=)]TJ /F8 11.955 Tf 9.3 0 Td[(8)]TJ /F8 11.955 Tf 11.96 0 Td[(38)]TJ /F8 11.955 Tf 11.96 0 Td[(328 (5) ... (5) Dn)]TJ /F5 7.97 Tf 6.58 0 Td[(1=)]TJ /F8 11.955 Tf 9.3 0 Td[(8n)]TJ /F5 7.97 Tf 6.59 0 Td[(4Xk=03k=)]TJ /F8 11.955 Tf 9.29 0 Td[(4(3(n)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F8 11.955 Tf 11.95 0 Td[(9) 9 (5) Also,thersttermof 5 hasapattern.Again,westartwithn=3. A3(1,1)=3 (5) A4(1,1)=32 (5) ... (5) An)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1,1)=3n)]TJ /F5 7.97 Tf 6.59 0 Td[(3 (5) 37

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Whenwesubstitute, 5 and 5 backinto 5 wehavethefollowingequation: Tn=)]TJ /F8 11.955 Tf 9.29 0 Td[(23n)]TJ /F5 7.97 Tf 6.58 0 Td[(2+3+Tn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(5)Whenn=3,weknowthatT3=)]TJ /F8 11.955 Tf 9.3 0 Td[(2,ourinitialcondition.Buildingfromourinitialconditionand 5 ,wehave, Tn=)]TJ /F8 11.955 Tf 9.3 0 Td[(3n)]TJ /F5 7.97 Tf 6.59 0 Td[(1+3n)]TJ /F8 11.955 Tf 11.96 0 Td[(2(5) FromthistheoremwehaveacorollaryabouttheefciencyofHSPn.ThiscorollaryusestheconceptoftraintracksinordertoinvokeatheorembyBandandBoyland[ 18 ].Agoodsourcefortraintracksinclude[ 15 ],[ 22 ]and[ 23 ].Supposef:M2!M2,thentheideaofatraintrackforfisanembeddedgraphonthesurfaceM2thatsatisesseveralconditionsandisinvariantunderf. Corollary1. LetHSPbethestirringprotocoldenedasabove.Thenlimsupn!1E(HSPn)ln(3) Proof. ByadirectcalculationthetraintrackofeachHSPnhasasingularitystructurewhichconsistsofjustone-prongsatthepuncturesandtheusualsingularityatinnity.BythepaperbyBandandBoyland[ 18 ](theorem5.1inthepaper),weseethatthisimpliestheBurauestimateforHSPnissharpforthesubstitutiont=)]TJ /F8 11.955 Tf 9.3 0 Td[(1. Figure5-2. TrainTrackforthecasen=5.Thereddiskdenotesthestirrer. Asbefore,let[HSPnbetheBuraurepresentationevaluatedatt=)]TJ /F8 11.955 Tf 9.3 0 Td[(1.Let([HSPn)denotethespectralradiusof[HSPn.Let1,,n)]TJ /F5 7.97 Tf 6.58 0 Td[(1denotetheeigenvaluesof[HSPn. 38

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Notethatthereareonlyn)]TJ /F8 11.955 Tf 13.08 0 Td[(1eigenvaluessinceweareusingthereducedBuraurepresentation.Withoutlossofgenerality,assumethat([HSPn)=j1j.jTr([HSPn)j=j1+2+n)]TJ /F5 7.97 Tf 6.59 0 Td[(1jj1j+j2j+jn)]TJ /F5 7.97 Tf 6.58 0 Td[(1j(n)]TJ /F8 11.955 Tf 11.96 0 Td[(1)j1jThusbytheprevioustheorem 5.1 wehave,([HSPn)=j1jjTr([HSPn)j n)]TJ /F8 11.955 Tf 11.95 0 Td[(1=)]TJ /F8 11.955 Tf 9.3 0 Td[(3n)]TJ /F5 7.97 Tf 6.58 0 Td[(1+3n)]TJ /F8 11.955 Tf 11.95 0 Td[(2 n)]TJ /F8 11.955 Tf 11.96 0 Td[(1=3n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 11.96 0 Td[(3n+2 n)]TJ /F8 11.955 Tf 11.95 0 Td[(1Nowwehaveaboundfortheefciency.limsupn!1E(HSPn)=limsupn!1log(([HSPn)) n)]TJ /F8 11.955 Tf 11.96 0 Td[(1limn!1log3n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F8 11.955 Tf 11.96 0 Td[(3n+2 n)]TJ /F8 11.955 Tf 11.96 0 Td[(1 n)]TJ /F8 11.955 Tf 11.95 0 Td[(1=limn!1log(3n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 11.95 0 Td[(3n+2))]TJ /F8 11.955 Tf 11.96 0 Td[(log(n)]TJ /F8 11.955 Tf 11.96 0 Td[(1) n)]TJ /F8 11.955 Tf 11.96 0 Td[(1=log(3) Toobtainequalityinthecorollary,weneedtoknowmoreabouttheeigenvalues.Forexample,ifknewthatthesumoveralltheeigenvalues,excepttheonethatisthespectralradius,grewlinearly,thenwewouldknowthattheasymptoticlimitoftheefciencyofHSPnisexactlylog(3).Itturnsoutthattherstfewcharacteristicpolynomialsof[HSPnarepalindromic/anti-palindromicSalempolynomials.AgoodreferenceforSalempolynomialsis[ 24 ] Denition7(palindromic/anti-palindromicpolynomials). Apolynomialp(x)2Z[x]issaidtopalindromicifitsatisesp(x)=xmp(1=x) 39

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wheremisthedegreeofp(x).Ifthepolynomialsatisesp(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(xmp(1=x)wheremisthedegreeofp(x)thenp(x)issaidtobeanti-palindromic.Inotherwords,ifthepolynomialisinstandardformandthecoefcientsarethesamewhenreadfrontwardsaswellasbackwardswesaythepolynomialispalindromic.Anti-palindromicisthesameaspalindromic,exceptthatthecoefcientsdifferbyanegativesign. Denition8(Salemnumber). ASalemnumberisarealalgebraicinteger,greaterthan1,withthepropertythatallitsGaloisconjugateslieonorwithintheunitcircle,andatleastoneGaloisconjugateliesontheunitcircle.AnirreduciblepolynomialthathasarootthatisaSalemnumberiscalledaSalemPolynomial.Letfn(x)bethecharacteristicpolynomialfor[HSPn.Herearejusttherstfewpolynomials:f3(x)=1+17x)]TJ /F8 11.955 Tf 11.95 0 Td[(17x2)]TJ /F3 11.955 Tf 11.95 0 Td[(x3=)]TJ /F8 11.955 Tf 9.3 0 Td[((x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)(1+18x+x2)f4(x)=1+68x)]TJ /F8 11.955 Tf 11.95 0 Td[(122x2+68x3+x4f5(x)=1+227x)]TJ /F8 11.955 Tf 11.95 0 Td[(542x2+542x3)]TJ /F8 11.955 Tf 11.95 0 Td[(227x4)]TJ /F3 11.955 Tf 11.95 0 Td[(x5=)]TJ /F8 11.955 Tf 9.3 0 Td[((x)]TJ /F8 11.955 Tf 11.95 0 Td[(1)(1+228)]TJ /F8 11.955 Tf 11.95 0 Td[(314x2+228x3+x4)Itappearsthatthecharacteristicpolynomialsareeitherpalindromicoranti-palindromicbasedontheirparity.ItturnsoutthatanyBuraumatrixwitht=)]TJ /F8 11.955 Tf 9.3 0 Td[(1enjoysthisproperty.LetBn(t)beanbynmatrixandBuraurepresentation.LetBn(t)=Bn(1=t),andBn(t)=Bn(t)TwhereTisthetranspose.Thus,Bn()]TJ /F8 11.955 Tf 9.3 0 Td[(1)=Bn()]TJ /F8 11.955 Tf 9.3 0 Td[(1)WemayassumethatBisthereducedBuraurepresentation(n)]TJ /F8 11.955 Tf 11.95 0 Td[(1)by(n)]TJ /F8 11.955 Tf 11.96 0 Td[(1). 40

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Theorem5.2(Squire). TheBurauRepresentationisUnitary,i.e.Thereexistsanon-singularmatrixJ2GL(n)]TJ /F8 11.955 Tf 11.95 0 Td[(1,Z)suchthatBJB=JTheproofmaybefoundinthepaperbySquier[ 19 ] Theorem5.3. ThecharacteristicpolynomialofBwhent=)]TJ /F8 11.955 Tf 9.3 0 Td[(1iseitherpalindromicoranti-palindromicdependingontheparityofn)]TJ /F8 11.955 Tf 11.95 0 Td[(1. Proof. char(B)(x)=det(B)]TJ /F3 11.955 Tf 11.96 0 Td[(xI)=det(J)]TJ /F5 7.97 Tf 6.59 0 Td[(1(B))]TJ /F5 7.97 Tf 6.59 0 Td[(1J)]TJ /F3 11.955 Tf 11.95 0 Td[(xI)=det((B))]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xI)=det((B))]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(BxI(B))]TJ /F5 7.97 Tf 6.59 0 Td[(1)=det(I)]TJ /F3 11.955 Tf 11.96 0 Td[(xB)det((B))]TJ /F5 7.97 Tf 6.58 0 Td[(1)=det(B))]TJ /F5 7.97 Tf 6.59 0 Td[(1det(I)]TJ /F3 11.955 Tf 11.95 0 Td[(xB)=det(B))]TJ /F5 7.97 Tf 6.59 0 Td[(1det()]TJ /F3 11.955 Tf 9.3 0 Td[(x(B)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F5 7.97 Tf 6.58 0 Td[(1I))=()]TJ /F3 11.955 Tf 9.3 0 Td[(x)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1 det(B)det(B)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F5 7.97 Tf 6.59 0 Td[(1I)Sodet(B)=det(BT)=det(B)=det(k1k2...km)=det(k1)det(km)Whent=)]TJ /F8 11.955 Tf 9.3 0 Td[(1,thendet(ki)=1,sodet(B)=1.LetB)]TJ /F5 7.97 Tf 6.59 0 Td[(1denoteBwitht=)]TJ /F8 11.955 Tf 9.29 0 Td[(1.Thus, 41

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char(B)]TJ /F5 7.97 Tf 6.59 0 Td[(1)(x)=()]TJ /F3 11.955 Tf 9.3 0 Td[(x)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1det(B)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F5 7.97 Tf 6.59 0 Td[(1I)=()]TJ /F3 11.955 Tf 9.3 0 Td[(x)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1det(BT)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F5 7.97 Tf 6.59 0 Td[(1I)=()]TJ /F3 11.955 Tf 9.3 0 Td[(x)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1det(B)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)]TJ /F5 7.97 Tf 6.59 0 Td[(1I)T=()]TJ /F3 11.955 Tf 9.3 0 Td[(x)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1det(B)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F5 7.97 Tf 6.59 0 Td[(1I)=()]TJ /F3 11.955 Tf 9.3 0 Td[(x)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1det(B)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)]TJ /F5 7.97 Tf 6.59 0 Td[(1I)=()]TJ /F3 11.955 Tf 9.3 0 Td[(x)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1char(B)]TJ /F5 7.97 Tf 6.58 0 Td[(1)(1=x)Letp(x)=char(B)(x),thenwehavetwocases. 1. Ifn)]TJ /F8 11.955 Tf 11.96 0 Td[(1iseven,thenp(x)=xn)]TJ /F5 7.97 Tf 6.59 0 Td[(1p(1=x)andpispalindromic. 2. Ifn)]TJ /F8 11.955 Tf 11.96 0 Td[(1isodd,thenp(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(xn)]TJ /F5 7.97 Tf 6.59 0 Td[(1p(1=x)andpisanti-palindromic. Weshouldnotethatifp(x)isananti-palindromicpolynomial,thenp(1)=)]TJ /F3 11.955 Tf 9.3 0 Td[(p(1)impliesthatp(1)=0.Thus,ifn)]TJ /F8 11.955 Tf 12.56 0 Td[(1isodd,thenchar(B)(x)=(x)]TJ /F8 11.955 Tf 12.56 0 Td[(1)q(x)forsomepolynomialq(x). Theorem5.4. SupposeBisaBurauMatrixwitht=)]TJ /F8 11.955 Tf 9.3 0 Td[(1thatis(n)]TJ /F8 11.955 Tf 12.14 0 Td[(1)(n)]TJ /F8 11.955 Tf 12.13 0 Td[(1)wheren)]TJ /F8 11.955 Tf 11.95 0 Td[(1isodd.Thenchar(B)(x)=(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)q(x)whereq(x)ispalindromic. Proof. First,weseethatchar(B)(x)=(x)]TJ /F8 11.955 Tf 10.99 0 Td[(1)q(x)andthatthedegreeofqisn)]TJ /F8 11.955 Tf 11 0 Td[(2,thus,char(B)(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(xn)]TJ /F5 7.97 Tf 6.59 0 Td[(1char(B)(1=x)(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)q(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(xn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1=x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)q(1=x)x(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)q(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(xn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)q(1=x)x(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)q(x)=xn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)q(1=x)(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)q(x)=xn)]TJ /F5 7.97 Tf 6.59 0 Td[(2(x)]TJ /F8 11.955 Tf 11.96 0 Td[(1)q(1=x)q(x)=xn)]TJ /F5 7.97 Tf 6.59 0 Td[(2q(1=x) 42

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Thus,everystirringprotocolwillhaveaBuraurepresentationwitht=)]TJ /F8 11.955 Tf 9.3 0 Td[(1whosecharacteristicpolynomialsarepalindromic/anti-palindromicpolynomials.However,therstfewcharacteristicpolynomialsfor[HSParenicerstill.TheyareSalempolynomials.Usingthetechniquefrom[ 24 ],weseethatadirectcomputationshowsthattherstfewcharacteristicpolynomialsoftheBuraurepresentationwitht=)]TJ /F8 11.955 Tf 9.3 0 Td[(1forHSPareSalempolynomials. Conjecture2. ThecharacteristicpolynomialoftheBuraurepresentationwitht=)]TJ /F8 11.955 Tf 9.3 0 Td[(1ofHSPnisaSalempolynomial.If[HSPnhasacharacteristicpolynomialthatisaSalempolynomialthenwewouldhavesharpnessincorollary 1 .SinceHSPisconstructedinductively,asseenabove,itwouldbeniceifthecharacteristicpolynomialsalsoenjoyedthatproperty. Conjecture3. Letgn(x)bethecharacteristicpolynomialfor[HSP(n),thengn+1(x)=gn(x)q(x)+gn)]TJ /F5 7.97 Tf 6.59 0 Td[(1(x)p(x)whereq(x)=)]TJ /F8 11.955 Tf 9.3 0 Td[(4(x)]TJ /F8 11.955 Tf 11.95 0 Td[(1)andp(x)=)]TJ /F8 11.955 Tf 9.3 0 Td[((3x2)]TJ /F8 11.955 Tf 11.95 0 Td[(10x+3).Thisconjecturewasbasedontheassumptionthatthecharacteristicpolynomialswereinductiveandhasbeenveriedfortherst20polynomials. 5.2GearabilityofHSPOneconcernaboutuidmixingandstirringprotocolsisthepracticalityofthestirringdeviceitself.Forexample,astirringprotocolwouldbeuselessifthemachineusedtostirtheuidwastoocomplicated.ToquoteKobayashiandUmedain[ 25 ]...fromtheviewpointofpracticalmixing,realizingsuchadeviceviasimplemechanism,ormechanismrequiringlessenergyisanimportantissue. Denition9(Gearable). Astirringprotocoliscalledgearableifadeviceusinganitesetofgearscanmoveastirringrodalongthepathofsaidstirringprotocol. 43

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WewillnowshowthatHSPnisinfactgearablebyshowingthatthepathofthestirrerfollowsthatofahypotrochoid.Supposewehavetwocirclesc1andc2wherec2iscenteredattheoriginandtheradiusofc1israndc2isRwhereR>r.Wesupposethatc1goesalongtheinsideofc2inacounterclockwisedirectionwithoutslipping.Apointpistheddistancefromthecenterofc1.Asc1travelsaroundc2,thepathofpisthepathofthehypotrochoid.Belowaretheparametricequationswheregoesfrom0to2.x()=(R)]TJ /F3 11.955 Tf 11.96 0 Td[(r)cos()+dcosR)]TJ /F3 11.955 Tf 11.96 0 Td[(r ry()=(R)]TJ /F3 11.955 Tf 11.96 0 Td[(r)sin())]TJ /F3 11.955 Tf 11.96 0 Td[(dsinR)]TJ /F3 11.955 Tf 11.96 0 Td[(r rInthecaseofHSPn+1,wehaveR=1,r=1=nandd=2=n.Thegure 5-3 isanexampleofHSP8asahypotrochoid.Nowwecanexplaintheencryptedlettersinthe Figure5-3. HSP(8):The7obstaclesarethebluedotsandthepathofthefundamentalstirrerisred. HSPwhichstandforHypotrochoidStirringProtocol.Infact,KobayashiandUmedain[ 25 ]actuallystudieddifferenttypesofhypotrochoidstirringprotocolsspecicallybecauseofthepracticality. 44

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CHAPTER6UPPERBOUNDS 6.1ConstructionofUpperBoundGivenafundamentalstirringprotocolwithn)]TJ /F8 11.955 Tf 12.15 0 Td[(1obstaclesandastirrer,wewillnowgiveanexplicitupperboundfortheefciencywhichwedenedaboveasE(n)=supfE():isafundamentalstirringprotocolwithn)]TJ /F8 11.955 Tf 11.96 0 Td[(1xedholesg.Eventhoughtherearen)]TJ /F8 11.955 Tf 12.04 0 Td[(1xedholesorobstacles,thestirreralsocountsanobstacle.SotheninE(n)iscountingalltheobstaclesintheuid.Wecomputedalowerboundinthepreviouschapter,namely,log((3n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F8 11.955 Tf 11.96 0 Td[(3n+2)=(n)]TJ /F8 11.955 Tf 11.96 0 Td[(1)) n)]TJ /F8 11.955 Tf 11.95 0 Td[(1log((HSPn)) n)]TJ /F8 11.955 Tf 11.96 0 Td[(1E(n).Theconstructionoftheupperboundusesthedenitionofthegeneralizedspectralradius.LetMbeanitesetofmatricesanddenotethespectralradiusofamatrix,i.e.themagnitudeofthelargesteigenvalue.Denek(M)=maxM1,...,Mk2M(M1Mk)1=kInotherwords,foraxedkvalue,weconsiderallproductsthatusekmatricesfromthesetM.SinceMisaniteset,thenthesetofallproductsoflengthkisalsonite.Fromthisnewsetofmatrices,takethespectralradiusofeachmatrixandndthemaximum.Thatmaximumvalueisk(M). Denition10(GeneralizedSpectralRadius). (M)=limsupk!1k(M)Thisdenitionrstappearsin[ 26 ].Forpropertiesofmatricesthatareusedinthissection,pleaserefertotheappendix A.1 45

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Wewillalsoneedthefollowingpropertyoflogarithms.Thispropertyiscommonlyusedindealingwithtopologicalentropyandagoodreferenceis[ 27 ]. Lemma1. Supposefangandfbngaretwosequencesofpositiverealnumberssuchthatlimn!1(logan)=n=aandlimn!1(logbn)=bexist.Thenlimn!1log(an+bn)=n=max(a,b). Proof. Sinceanandbnarepositiverealnumberswehavethatlog(an+bn)=nlog(an)=nandlog(an+bn)=nlog(bn)=nforeachn.Thuslimn!1log(an+bn)=nmax(a,b).Toshowthereverseinequality,xc>max(a,b).Thus,c>log(an)=nandc>log(an)=nforeachnimpliesthatanmax(a,b).Wemusthavelimn!1log(an+bn)=nmax(a,b). Theorem6.1. E(n)log(3) Proof. Toobtainthisupperbound,itisconvenienttoreplacethegeneratorsofthefundamentalgroupofthe(n+1)punctureddisk.Thenewgeneratorsiinthecasen=6areshowninthegure 6-1 .Ingure 6-2 weshow4actingonthenewgenerators.Recallthat4sendsthefundamentalstirreraroundthe4thobstacle.Ingeneral,wehavetwocases: (j)(j)=(j)]TJ /F5 7.97 Tf 6.58 0 Td[(1j)]TJ /F5 7.97 Tf 6.59 0 Td[(2...2)]TJ /F5 7.97 Tf 6.58 0 Td[(11n...j+1))]TJ /F5 7.97 Tf 6.58 0 Td[(1j(j)]TJ /F5 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 7.97 Tf 6.59 0 Td[(2...2)]TJ /F5 7.97 Tf 6.59 0 Td[(11n...j+1) (6) (j)(k)=k,whenk6=j (6) 46

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Figure6-1. Thenewgeneratorsfor1(M2)withn=6;thesquareisthefundamentalstirrerandthecirclesaretheobstacles Figure6-2. The4actingonthenewgeneratingsetwithn=6 47

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If:1(M2)!1(M2)istheinducedactionofthepseudo-Anosovbraidon1(M2),thenbyBowen[ 4 ]andFathi[ 5 ]wehaveh()=supg21(M2)limk!1log(L(kg)) kwhereL(kg)denotesthenumberofgeneratorsinthereducedwordofkg.Dene~ni(g):1(M2)!Nwhere~ni(g)isthenumberofoccurrencesofthegeneratorsiand)]TJ /F5 7.97 Tf 6.59 0 Td[(1iinthereducedwordforg.Let~n(g)=(~n1(g),...,~nn(g)).Thus,L(g)=k~n(g)k1.Wewilluseasimplerformofh()fromtheformulain 6.1 namely,()=max1,...,nlimk!1log(k~n(ki)k1) k.Wewillnowshowthat()=h(),i.e.wecantakethemaximumoverthegeneratorsinsteadofthesupremumoverallwords.Itisobviousthat()h()since1,...,n21(M2).Nowwewilluselemma 6.1 toshowthereverseinequality.Letg21(M2)beanarbitraryelementwhereg=i1i1...ijijwheren2f1g.WealsoneedtopointoutthatL(k(i))=L(k()]TJ /F5 7.97 Tf 6.59 0 Td[(1i))foreachibecausethe(i)hasthesamenumberofoccurrencesofthegeneratorsas()]TJ /F5 7.97 Tf 6.58 0 Td[(1i)foreachi.Thus,k~n(k(i))k1=k~n(k()]TJ /F5 7.97 Tf 6.59 0 Td[(1i))k1foreachi.limk!1log(L(k(g))) k=limk!1log(L(k(i1i1...ijij))) klimk!1log(L(k(i1i1)++L(k(ijij)))) k=limk!1log(L(k(i1)++L(k(ij)))) kmaxi1,...,ijlimk!1logL(k(i)) k(Bylemma 6.1 .)max1,...,nlimk!1log(k~n(ki)k1) k(Maximumoverallgenerators.)=() 48

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Sincewasarbitrary,weseethath()().Sowehavethath()=(),i.e.h()=max1,...,nlimk!1log(k~n(ki)k1) k. Denition11(IncidenceMatrix). Usingthespeciedsetofgeneratorsfor1asin 6-1 andforanyhomomorphism:1(M2)!1(M2)denetheincidencematrixAbyAi,j=~nj(i).LetAbetheincidencematrixfor.Noticethatthejthcomponentof~n(g)Aisthenumberofoccurrencesofjand)]TJ /F5 7.97 Tf 6.58 0 Td[(1jbeforeanycancellationsaredone.Thus,~n(g)A~n(g)where[a1,...,am][b1,...,bn]meansaibiforalli.Whenweiterate,wegetforallk>0,~n(g)Ak~n(kg).Takingone-norms(everythingispositive)wehavek~n(kg)k1k~n(g)Akk1k~n(g)k1kAkk1.Hence,limk!1k~n(kg)k1=k1limk!1k~n(g)k1=k1kAkk1=k1=(A)whereisthespectralradiusofthematrixA.Thus,foranyfromafundamentalstirringprotocol,ifitsincidencematrixisAthenh()log((A)).Recallthatthefundamentalstirreriscountedasthersthole,andidenotesbraidtheelementsendingtherststringaroundtheithstringandi2f2,...,ng.Using( 6 ) 49

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and( 6 )wehavethatcorrespondingincidencematrixis:Ai=266666666666666666410...000...0010...00...0...22...212...200...001...0...00...000...13777777777777777775(i)]TJ /F8 11.955 Tf 11.95 0 Td[(1)throwAlsonotethatiand)]TJ /F5 7.97 Tf 6.59 0 Td[(1ihavethesameincidencematricessince~nj(ii)=~nj()]TJ /F5 7.97 Tf 6.58 0 Td[(1ii)foreveryiandj.Foreaseofnotation,letAi=Ai)]TJ /F14 5.978 Tf 5.75 0 Td[(1soA1=A2,forexample.LetM=fA1,...,An)]TJ /F5 7.97 Tf 6.58 0 Td[(1g.Letbeafundamentalstirringprotocolwithwordlengthmso=k1...kmandletAdenotetheincidencematrixfor.Thenforeachiandjwehave,(A)ij=~nj(i)=~nj(k1...km(i))~nj(k1i)~nj(kmi)Since~niscountingbeforecancellations=(Ak1)ij(Akm)ij.Inotherwords,AAk1Akmwheremeansforeachij-thposition.Then(A)kAk1kAi1k1...kAimk13mSincewasanarbitraryfundamentalstirringprotocolwithlengthmthen,m(M)=maxA1,...,Am2M(AmA1)1=m(3m)1=m=3and(M)=limsupk!1k(M)3. 50

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Wenowhaveanupperboundforthetopologicalefciency(recallthedenitionoftopologicalefciencyfromsection 3.2 )ofanyarbitrarystirringprotocol,namely,E()=h() #log((A)) #.Since#isnumberofalphageneratorsusedtoconstruct,then#isalsothenumberofincidencematricesusedtoconstructA.Thus,E()log((A)1 #)log((M))log(3)Thisistrueforeveryfundamentalstirringprotocolandbytakingthesupovereveryfundamentalstirringprotocolwehave,E(n)log(3). 6.2SpectralRadiusofincidenceMatricesLet HSPn=An...A2.So HSPnistheproductoftheincidencematricesofthestirrersinthesameorderastheHSP(n).Wewillexploresomepropertiesof HSPn. HSPn=26666666666412...201...0001...0.........000...137777777777526666666666410...0010...0001...0.........000...137777777777526666666666410...0212...2001...0.........222...1377777777775 HSPn=266666666666666423n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 11.95 0 Td[(123n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 11.95 0 Td[(223023n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 11.96 0 Td[(2231...23n)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 11.95 0 Td[(223n)]TJ /F5 7.97 Tf 6.58 0 Td[(223n)]TJ /F5 7.97 Tf 6.59 0 Td[(223n)]TJ /F5 7.97 Tf 6.59 0 Td[(2...23n)]TJ /F5 7.97 Tf 6.59 0 Td[(323n)]TJ /F5 7.97 Tf 6.59 0 Td[(3...23n)]TJ /F5 7.97 Tf 6.59 0 Td[(4.........23...2302222...13777777777777775Wewilllet( HSPn)denotethespectralradiusof HSPnandweletf1,2,...,ngdenotetheeigenvaluesof HSPnand1isthePerron-Frobeniusvalue,i.e.( HSPn)=1 51

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First,wewillcomputethecolumnsums.LetCidenotethecolumnsumfortheithcolumn.Wenoticethat C1=23n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F8 11.955 Tf 11.96 0 Td[(1+23n)]TJ /F5 7.97 Tf 6.58 0 Td[(2++23+2=)]TJ /F8 11.955 Tf 9.3 0 Td[(1+2n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xj=03j=3n)]TJ /F8 11.955 Tf 11.95 0 Td[(2 (6) Cn=23n)]TJ /F5 7.97 Tf 6.58 0 Td[(2+23n)]TJ /F5 7.97 Tf 6.59 0 Td[(3++230+1=1+2n)]TJ /F5 7.97 Tf 6.59 0 Td[(2Xj=03j=3n)]TJ /F5 7.97 Tf 6.58 0 Td[(1 (6) Thethingtonoteisthatcolumnsumsaredecreasing.SoC1C2Cn.Next,weconsiderthat( HSPn)=( HSPnT)where HSPnTdenotesthetransposeof HSPn.From[ 28 ]anddenition 17 fromtheappendixand,wehave Theorem6.2. SupposethatMisannnnon-negativeprimitivematrix.Then,mininXj=1mij(M)maxinXj=1mijwithequalityoneithersideimplyingequalitythroughout.Asimilarpropositionholdsforcolumnsums.Soinourcase,thisimpliesminjnXi=1( HSPnT)i,j( HSPnT)maxjnXi=1( HSPnT)i,jwhichimpliesthatCn( HSPn)C13n)]TJ /F5 7.97 Tf 6.58 0 Td[(2( HSPn)3n)]TJ /F8 11.955 Tf 11.95 0 Td[(2. Denition12(Pisotnumber). APisotnumberisarealalgebraicinteger,greaterthan1,withthepropertythatallitsGaloisconjugatesliewithintheunitcircle.AnirreduciblepolynomialthathasarootthatisaPisotnumberiscalledaPisotPolynomial. Conjecture4. Thematrix HSPnhasacharacteristicpolynomialwhoserootsarePisot.Weseethat( HSPn)j,j0foreachj.Thus,wehavej23n)]TJ /F5 7.97 Tf 6.58 0 Td[(1j=j( HSPn)1,1jjTr( HSPn)j=j1+2++njj1j+j2j++jnj 52

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Supposingthattheconjectureistrue,thenjij1foreachi2.Sowehave23n)]TJ /F5 7.97 Tf 6.58 0 Td[(1j1j+(n)]TJ /F8 11.955 Tf 11.95 0 Td[(1)Thus,23n)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(n)]TJ /F8 11.955 Tf 11.96 0 Td[(1( HSPn).LetMbethesetoftheincidencematricesforthefundamentalstirringprotocolsiwherei2f2,...,ng,i.e.M=fA2,...,Amg.Foreaseofnotation,wewilluseAi=Ai)]TJ /F5 7.97 Tf 6.58 0 Td[(1andAn=Am,soM=fA1,...,Ang.Wehavealreadyseenthat(M)3.Wewouldliketocompute(M)directly.Inordertoprovethisweneedtosetupsomemachinery.Themainideaforcomputing(M)isthatforanypositivematrixN,AiactsverysimplyonthevectorofcolumnssumsofNforeachi.Weusethejointspectralradiuswhichwewilldenenow.Recallthedenitionofaninducedmatrixnormfromdenition 19 intheappendix.LetMbeanitesetofmatricesandkkdenoteaninducedmatrixnorm.Denek(M,kk)=maxA1,...,Ak2MkM1Mkk1=k Denition13(JointSpectralRadius). (M,kk)=limsupk!1k(M)TheJointSpectralRadiusisindependentofthenormthatwechoose[ 29 ].Soforsimplicitywewilluse(M)fortheJointSpectralRadiusforthesetM.Also,ifMisanitesetofmatrices,then(M)=(M)whichwasrstprovedin[ 30 ].Denec:GLn(Z)!Znwherec(M)isthecolumnsumsofM. Example2. Bydirectcalculationwehave,c(Ai)=(3,3,...,3,1,3,...,3)wherethe1isintheithplace. Lemma2. LetM2GLn(Z)beanarbitrarymatrix,andAi2M,thenc(MAi)=c(M)Ai. 53

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Proof. Fixi2f1,...,ngandconsiderMAi.MAi=266666666664m11m12...m1n......mi1min......mn1mn2mnn37777777777526666666666410...0......222122......001377777777775=266666666664m11+2m1i...m1i...m1n+2m1i......mi1+2miimiimin+2mii......mn1+2mni...mni...mnn+2mni377777777775Bycomputingthecolumnsums,wehave c(MAi)=(c1(M)+2ci(M),...,ci)]TJ /F5 7.97 Tf 6.58 0 Td[(1(M)+2ci(M),ci,ci+1(M)+2ci(M),......,cn(M)+2ci(M)). (6) Next,wewillcomputethec(M)Ai. c(M)Ai=(c1(M),...cn(M))26666666666410...0......222122......001377777777775=(c1(M)+2ci(M),...,ci)]TJ /F5 7.97 Tf 6.58 0 Td[(1(M)+2ci(M),ci,cj+1(M)+2ci(M),......,cn(M)+2ci(M)). (6) Thus,( 6 )and( 6 )arethesamesoc(MAi)=c(M)Aiandsinceiwasarbitrary,ourlemmaistrue. 54

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Fixn3andletOnbealllistsofnpositiveintegersgivenindecreasingorder,soOn=f~a2(Z+)n:a1a2...an>0g Denition14(SortingFunction). LetS:(Z+)n!Onbethesortingfunction,i.e.itsendsavectorofpositiveintegerstothelistofitsentriessortedinnonincreasingorder. Example3. Let~a=(1,2,3)thenS(~a)=(3,2,1).Let~a2(Z+)n,thenAi~aT(Z+)n.Now,wedeneWj:On!OnwhereWj(~a):=SAj(~aT).WewillalsoneedthestandardorderonOn,namely,~a~b()ajbjforallj~a<~b()aj~0and~b>~0.Thenthefollowingaretrue. (a) ~a~b)Wj(~a)Wj(~b)forallj (b) ~a>~b)Wj(~a)>Wj(~b)forallj (c) jak)Wj(~a)>Wk(~b) (e) jkand~a>~b)Wj(~a)>Wk(~b) Proof. Assumethat~a=(a1,a2,...,am)and~b=(b1,b2,...,bm)andai>0andbi>0. 55

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Forparta)suppose~a~b,thenaibiforeachi.Wj(~a)=SAj(~aT)=S(a1+2aj,...,aj)]TJ /F5 7.97 Tf 6.59 0 Td[(1+2aj,aj,aj+1+2aj,...,am+2aj)=(a1+2aj,...,aj)]TJ /F5 7.97 Tf 6.59 0 Td[(1+2aj,aj+1+2aj,am+2aj,...,aj)Wj(~b)=SAj(~bT)=S(b1+2bj,...,bj)]TJ /F5 7.97 Tf 6.58 0 Td[(1+2bj,aj,bj+1+2bj,...,bm+2bj)=(b1+2bj,...,bj)]TJ /F5 7.97 Tf 6.58 0 Td[(1+2bj,bj+1+2bj,bm+2bj,...,bj)Sinceaibiforeachi,thenai+2ajbi+2bjforeachi.Thus,Wj(~a)Wj(~b).Partb)issimilartoparta).Forpartc)supposewehave~awherejbji. Proposition6.1. Given~a2On,ifsisalength-kstrategyandisstandardfor~aands6=1,thenW1(~a)>Ws(~a). 56

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Proof. Lets=(j1,...,jk)andletibetherstindexwhereji>1.Let~b=Wji)]TJ /F14 5.978 Tf 5.76 0 Td[(1,...,j1.Since~sisstandardfor~a,thenbjiWji(~b),i.e.W1,...,1(~a)>Wji,...,j1(~b).Sincejn1foralln>i,thenbylemma 3 e ,thenW1,...,1(~a)>Ws,asrequired. Corollary2. Ifsisanylengthkstrategyfor~athenW1Ws. Proof. If1isthestandardstrategyderivedfroms,thenW1(~a)=Ws(~a)bylemma 3 a .Otherwiseusethepreviousproposition. Theorem6.3. LetM=fA1,...,Angbeasdenedabove.Thenthejointspectralradius(M)=( HSPn),i.e.thespectralradiusofMisrealizedby HSPn. Proof. Fixw>0.WeconsideranarbitrarynumberwofmatricesfromthesetM.LetN=Ai1Aiw.Bythedivisionalgorithm,wehavew=m_n+jwhere0j
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wherew=mn+j.Inparticular,sinceforapositivematrix,c(A)=kAk1andc(~aA)=c(IA)=c(A),wehavethatkA1AnA1AnA1AjkkAi1Aiwk1.Since,Ai1Aiwisconstructedfromw=mn+jelementsandwasarbitrary,thenbydenition,w(M)=maxAj1,...,AjwkAj1,...,Ajwk1=kA1,...An,A1,...,An,...,A1,...,Ajk1=w1.Thus,(M)=limk!1k(M)=limk!1k(A1,...An)kk1=k1=(A1,...,An). ByBergerandWangin[ 30 ],itwasshownthatthegeneralizedspectralradiusandthejointspectralradiusareequalforboundedsetsofmatrices.Inourcase,wehavethat,(M)=(M). Corollary3. log((HSPn)) n)]TJ /F8 11.955 Tf 11.96 0 Td[(1E(n)log(( HSPn)). 58

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APPENDIX:SOMEGENERALMATRIXLEMMAS A.1GeneralMatrixTheoryMostofthetheoremsdiscussedareclassicaltheoremsandcanbefoundinthebooks[ 28 ],[ 31 ],and[ 32 ].LetMbeasquarematrixwhereM=fmijg.WewillwriteM0whenmij0foreachiandjandlikewiseM>0whenmij>0foreachiandj. Denition17(PrimitiveMatrix). Asquarenon-negativematrixMissaidtobeprimitiveifthereexistsapositiveintegerksuchthatMk>0. Denition18(VectorNorm). AnormkkinavectorspaceVoveraeldFisafunctionkk:V!Rthatsatisesthefollowingforallx,y2Vandc2F: 1. kcxk=jcjkxk, 2. kxk=0ifandonlyifx=0, 3. kxk0,and 4. kx+ykkxk+kyk(TriangleInequality). Denition19(InducedMatrixNorm). AvectornormthatisdenedonCm,inducesamatrixnormonCmnbykMk=supkxk6=0kMxk kxkwhereM2Cmn. Denition20(Spectrum). Thesetf1,,ngofeigenvaluesofasquarematrixMiscalledthespectrumofM. Denition21(SpectralRadius). LetMbeasquarematrixwitheigenvaluesf1,,ng,thenthespectralradius(M)isgivenby(M)=max1injij. TheoremA.4. SupposeMisasquarematrixandkkisaninducedmatrixnorm,then(M)kMk 59

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Proof. SupposeMhasaneigenvaluejwithacorrespondingeigenvectorvj,sothatMvj=jvj.ThenwehavekMk=supkxk6=0kMxk kxkkMvjk kvjk=kjvjk kvjk=jjj.SincethisistrueforalleigenvaluesofM,thenkMk(M). Thefollowingtheoremcanbefoundin[ 33 ],Corollary5.6.14. TheoremA.5(Gelfand'sformula). SupposeMisasquarematrixwithrealentriesandkkisamatrixnorm,thenlimn!1kMnk1=n=(M) A.2SpectralRadiusPropertiesThesenextsetofdenitionsandtheoremscanbefoundby[ 29 ].LetMbeanitesetofsquarematricesanddenotethespectralradiusofamatrix,i.e.themagnitudeofthelargesteigenvalue.Denek(M)=maxA1,...,Ak2M(A1Ak)1=k Denition22(generalizedspectralradius). (M)=limsupk!1k(M) TheoremA.6. LetM=fM1,,Mngbeasetofsquarematriceswithnonnegativeentries.LetSbethematrixwhoseentriesarethecomponentwisemaximumoftheentriesofthematricesM.Then(S) n(M)(S). Corollary4. LetM=fM1,,Mngbeasetofsquarematriceswithnonnegativeentries.Then0(M)<1. 60

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REFERENCES [1] D.Rolfsen,NewdevelopmentsinthetheoryofArtin'sbraidgroups,in:ProceedingsofthePacicInstitutefortheMathematicalSciencesWorkshopInvariantsofThree-Manifolds(Calgary,AB,1999),Vol.127,2003,pp.77.URL http://dx.doi.org/10.1016/S0166-8641(02)00054-8 [2] J.S.Birman,Braids,links,andmappingclassgroups,PrincetonUniversityPress,Princeton,N.J.,1974,annalsofMathematicsStudies,No.82. [3] E.Artin,Theoriederzopfe,in:AbhandlungenausdemMathematischenSeminarderUniversitatHamburg,Vol.4,Springer,1925,pp.47. [4] R.Bowen,Entropyforgroupendomorphismsandhomogeneousspaces,Trans.Amer.Math.Soc.153(1971)401. [5] TravauxdeThurstonsurlessurfaces,Vol.66ofAsterisque,SocieteMathematiquedeFrance,Paris,1979,seminaireOrsay,WithanEnglishsummary. [6] W.P.Thurston,Onthegeometryanddynamicsofdiffeomorphismsofsurfaces,Bull.Amer.Math.Soc.(N.S.)19(2)(1988)417.URL http://dx.doi.org/10.1090/S0273-0979-1988-15685-6 [7] M.Handel,Globalshadowingofpseudo-Anosovhomeomorphisms,ErgodicTheoryDynam.Systems5(3)(1985)373.URL http://dx.doi.org/10.1017/S0143385700003011 [8] M.D.Finn,S.M.Cox,H.M.Byrne,Mixingmeasuresforatwo-dimensionalchaoticStokesow,J.Engrg.Math.48(2)(2004)129.URL http://dx.doi.org/10.1023/B:ENGI.0000011930.55539.69 [9] J.M.Ottino,Thekinematicsofmixing:stretching,chaos,andtransport,CambridgeTextsinAppliedMathematics,CambridgeUniversityPress,Cambridge,1989. [10] A.J.Chorin,J.E.Marsden,Amathematicalintroductiontouidmechanics,3rdEdition,Vol.4ofTextsinAppliedMathematics,Springer-Verlag,NewYork,1993. [11] W.Burau,UberZopfgruppenundgleichsinnigverdrilteVerkettungen,Abh,Math.Sem.HanischenUniv11(1936)171. [12] S.Bigelow,TheBuraurepresentationisnotfaithfulforn=5,Geom.Topol.3(1999)397(electronic).URL http://dx.doi.org/10.2140/gt.1999.3.397 61

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BIOGRAPHICALSKETCH JasonHarringtonwasborninLouisville,Kentucky.HeearnedhisB.A.atWesternKentuckyUniversitywherehemethisbeautifulwifeLeslie.WhileworkingonthisdissertationattheUniversityofFloridatheyhadtheirrstsonDaniel.Inadditiontomathematics,healsoenjoyscomputerprogrammingandreading. 64