ChaoticBilliardsNikolaiChernovRobertoMarkarianDepartmentofMathematics, - PDF document

ChaoticBilliardsNikolaiChernovRobertoMarkarianDepartmentofMathematics,UniversityofAlabamaatBirming-ham,Birmingham,AL35294,USAE-mailaddress:chernov@math.uab.eduInstitutodeMatematicayEstadstica\Prof.Ing.RafaelLa-guardia"FacultaddeIngeniera,UniversidaddelaRepublica,C.C.30,Montevideo,UruguayE-mailaddress:roma@fing.edu.uyToYakovSinaiontheoccasionofhis70thbirthday Theauthorsaregratefultomanycolleagueswhohavereadthemanuscriptandmadenumeroususefulremarks,inparticularP.Balint,D.Dolgopyat,C.Liverani,G.DelMagno,andH.-K.Zhang.ItisapleasuretoacknowledgethewarmhospitalityofIMPA(RiodeJaneiro),wherethenalversionofthebookwasprepared.Wealsothanktheanonymousrefereesforhelpfulcomments.Lastbutnottheleast,thebookwaswrittenatthesuggestionofSergeiGelfandandthankstohisconstantencouragement.TherstauthorwaspartiallysupportedbyNSFgrantDMS-0354775(USA).ThesecondauthorwaspartiallysupportedbyaProyectoPDT-Conicyt(Uruguay). ContentsPrefaceviiSymbolsandnotationixChapter1.Simpleexamples11.1.Billiardinacircle11.2.Billiardinasquare51.3.Asimplemechanicalmodel91.4.Billiardinanellipse111.5.Achaoticbilliard:pinballmachine15Chapter2.Basicconstructions192.1.Billiardtables192.2.Unboundedbilliardtables222.3.Billiard\row232.4.Accumulationofcollisiontimes242.5.Phasespaceforthe\row262.6.Coordinaterepresentationofthe\row272.7.Smoothnessofthe\row292.8.Continuousextensionofthe\row302.9.Collisionmap312.10.Coordinatesforthemapanditssingularities322.11.Derivativeofthemap332.12.Invariantmeasureofthemap352.13.Meanfreepath372.14.Involution38Chapter3.Lyapunovexponentsandhyperbolicity413.1.Lyapunovexponents:generalfacts413.2.Lyapunovexponentsforthemap433.3.Lyapunovexponentsforthe\row453.4.Hyperbolicityastheoriginofchaos483.5.Hyperbolicityandnumericalexperiments503.6.Jacobicoordinates513.7.Tangentlinesandwavefronts523.8.Billiard-relatedcontinuedfractions553.9.Jacobianfortangentlines573.10.Tangentlinesinthecollisionspace583.11.Stableandunstablelines593.12.Entropy60iii ivCONTENTS3.13.Provinghyperbolicity:conetechniques62Chapter4.Dispersingbilliards674.1.Classicationandexamples674.2.Anothermechanicalmodel694.3.Dispersingwavefronts714.4.Hyperbolicity734.5.Stableandunstablecurves754.6.ProofofProposition4.29774.7.Morecontinuedfractions834.8.Singularities(localanalysis)864.9.Singularities(globalanalysis)884.10.SingularitiesfortypeBbilliardtables914.11.Stableandunstablemanifolds934.12.Sizeofunstablemanifolds954.13.Additionalfactsaboutunstablemanifolds974.14.ExtensiontotypeBbilliardtables99Chapter5.Dynamicsofunstablemanifolds1035.1.Measurablepartitionintounstablemanifolds1035.2.u-SRBdensities1045.3.Distortioncontrolandhomogeneitystrips1075.4.Homogeneousunstablemanifolds1095.5.SizeofH-manifolds1115.6.Distortionbounds1135.7.Holonomymap1185.8.Absolutecontinuity1215.9.Twogrowthlemmas1255.10.Proofsoftwogrowthlemmas1275.11.Thirdgrowthlemma1325.12.Fundamentaltheorem136Chapter6.Ergodicproperties1416.1.History1416.2.Hopf'smethod:heuristics1416.3.Hopf'smethod:preliminaries1436.4.Hopf'smethod:mainconstruction1446.5.Localergodicity1476.6.Globalergodicity1516.7.Mixingproperties1526.8.Ergodicityandinvariantmanifoldsforbilliard\rows1546.9.Mixingpropertiesofthe\rowand4-loops1566.10.Using4-loopstoproveK-mixing1586.11.Mixingpropertiesfordispersingbilliard\rows160Chapter7.Statisticalproperties1637.1.Introduction1637.2.Denitions1637.3.Historicoverview1677.4.Standardpairsandfamilies169 CONTENTSv7.5.Couplinglemma1727.6.Equidistributionproperty1757.7.Exponentialdecayofcorrelations1767.8.CentralLimitTheorem1797.9.Otherlimittheorems1847.10.Statisticsofcollisionsanddiusion1867.11.SolidrectanglesandCantorrectangles1907.12.A`magnet'rectangle1937.13.Gaps,recovery,andstopping1977.14.Constructionofcouplingmap2007.15.Exponentialtailbound205Chapter8.Bunimovichbilliards2078.1.Introduction2078.2.Defocusingmechanism2078.3.Bunimovichtables2098.4.Hyperbolicity2108.5.Unstablewavefrontsandcontinuedfractions2148.6.Somemorecontinuedfractions2168.7.Reductionofnonessentialcollisions2208.8.Stadia2238.9.Uniformhyperbolicity2278.10.Stableandunstablecurves2308.11.Constructionofstableandunstablemanifolds2328.12.u-SRBdensitiesanddistortionbounds2358.13.Absolutecontinuity2388.14.Growthlemmas2428.15.Ergodicityandstatisticalproperties248Chapter9.Generalfocusingchaoticbilliards2519.1.Hyperbolicityviaconetechniques2529.2.Hyperbolicityviaquadraticforms2549.3.Quadraticformsinbilliards2559.4.Constructionofhyperbolicbilliards2579.5.Absolutelyfocusingarcs2609.6.Continuedfractions2659.7.Singularities2669.8.ApplicationofPesinandKatok-Strelcyntheory2709.9.Invariantmanifoldsandabsolutecontinuity2739.10.Ergodicityvia`regularcoverings'274Afterword279AppendixA.Measuretheory281AppendixB.Probabilitytheory291AppendixC.Ergodictheory299Index309 viCONTENTSBibliography311 PrefaceBilliardsaremathematicalmodelsformanyphysicalphenomenawhereoneormoreparticlesmoveinacontainerandcollidewithitswallsand/orwitheachother.Thedynamicalpropertiesofsuchmodelsaredeterminedbytheshapeofthewallsofthecontainer,andtheymayvaryfromcompletelyregular(integrable)tofullychaotic.Themostintriguing,thoughleastelementary,arechaoticbilliards.TheyincludetheclassicalmodelsofhardballsstudiedbyL.BoltzmannintheXIXcentury,theLorentzgasintroducedtodescribeelectricityin1905,aswellasmoderndispersingbilliardtablesduetoYa.Sinaiandthestadium.Mathematicaltheoryofchaoticbilliardswasbornin1970whenYa.Sinaipub-lishedhisseminalpaper[Sin70];andnowitisonly35yearsold.Butduringtheseyearsitgrewanddevelopedataremarkablespeed,andbecameawell-establishedand\rourishingareawithinthemoderntheoryofdynamicalsystemsandstatisticalmechanics.Itisnosurprisethatmanyyoungmathematiciansandscientistsattempttolearnchaoticbilliards,inordertoinvestigatesomeofthemorexplorerelatedphys-icalmodels.Butsuchstudiesareusedtobeprohibitivelydicultfortoomanyanoviceandanoutsider,notonlybecausethesubjectitselfisintrinsicallyquitecomplex,buttoalargeextendbecauseofthelackofcomprehensiveintroductorytexts.True,thereareexcellentbookscoveringgeneralmathematicalbilliards[Ta95,KT91,KS86,GZ90,CFS82],butthesebarelytouchuponchaoticmodels.Therearesurveysdevotedtochaoticbilliardsaswell,see[Sin00,Sz00,CM03],butthoseareexpository,theyonlysketchselectiveargumentsandrarelygodownto`nutsandbolts'.Forthereaderswhowanttolook`underthehood'andbecomeprofessional(andwespeakofgraduatestudentsandyoungresearchershere),thereisnotmuchchoiceleft:eitherlearningfromtheiradvisorsorotherexpertsbywayofpersonalcommunication,orreadingtheoriginalpublications(mostofthemverylongandtechnicalarticlestranslatedfromRussian).Thenstudentsquicklydiscoverthatsomeessentialfactsandtechniquescanonlybefoundinthemiddleoflongdensepapers.Worseyet,someofthosefactshaveneverevenbeenpublished{theyexistasfolklore.Thisbookattemptstopresentthefundamentalsofthemathematicaltheoryofchaoticbilliardsinasystematicway.Wecoverallthebasicfacts,providefullproofs,intuitiveexplanationsandplentyofillustrations.Ourbookcanbeusedbystudentsandself-learners{itstartswiththemostelementaryexamplesandformaldenitions,andthentakesthereaderstepbystepintothedepthofSinai'stheoryofhyperbolicityandergodicityofchaoticbilliards,aswellasmorerecentachievementsrelatedtotheirstatisticalproperties(decayofcorrelationsandlimittheorems).vii viiiPREFACEThereadershouldbewarnedthatourbookisdesignedforactivelearning.Itcontainsplentyexercisesofvariouskinds;someconstitutesmallstepsintheproofsofmajortheorems,someotherspresentinterestingexamplesandcounterexamples,yetothersaregivenforthereader'spractice(someexercisesareactuallyquitechallenging).Thereaderisstronglyencouragedtodoexerciseswhenreadingthebook,asthisisthebestwaytograspthemainconceptsandeventuallymasterthetechniquesofbilliardtheory.Thebookisrestrictedtotwo-dimensionalchaoticbilliards,primarilydispersingtablesbySinaiandcircular-arc-tablesbyBunimovich(withsomeotherplanarchaoticbilliardsreviewedinthelastchapter).Wehaveseveralcompellingreasonsforsuchaconnement.First,Sinai'sandBunimovich'sbilliardsaretheoldestandbestexplored(forinstance,statisticalpropertiesareestablishedonlyforthemandfornootherbilliardmodel);thecurrentknowledgeofotherchaoticbilliardsismuchlesscomplete;theworkonsomeofthem(mostnotably,hardballgases)iscurrentlyunderwayandshouldbeperhapsthesubjectoffuturetextbooks.Second,thetwoclassespresentedhereconstitutethecoreoftheentiretheoryofchaoticbilliards,allitsapparatusisbuiltupontheoriginalworksbySinaiandBunimovich;buttheirfundamentalworksarehardlyaccessibletotoday'sstudentsorresearchers;therehavebeennoattemptstoupdateorrepublishtheirresultssincethemiddle1970s(afterGallavotti'sbook[Ga74]).Ourbookmakessuchanattempt.Wedonotcoverpolygonalbilliards,eventhoughsomeofthemaremildlychaotic(ergodic);forsurveysofpolygonalbilliardssee[Gut86,Gut96].Weassumethatthereaderisfamiliarwithstandardgraduatecoursesinmath-ematics{linearalgebra,measuretheory,topology,Riemanniangeometry,complexanalysis,probabilitytheory.Wealsoassumeknowledgeofergodictheory;althoughthelatterisnotastandardgraduatecourse,itisabsolutelynecessaryforreadingthisbook;wedonotattempttocoverithere,though,astherearemanyexcellenttextsaround[Wa82,Man83,KH95,Pet83,CFS82,Sin00,BrS02,Dev89,Sin76](seealsoourpreviousbook[CM03]).Forthereader'sconvenience,wepro-videbasicdenitionsandfactsfromergodictheory,probabilitytheory,andmeasuretheoryinAppendices. SymbolsandnotationDbilliardtableSection2.1boundaryofthebilliardtable2.1+unionofdispersingcomponentsoftheboundary2.1unionoffocusingcomponentsoftheboundary2.10unionofneutral(\rat)componentsoftheboundary2.1~regularpartoftheboundaryofbilliardtable2.1Cornerpointsonbilliardtable2.1`degreeofsmoothnessoftheboundary=@D2.1nnormalvectortotheboundaryofbilliardtable2.3Ttangentvectortotheboundaryofbilliardtable2.6K(signed)curvatureoftheboundaryofbilliardtable2.1tbilliard\row2.5\nthephasespaceofthebilliard\row2.5~\nPartofphasespacewheredynamicsisdenedatalltimes2.5q;vprojectionsof\ntothepositionandvelocitysubspaces2.5!angularcoordinateinphasespace\n2.6;Jacobicoordinatesinphasespace\n3.6\ninvariantmeasureforthe\rowt2.6Fcollisionmaporbilliardmap2.9Mcollisionspace(phasespaceofthebilliardmap)2.9~MpartofMwherealliterationsofFaredened2.9^MpartofMwherealliterationsofFaresmooth2.11r;'coordinatesinthecollisionspaceM2.10invariantmeasureforthecollisionmapF2.12S0boundaryofthecollisionspaceM2.10S1singularitysetforthemapF12.10SnsingularitysetforthemapFn2.11S1sameas[n1Sn4.11Qn(x)connectedcomponentofMnSncontainingx4.11V(=d'=dr)slopeofsmoothcurvesinM3.10returntime(intercollisiontime)2.9meanreturntime(meanfreepath)2.12(i)xLyapunovexponentatthepointx3.1Esx;Euxstableandunstabletangentsubspacesatthepointx3.1Csx;Cuxstableandunstableconesatthepointx3.13(minimal)factorofexpansionofunstablevectors4.4Bthecurvatureofwavefronts3.7ix xSYMBOLSANDNOTATIONRcollisionparameter3.6Hkhomogeneitystrips5.3Sklinesseparatinghomogeneitystrips5.3k0minimalnonzeroindexofhomogeneitystrips5.3MHnewcollisionspace(unionofhomogeneitystrips)5.4hholonomymap5.7Iinvolutionmap2.14mLebesguemeasureonlinesandcurves5.9jWjlengthofthecurveW4.5jWjplengthofthecurveWinthep-metric4.5JWFn(x)JacobianoftherestrictionofFntothecurveWatthepointx2W5.2rW(x)distancefromx2WtothenearestendpointofthecurveW4.12rn(x)distancefromFn(x)tothenearestendpointofthecomponentofFn(W)thatcontainsFn(x)5.9pW(x)distancefromx2WtothenearestendpointofWinthep-metric4.13W(x)u-SRBdensityonunstablemanifoldW5.2`sameorderofmagnitude'4.3Lceilingfunctionforsuspension\rows2.9 CHAPTER1SimpleexamplesWestartwithafewsimpleexamplesofmathematicalbilliards,whichwillhelpusintroducebasicfeaturesofbilliarddynamics.Thischapterisforthecompletebeginner.Thereaderfamiliarwithsomebilliardsmaysafelyskipit{alltheformaldenitionswillbegiveninChapter2.1.1.BilliardinacircleLetDdenotetheunitdiskx2+y21.Letapoint-like(dimensionless)particlemoveinsideDwithconstantspeedandbounceoitsboundary@Daccordingtotheclassicalruletheangleofincidenceisequaltotheangleofre\rection,seebelow.Denotebyqt=(xt;yt)thecoordinatesofthemovingparticleattimetandbyvt=(ut;wt)itsvelocityvector.Thenitspositionandvelocityattimet+scanbecomputedbyxt+s=xt+utsut+s=utyt+s=yt+wtswt+s=wt(1.1)longastheparticlestaysinsideD(makesnocontactwith@D).Whentheparticlecollideswiththeboundary@D=fx2+y2=1g,itsvelocityvectorvgetsre\rectedacrossthetangentlineto@Datthepointofcollision,seeFig.1.1.PSfragreplacements Figure1.1.Billiardmotioninacircle.Exercise1.1.Showthatthenew(postcollisional)velocityvectorisrelatedtotheold(precollisional)velocitybytherule(1.2)vnew=vold2hvold;nin;wheren=(x;y)istheunitnormalvectortothecirclex2+y2=1andhv;ni=ux+wydenotesthescalarproduct.1 21.SIMPLEEXAMPLESAfterthere\rection,theparticleresumesitsfreemotion(1.1)insidethediskD,untilthenextcollisionwiththeboundary@D.Thenitbouncesoagain,andsoon.Themotioncanbecontinuedindenitely,bothinthefutureandthepast.Forexample,iftheparticlerunsalongadiameterofthedisk,itsvelocityvectorwillgetreversedateverycollision;andtheparticlewillkeeprunningbackandforthalongthesamediameterforever.OtherexamplesofperiodicmotionareshownonFig.1.2,wheretheparticletraversesthesidesofsomeregularpolygons.Figure1.2.Periodicmotioninacircle.Inthestudiesofdynamicalsystems,theprimarygoalistodescribetheevolu-tionofthesystemoverlongtimeperiodsanditsasymptoticbehaviorinthelimitt!1.Wewillfocusonsuchadescription.Letusparameterizetheunitcirclex2+y2=1bythepolar(counterclockwise)angle2[0;2](sinceisacycliccoordinate,itsvalues0and2areidentied).Also,denoteby 2[0;]theangleofre\rectionasshownonFig.1.1.Remark1.2.Wenotethatisactuallyanarclengthparameteronthecircle@D;whenstudyingmoregeneralbilliardtablesDwewillalwaysparameterizetheboundary@Dbyitsarclength.Insteadof ,are\rectioncanalsobedescribedbytheangle'==2 2[=2;=2]thatthepostcollisionalvelocityvectormakeswiththeinwardnormalto@D.Infact,allprincipalformulasinthisbookwillbegivenintermsof',ratherthan ,butforthemomentweproceedwith .Foreveryn2Z,letndenotethenthcollisionpointand nthecorrespondingangleofre\rection.Exercise1.3.Showthatn+1=n+2 n(mod2) n+1= n(1.3)alln2Z.Wemaketwoimportantobservationsnow:Allthedistancesbetweenre\rectionpointsareequal.Theangleofre\rectionremainsunchanged. 1.1.BILLIARDINACIRCLE3Corollary1.4.Let(0; 0)denotetheparametersoftheinitialcollision.Thenn=0+2n 0(mod2) n= 0:Everycollisionischaracterizedbytwonumbers:(thepoint)and (theangle).Allthecollisionsmakethecollisionspacewithcoordinatesand onit.Itisacylinderbecauseisacycliccoordinate,seeFig.1.3.WedenotethecollisionspacebyM.Themotionoftheparticle,fromcollisiontocollision,correspondstoamapF:M!M,whichwecallthecollisionmap.Foracircularbilliarditisgivenbyequations(1.3).ObservethatFleaveseveryhorizontallevelC =f =constgofthecylinderMinvariant.Furthermore,therestrictionofFtoC isarotationofthecircleC throughtheangle2 .Theangleofrotationcontinuouslychangesfromcircletocircle,growingfrom0atthebottomf =0gto2atthetopf =g(thusthetopandbottomcirclesareactuallykeptxedbyF).ThecylinderMis\twistedupward"(\unscrewed")bythemapF,seeFig.1.3.PSfragreplacements0MF Figure1.3.ActionofthecollisionmapFonM.Rigidrotationofacircleisabasicexampleinergodictheory,cf.AppendixC.ItpreservestheLebesguemeasureonthecircle.Rotationsthroughrationalanglesareperiodic,whilethosethroughirrationalangles{ergodic.Exercise1.5.Showthatif isarationalmultipleof,i.e. ==m=n(irreduciblefraction),thentherotationofthecircleC isperiodicwith(minimal)periodn;everypointonthatcircleisperiodicwithperiodn,i.e.Fn(; )=(; )forevery02.If =isirrational,thentherotationofC isergodicwithrespecttotheLebesguemeasure.Furthermore,itisuniquelyergodic,whichmeansthatthein-variantmeasureisunique.Asaconsequence,foreverypoint( ;)2C itsimagesf+2n ;n2Zgaredenseanduniformlydistributed1onC ;thislastfactissometimesreferredtoasWeyl'stheorem[Pet83,pp.49{50].1Asequenceofpointsxn2ConacircleCissaidtobeuniformlydistributedifforanyintervalICwehavelimN!1#fn:0nN;an2Ig=N=length(I)=length(C). 41.SIMPLEEXAMPLESExercise1.6.Showthateverysegmentoftheparticle'strajectorybetweenconsecutivecollisionsistangenttothesmallercircleS =fx2+y2=cos2 gconcentrictothediskD.Showthatif =isirrational,thetrajectorydenselyllstheringbetween@DandthesmallercircleS (seeFig.1.4).Remark:onecanclearlyseeonFig.1.4thattheparticle'strajectorylooksdenserneartheinnerboundaryofthering(it\focuses"ontheinnercircle).Iftheparticle'strajectorywerethepathofalaserrayandtheborderoftheunitdiskwereaperfectmirror,thenitwouldfeel\veryhot"there,ontheinnercircle.Forthisreason,theinnercircleiscalledacaustic(whichmeans\burning"inGreek).Figure1.4.Anonperiodictrajectory.Exercise1.7.CanthetrajectoryofthemovingparticlebedenseistheentirediskD?(Answer:No.)Exercise1.8.DoesthemapF:M!Mpreserveanyabsolutelycontinuousinvariantmeasured=f(; )dd onM?Answer:anymeasurewhosedensityf(; )=f( )isindependentofisF-invariant.Next,wecanxthespeedofthemovingparticleduetothefollowingfacts.Exercise1.9.Showthatkvtk=const,sothatthespeedoftheparticleremainsconstantatalltimes.Exercise1.10.Showthatifwechangethespeedoftheparticle,saywesetkvknew=ckvkoldwithsomec�0,thenitstrajectorywillremainunchanged,uptoasimplerescalingoftime:qnewt=qoldctandvnewt=voldctforallt2R.Thus,thespeedoftheparticleremainsconstantanditsvalueisnotimportant.Itiscustomarytosetthespeedtoone:kvk=1.Thenthevelocityvectorattimetcanbedescribedbyanangularcoordinate!tsothatvt=(cos!t;sin!t)and!t2[0;2]withtheendpoints0and2beingidentied.Now,thecollisionmapF:M!Mrepresentscollisionsonly.TodescribethemotionoftheparticleinsideD,letusconsiderallpossiblestates(q;v),whereq2Disthepositionandv2S1isthevelocityvectoroftheparticle.Thespaceofallstates(calledthephasespace)isthenathree-dimensionalmanifold\n:=DS1,whichis,ofcourse,asolidtorus(doughnut).Themotionofthebilliardparticleinducesacontinuousgroupoftransforma-tionsofthetorus\nintoitself.Precisely,forevery(q;v)2\nandeveryt2Rthebilliardparticlestartingat(q;v)willcometosomepoint(qt;vt)2\nattimet. 1.2.BILLIARDINASQUARE5Thuswegetamap(q;v)7!(qt;vt)on\n,whichwedenotebyt.Thefamilyofmapsftgisagroup,i.e.ts=t+sforallt;s2R.Thisfamilyiscalledthebilliard\rowonthephasespace.Letusconsideramodicationofthecircularbilliard.DenotebyD+theupperhalfdiskx2+y21,y0,andletapointparticlemoveinsideD+andbounceo@D+.(Adelicatequestionariseshere:whathappensiftheparticlehits@D+at(1;0)or(1;0),sincethereisnotangentlineto@D+atthosepoints?Weaddressthisquestioninthenextsection.)PSfragreplacementsD+DFigure1.5.Billiardintheupperhalfcircle.AsimpletrickallowsustoreducethismodeltoabilliardinthefullunitdiskD.DenotebyDtheclosureofDnD+,i.e.themirrorimageofD+acrossthexaxisL=fy=0g.WhentheparticlehitsL,itstrajectorygetsre\rectedacrossL,butwewillalsodrawitscontinuation(mirrorimage)belowL.ThelatterwillevolveinDsymmetricallytotherealtrajectoryinD+,untilthelatterhitsLagain.ThenthesetwotrajectorieswillmergeandmovetogetherinD+forawhile,untilthenextcollisionwithL,atwhichtimetheysplitagain(onegoesintoDandtheotherintoD+),etc.Itisimportantthatthesecond(imaginary)trajectoryneveractuallygetsre-\rectedothelineL,itjustcrossesLeverytime,thusitevolvesasabilliardtrajectoryinthefulldiskDasdescribedabove.Thus,thepropertiesofbilliardtrajectoriesinD+canbeeasilyderivedfromthosediscussedaboveforthefulldiskD.Thistypeofreductionisquitecommoninthestudiesofbilliards.Exercise1.11.Provethatperiodictrajectoriesinthehalf-diskD+correspondtoperiodictrajectoriesinthefulldiskD.Note,however,thattheperiod(thenumberofre\rections)maydier.Exercise1.12.Investigatethebilliardmotioninaquarteroftheunitdiskx2+y21,x0,y0.1.2.BilliardinasquareHerewedescribeanothersimpleexample{abilliardintheunitsquareD=f(x;y):0x;y1g,seeFig.1.6.Thelawsofmotionarethesameasbefore,butthissystempresentsnewfeatures.Firstofall,whenthemovingparticlehitsavertexofthesquareD,there\rectionrule(1.2)doesnotapply(thereisnonormalvectornatavertex).Theparticle 61.SIMPLEEXAMPLESFigure1.6.Billiardinasquare.thenstopsanditstrajectoryterminates.Wewilldiscussthisexceptionalsituationlater,rstweconsiderregulartrajectoriesthatneverhitthevertices.Letvt=(ut;wt)denotethevelocityvectorofthemovingparticleattimet(inthex;ycoordinates).IfithitsaverticalsideofDattimet,thenutchangessign(ut+0=ut0)andwtremainsunchanged.IftheparticlehitsahorizontalsideofD,thenwtchangessign(wt+0=wt0)andutremainsunchanged.Thus,(1.4)ut=(1)mu0andwt=(1)nw0;wheremandndenotethenumberofcollisionswithverticaland,respectively,horizontalsidesofDduringthetimeinterval(0;t).Exercise1.13.Showthatifu06=0andw06=0(andassumingtheparticleneverhitsavertex),thenallthefourcombinations(u0;w0)appearalongtheparticle'strajectoryinnitelymanytimes.NextwemakeuseofthetrickshownonFig.1.5.Insteadofre\rectingthetrajectoryofthebilliardparticleinasideof@D,were\rectthesquareDacrossthatsideandlettheparticlemovestraightintothemirrorimageofD.Ifwekeepdoingthisateverycollision,ourparticlewillmovealongastraightlinethroughthemultiplecopiesofDobtainedbysuccessivere\rections(theparticle\pierces"achainofsquares,seeFig.1.7).Thisconstructioniscalledtheunfoldingofthebilliardtrajectory.TorecovertheoriginaltrajectoryinD,onefoldstheresultingstringofadjacentcopiesofDbackontoD.WedenotethecopiesofDby(1.5)Dm;n=f(x;y):mxm+1;nyn+1gExercise1.14.Showthatifmandnareeven,thenthefoldingproceduretransformsDm;nbackontoD=D0;0bytranslationsx7!xmandy7!yn,thuspreservingorientationofbothxandy.Ifmisodd,thentheorientationofxisreversed(precisely,x7!m+1x).Ifnisodd,thentheorientationofyisreversed(precisely,y7!n+1y).Observethattheserulesdonotdependontheparticulartrajectorythatwasoriginallyunfolded.ThesquaresDm;nwithm;n2Ztile,likeblocks,theentireplaneR2.Anyregularbilliardtrajectoryunfoldsintoadirectedstraightlineontheplane,andanydirectedline(whichavoidsthesitesoftheintegerlattice)foldsbackintoabilliardtrajectory.AtrajectoryhitsavertexofDithecorrespondinglinerunsintoasiteoftheintegerlattice. 1.2.BILLIARDINASQUARE7PSfragreplacementsDD10D11D21D31D41D42D52Figure1.7.Unfoldingabilliardtrajectory.ThestructureofblocksDm;nwiththerespectivefoldingrulesisclearlyperiodic,inwhichthe22squareK2=f(x;y):0x;y2gplaystheroleofafundamentaldomain{theentireplaneiscoveredbyparalleltranslationsofK2.ThusthestandardprojectionofR2ontoK2transformsunfoldedtrajectoriesintodirectedstraightlinesonthe22torusTor2(thelatterisobtainedbyidentifyingtheoppositesidesofthesquareK2).ThebilliardintheunitsquareDthusreducestothesimplelinear\rowona\rat22torusTor2,inwhichpointsmovewithconstant(unit)velocityvectors.Thelinear\rowona\rattorusisoneofthestandardexamplesinergodictheory,cf.AppendixCand[KH95,Pet83,Sin76].Itsmainpropertiesarethese:ifatrajectoryhasrationalslopedy=dx2Q,thenitisperiodic(itrunsalongaclosedgeodesic);ifatrajectoryhasirrationalslopedy=dx=2Q,thenitisdense(itsclosureisthewholetorus).ThistranslatesintothefollowingalternativeforregularbilliardtrajectoriesintheunitsquareD:Corollary1.15.Ifw0=u02Q,thenthecorrespondingregularbilliardtrajec-toryintheunitsquareDisperiodic.Ifw0=u0=2Q,thenthecorrespondingregularbilliardtrajectoryisdense.Exercise1.16.ExtendthisresulttothebilliardinarectangleRwithsidesaandb.Answer:aregularbilliardtrajectoryinRisperiodici(aw0)=(bu0)2Q,otherwiseitisdense.Hint:transformtherectangleintotheunitsquarebyscalingthecoordinates:(x;y)7!(x=a;y=b).ArguethatthebilliardtrajectoriesinRwillbethustransformedintothoseinD.Exercise1.17.Extendtheaboveresulttobilliardsinthefollowingpolygons:anequilateraltriangle,arightisoscelestriangle,arighttrianglewiththeacuteangle=6,andaregularhexagon.Whatiscommonaboutthesepolygons?(Notethatthebilliardinahexagondoesnotreducetoageodesic\rowonatorus;doesitreducetoageodesic\rowonanothermanifold?). 81.SIMPLEEXAMPLESThephasespaceofthebilliardsystemintheunitsquareDisthethree-dimensionalmanifold\n=DS1,cf.theprevioussection.Thebilliard\rowtisdenedforalltimes1t1onregulartrajectories.Onexceptionaltrajectories(whichhitavertexofDatsometime),the\rowisonlydeneduntilthetrajectoryterminatesinavertex.Exercise1.18.Showthatthesetofexceptionaltrajectoriesisacountableunionof2Dsurfacesin\n.Weseethatthesetofexceptionaltrajectoriesisnegligibleinthetopologicalandmeasure-theoreticsense(ithaszeroLebesguemeasureandisanFset,i.e.acountableunionofnowheredenseclosedsubsets),butstillitspresenceisbother-some.Forthebilliardinasquare,though,onecangetridofthemaltogetherbyextendingthebilliard\rowbycontinuity:Exercise1.19.Showthatthe\rowtcanbeuniquelyextendedbycontinuitytoallexceptionaltrajectories.InthatcaseeverytrajectoryhittingavertexofDwillsimplyreverseitscourseandrunstraightback,seeFig.1.8.Figure1.8.Extensionofthe\rownearavertex.Theaboveextensiondenesthebilliard\rowtontheentirephasespace\nandmakesitcontinuouseverywhere.Wewillassumethisextensioninwhatfollows.Weremark,however,thatingenericbilliardssuchniceextensionsarerarelypossible{seeSection2.8.Now,theactionofthe\rowtonthephasespace\ncanbefullydescribedasfollows.Foreveryunitvectorv0=(u0;w0)2S1,considerthesetLv0=f(q;v)2\n:q2D;v=(u0;w0)g(thetwosignsare,ofcourse,independent).Dueto(1.4),eachsetLv0remainsinvariantunderthe\rowt.Supposerstthatu06=0andv06=0,thenLv0istheunionoffoursquares,obtainedby\slicing"\natthefour\levels"correspondingtothevectors(u0;w0),seeFig.1.9.Exercise1.20.CheckthatthefoursquaresconstitutingthesetLv0canbegluedtogetheralongtheirboundariesandobtainasmoothclosedsurfacewithoutboundary(a22torus)T20,onwhichthebilliard\rowwillcoincidewiththelinear\rowalongthevectorv0(i.e.the\rowonT20willbedenedbydierentialequations_x=u0,_y=w0).Hint:theassemblyofthetorusT2v0fromthesquaresofLv0isverysimilartothereductionofthebilliarddynamicsinDtothegeodesic\rowonthe22torusdescribedabove(infact,thesetwoproceduresareequivalent). 1.3.ASIMPLEMECHANICALMODEL9PSfragreplacementsv0Figure1.9.FoursquaresconstitutingLv0.Now,itisastandardfactinergodictheory,cf.AppendixC,thatthelinear\rowona2Dtorusdenedby_x=u0,_y=w0isperiodicifw0=u02Qandergodic(furthermore,uniquelyergodic)ifw0=u0=2Q.Inthelattercaseeverytrajectoryisdenseanduniformlydistributed2onthetorus.Inthetworemainingcases(rstu0=0,andsecondw0=0)thesetLv0consistsofjusttwosquares.Weleavetheiranalysistothereaderasaneasyexercise.Thisfullydescribestheactionofthe\rowt:\n!\nforthebilliardintheunitsquare.1.3.AsimplemechanicalmodelAsamotivationforthestudyofbilliards,oneusuallydescribesasimplemodeloftwomovingparticlesinaone-dimensionalcontainer.Itreducestoabilliardinarighttriangle,whichissimilartoabilliardinasquare.Wedescribethismodelhere,seealso[CFS82,CM03].Considerasystemoftwopointparticlesofmassesm1andm2onaunitinterval0x1.Theparticlesmovefreelyandcollideelasticallywitheachotherandwiththe`walls'atx=0andx=1.Letx1andx2denotethepositionsoftheparticlesandu1andu2theirvelocities.Sincetheparticlescollideuponcontact,theirpositionsremainordered,weassumethatx1x2atalltimes.PSfragreplacements01x1x2Figure1.10.Twoparticlesinaunitinterval.Nextwedescribecollisions.Whenaparticlehitsawall,itsimplyreversesitsvelocity.Whenthetwoparticlescollidewitheachother,wedenotebyutheprecollisionalvelocityandbyu+ithepostcollisionalvelocityoftheithparticle,i=1;2.Thelawofelasticcollisionsrequirestheconservationofthetotalmomentum,i.e.m1u++m2u+=m1u1+m2u22Alinextona2DtorusTor2issaidtobeuniformlydistributedifforanyrectangleRTor2wehavelimT!1mft:0tT;xt2Rg
=T=area(R)=area(Tor2),heremistheLebesguemeasureonR. 101.SIMPLEEXAMPLESandthetotalkineticenergy,i.e.(1.6)m1[u+]2+m2[u+]2=m1[u1]2+m2[u2]2:Solvingtheseequationsgivesu+=u1+2m2m1+m2(uu1)andu+=u2+2m1m1+m2(uu2)(werecommendthereaderderivestheseformulasforanexercise).Notethatifm1=m2,thentheparticlessimplyexchangetheirvelocities:u+=u2andu+=u1.Thevariablesxianduiareactuallyinconvenient,wewillworkwithnewvari-ablesdenedby(1.7)qi=xipmiandvi=dqi=dt=uipmifori=1;2.Nowthepositionsoftheparticlesaredescribedbyapointq=(q1;q2)2R2(itiscalledacongurationpoint).Thesetofallcongurationpoints(calledthecongurationspace)istherighttriangleD=fq=(q1;q2):0q1=pm1q2=pm21g:Thevelocitiesoftheparticlesaredescribedbythevectorv=(v1;v2).Notethattheenergyconservationlaw(1.6)impliesthatkvk=const,thuswecansetkvk=1.Thestateofthesystemisdescribedbyapair(q;v).ThecongurationpointqmovesinDwithvelocityvectorv.Whentherstparticlecollideswiththewall(x1=0),thecongurationpointhitstheleftsideq1=0ofthetriangleD.Whenthesecondparticlecollideswiththewall(x2=1),thepointqhitstheuppersideq2=pm2=1ofD.Whentheparticlescollidewitheachother,thepointqhitsthehypotenuseq1=pm1=q2=pm2ofD.Exercise1.21.Provethatthevelocityvectorvchangesatcollisionssothatitgetsre\rectedat@Daccordingtothelaw`theangleofincidenceisequaltotheangleofre\rection'.Thus,themotionofthecongurationpointqisgovernedbythebilliardrules.Hencetheevolutionofthemechanicalmodeloftwoparticlesinaunitintervalreducestobilliarddynamicsinarighttriangle.Ifm1=m2,weobtainabilliardinarightisoscelestriangle,whichreadilyreducestoabilliardinasquare,seeExercise1.17.Forgenericmassratiom1=m2,weobtainabilliardinagenericrighttriangle,whichmayberathercomplicated(suchbilliardsarenotcoveredinourbook).OnecomplicationariseswhenabilliardtrajectoryhitsacornerpointofD.Hittingthevertexoftherightanglecorrespondstoaneventwhenbothparticlessimultaneouslycollidewithoppositewalls;thentheirfurthermotionisclearlywelldened,thusthebilliardtrajectorycaneasilybecontinued(cf.Exercise1.19).However,hittingthevertexofanacuteangleofDcorrespondstoaneventwhenbothparticlessimultaneouslycollidewiththesamewall(eitherx=0orx=1).Inthiscase,forgenericm1andm2,thebilliard\rowcannotbeextendedbycontinuity,asnearbytrajectorieshittingthetwoadjacentsidesindierentorderwillcomebacktoDalongdierentlines,seeFig.1.11. 1.4.BILLIARDINANELLIPSE11PSfragreplacementsq1q2DFigure1.11.TherighttriangleD;hittingthevertexofanacuteangle.Inmechanicalterms,hittingthevertexofanacuteangleofDcorrespondstoamultiplecollision.Suchexceptionaleventsusuallycannotberesolvedbythelawsofclassicalmechanics.1.4.BilliardinanellipseWeproceedtoyetanothersimpleexamplethatadmitsacompletelyelementaryanalysis{thebilliardinanellipsex2a2+y2b2=1withsomea�b�0.Infact,itwasthisexamplethatBirkhodescribedintheveryrstbookonmathematicalbilliardsin1927[Bi27,ChapterVIII].WedenotebyDthedomainboundedbytheellipse(itwillbeourbilliardtable).LetF1andF2denotethefocioftheellipse,observethattheylieonthexaxis.TheellipseisthelocusofpointsA2R2suchthatdist(A;F1)+dist(A;F2)=const:Exercise1.22.LetA2@DandLdenotethetangentlinetotheellipseatA.ProvethatthesegmentsAF1andAF2makeequalangleswithL.(ThisfactisknowninprojectivegeometryasPoncelettheorem.)Hint:re\rectthepointF2acrossthetangentlineLandshowthatitsimagewilllieonthelineAF1.Thus,ifabilliardtrajectorypassesthroughonefocus,thenitre\rectsatapointA2@Dontheellipseandrunsstraightintotheotherfocus.Suchatrajectorywillthenpassthroughafocusaftereveryre\rection,seeFig.1.12.PSfragreplacementsF1F2Figure1.12.Atrajectorypassingthroughthefoci. 121.SIMPLEEXAMPLESExercise1.23.ShowthateverytrajectorypassingthroughthefociF1andF2convergestothemajoraxisoftheellipse(thexaxis).Bytheway,themajorandtheminoraxesoftheellipseareclearlytwoperiodictrajectories{theyrunbackandforthbetweentheirendpoints.InSection1.1weusedthecoordinates andtodescribecollisionsinacircularbilliard,andthecycliccoordinatewasactuallythearclengthparameteronthecircle(Remark1.2).Hereweusetwocoordinates andr,where isthesameangleofre\rectionasinSection1.1andrisanarclengthparameterontheellipse.Wechoosethereferencepointr=0astherightmostpoint(a;0)ontheellipseandorientrcounterclockwise.Notethat0rj@Djand0 .ThecollisionspaceMisagainacylinderwhosebaseistheellipseandwhoseheightis.ItisshownonFig.1.13asarectangle[0;j@Dj][0;],butwekeepinmindthattheleftandrightsidesofthisrectanglemustbeidentied.Themotionofthebilliardparticle,fromcollisiontocollision,inducesthecollisionmapF:M!M.Exercise1.24.VerifythatthetrajectoriespassingthroughthefocilieonaclosedcurveonthesurfaceM.Determineitsshape.Answer:itisthe1-shapedcurveonFig.1.13thatseparatesthewhiteandgreyareas.Thus,thetrajectoriespassingthroughthefocimakeaspecial(one-dimensional)familyinM.PSfragreplacements r0j@DjFigure1.13.Thecollisionspaceofellipticbilliard.Exercise1.25.ShowthatifthetrajectoryofthebilliardparticlecrossesthesegmentF1F2joiningthefoci,thenitre\rectsat@Dandcrossesthissegmentagain.Similarly,ifthetrajectorycrossesthemajoraxisbeyondthesegmentF1F2,saytotheleftofit,thenafteroneormorere\rectionsat@Ditwillcrossthemajoraxistotherightofthissegment,etc.Thepreviousexerciseshowsthattherearetrajectoriesoftwotypes:thosecrossingtheinnersegmentF1F2ofthemajoraxisaftereveryre\rection(wecalltheminnertrajectories)andthosegoingaroundthissegment(wecallthemoutertrajectories). 1.4.BILLIARDINANELLIPSE13Exercise1.26.VerifythattheinnertrajectoriesllthewhiteareaonFig.1.13,andoutertrajectoriesllthegreyarea.Thefollowingisthemostimportantpropertyofellipticbilliards:Theorem1.27.ForeveryoutertrajectorythereisanellipsewithfociF1andF2thatistangenttoeachlinkofthattrajectory.ForeveryinnertrajectorythereisahyperbolawithfociF1andF2thatistangenttoeachlink(oritslinearextension)ofthattrajectory.Proof.Weonlyprovetherststatement(abouttheoutertrajectories),theproofofthesecondissimilar.TheargumentisprettyelementaryandillustratedonFig.1.14.HereA1AandA2Aaretwosuccessivelinksofanoutertrajectory.ThepointsB1andB2areobtainedbyre\rectingthefociF1andF2acrossthelinesA1AandA2A,respectively.Thefourangles\B1AA1,\A1AF1,\F2AA2and\A2AB2areequal.HencethetrianglesAB1F2andAB2F1arecongruent,inparticularjB1F2j=jB2F1j.ThereforejF1C1j+jF2C1j=jF1C2j+jF2C2j;whereC1andC2arethepointsofintersectionofA1AwithB1F2andA2AwithB2F1,respectively.Thus,thepointsC1andC2belongtothesameellipsewithfociF1andF2,andthelinesA1AandA2Aaretangenttothatellipse.PSfragreplacementsF1F2A1A2B1B2C1C2AFigure1.14.ProofofTheorem1.27.Ifeverylinkofabilliardtrajectoryistangenttoacertaingivencurve,thenthatcurveiscalledacaustic.Fig.1.15showsanellipticcausticforanoutertrajectoryandahyperboliccausticforaninnertrajectory.Theterm`caustic'isborrowedfromoptics,whereitmeansacurveonwhichlightraysfocusafterbeingre\rectedoamirror(wehaveseencausticsincircularbilliardsinSection1.1).Fig.1.15demonstratestheconcentrationofraysoncaustics(compareittoFig.1.4).AllthetrajectoriestangenttooneellipticcausticlieonaclosedcurveinthecollisionspaceM.Suchcurvesareshownas`horizontalwaves'inthewhiteareaonFig.1.13(rememberthattheleftandrightsidesoftherectangleneedbeidentied).EverysuchcurveisobviouslyinvariantunderthemapF. 141.SIMPLEEXAMPLESFigure1.15.Ellipticandhyperboliccausticsintheellipticbilliard.Exercise1.28.OneachinvariantcurvethemapFisconjugatetoarigidcirclerotationthroughsomeangle(thatangleiscalledtherotationnumber).Showthattherotationnumberchangescontinuouslyandmonotonicallywiththeinvariantcurve.Hint:considertwooutertrajectoriesstartingatthesamepointA02@Dbutwithdistinctellipticalcaustics;denotebyA0there\rectionpointsofthetrajectorywhoseellipticalcausticissmallerandbyA00nthoseoftheothertrajectory;observethatthesequencefA0gwillmovealongtheellipsefasterthanfA00ngdoes,seeFig.1.16.TheactionofthemapFoneachinvariantcurvecanbeanalyzedexplicitlyandtherotationnumbercanbecomputedanalytically,see[Be01,Sections2.5and3.2],butwewillnotgothatfar.PSfragreplacementsA0A02A001A002A0Figure1.16.Exercise1.28.Next,allthetrajectoriestangenttoonehyperboliccausticlieontwoclosedcurvesinM,oneinsideeachhalfofthe1-shapedgreydomain.SuchcurvesappearasovalsonFig.1.13.ThemapFtransformseachovalontoanidenticalovalwithintheotherhalfofthe1-shapedgreydomain.Thustheunionofthetwoidentical(symmetric)ovalswillbeinvariantunderF,andeachovalseparatelywillbeinvariantunderF2.Therefore,thecollisionspaceMofanellipticalbilliardiscompletelyfoliatedbyinvariantcurves.Inthissense,theellipticalbilliardissimilartothoseinacircleandinasquare.Inphysics,suchmodelsbelongtoaspecialclass:ifthephase 1.5.ACHAOTICBILLIARD:PINBALLMACHINE15spaceofasystemisfoliatedbyone-dimensionalinvariantsubmanifolds,thesystemissaidtobeintegrable;thedynamicsinsuchasystemiscompletelyregular.Thus,billiardsincircles,squaresandellipsesarecompletelyregular.1.5.Achaoticbilliard:pinballmachineThesimpleexamplesweregivenintheprevioussectionsforthesakeofintro-ductionofsomebasicfeaturesofbilliardstothenovicereader.Buttheyshouldnotbetakenastypical;infacttheirdynamicalcharacteristicsarequitespecialandinasenseoppositetothoseofchaoticbilliardsthatwillbecoveredintherestofthebook.Herewewilltakeaglimpseatsomethingthathappensinchaoticbilliards.Imagineyouareplayingapinballmachine.Asmallballshootsfromacannonintherightbottomcornerofarectangulartable,thenitbouncesotheedgesuntiliteitherhitsthetarget(thenyouwin)orfallsthroughanopeninginthebottom(goesdownthedrain,thenyoulose).Thetargetmightbeaspecialgureonthetablethatregistersthehitwhentheballtouchesit.Topreventdirecthits,assumethetargetisscreenedfromthecannon,andhittingthescreenisforbiddenbytherules.Thentheballhastobounceotheedgesbeforereachingthetarget,seeFig.1.17.Itisquiteanunusualpinballmachine,butforusitisagoodstartingexample.Figure1.17.Apinballmachine.Thetargetisthegreydiskscreenedfromthecannonbyadarkgreyarc.Supposeyoucanrotatethecannontochangetheangleatwhichtheballshootsout.Afteryoumissonce,youcanadjusttheshootingangleandsendtheballmoreaccuratelyintothetarget.Thisisarelativelyeasytask(illustratedonFig.1.17),asthetrajectoryoftheball(inarectangularbilliard)isverysimpleandpredictable.Also,youdonotneedtoaimwithabsoluteprecision:Exercise1.29.Supposethetargetisadiskofradiusrandthemovingballisapointparticle.LetLdenotethedistancecoveredbytheballfromthecannontothetarget.Showthatiftheshootingangleisobylessthanr=L(radians),thentheballstillhitsthetarget.Nowletusmakethetaskmorerealisticandchallengingbyinstallingsomebumpers(roundpillars)alloverthetable,seeFig.1.18,sothatourmovingball 161.SIMPLEEXAMPLESwillbouncebetweenthebumpersonitswaytothetarget(ordownthedrain).Anyonewhoplayedrealpinballmachinescaneasilyimaginesuchaprocess.Figure1.18.Apinballmachinewithbumpers(darkgreydisks).Woulditbeeasytoadjustthecannononthisnewtable?Obviously,not.Therouteoftheballiscomplicatedandalmostunpredictable,asitmaybounceodierentbumpers.Evenndingtherightsequenceofbumpersthattheballneedstohitbeforeitreachesthetarget(andavoidsthescreen)isnotasimpletask.Professionalbilliardplayerssolveasimilarproblemwhentryingthesendaballtoapocket,sothatithitsoneormoreotherballs.Furthermore,thecannonmustbeaimedwithalmostultimateprecision,asatinyerrorintheshootinganglemaysendtheballrollingdownalongacompletelywrongpath.ThisisillustratedonFig.1.19wherejusttwosuccessivebouncesareshown;itisquiteclearthattheinstabilityoftheball'smotionincreaseswitheverysubsequentre\rectionoabumper.Again,professionalbilliardplayersknowthatiftheirballneedstohitmorethanoneotherballsbeforesendingoneofthemintoapocket,theirtaskisverydicult.Furthermore,iftheballmusthitthreeormoreotherballs,thetaskisalmostimpossible.Arectangulartablewithroundbumpersinstalledonitisaclassicalexampleofachaoticbilliard.Themotionofthebilliardparticleonsuchatableiscomplicatedandunpredictable.Toanakedeye,itmaylooklikeawilddancebetweenthewallsofthetable,withoutanypatternorlogic(thatiswhypinballmachinesaresoattractive!).Thelackofpredictabilityischaracteristicforchaoticbilliards.Furthermore,slightchangesintheinitialpositionand/orvelocityoftheparticlequicklyleadtolargedeviations(suchasthatonFig.1.19),soafterjustafewcollisionswithbumperstwotrajectories,initiallyveryclosetogether,willseparateandmovefarapartfromeachother,asiftheyareunrelated.Thisinstability(alsoknownassensitivitytoinitialconditions)isanothercharacteristicfeatureofchaoticbilliards(andchaoticdynamicsingeneral).Inpracticalterms,thebestthingtheplayercandoinourgameistoshootrandomlyandwatchtheballrunningalloverthetablebouncingaroundbetweenbumpers{therewillalwaysbeachancethatithitsthetarget`byaccident'.Thisisessentiallyagameofchance,justlike\rippingacoin,rollingadie,orplayingcards.WewillseeinChapters6and7thatthemotioninachaoticbilliardisindeedessentiallyrandomandisbestdescribedintermsofprobabilitytheory. 1.5.ACHAOTICBILLIARD:PINBALLMACHINE17Figure1.19.Aballbouncingotwobumpers:aslighterrorintheinitialshootingangleresultsinadramaticde\rectionintheend.Ourtoyexampleactuallyhasalotincommonwithaclassicalmodelofstatis-ticalphysics,calledtheLorentzgas.Inthatmodel,asmallball(electron)bouncesbetweenlargexeddisks(molecules)thatmakearegularperiodic(crystalline)structure.WewillpresentitinChapter5.6.Wewillnotattempttogobeyondthisveryinformalintroductiontotherealmofchaoticbilliards,leavingallformalitiestillfurtherchapters.Interestedread-ersmayndamoreextensivedescriptionofchaoticbilliards,includingcomputerillustrations,in[Be01,Section1.1]. IndexAbsolutecontinuity,103,119{121,143,144,169,192,195,197,223,238,242,250,273,274,276Alignment(ofsingularitylines),87,110,232AlmostSureInvariancePrinciple(ASIP),184,185Autocorrelations,165Bernoulliproperty,69,141,153,154,157,160,163,207,278,304,305,307Birkhoergodictheorem,36,44,59,60,145,164,189,301,306Borel-Cantellilemma,97,112,293Boundedhorizon,26,27,35,37,67,72,88,189Brownianmotion,184Cardioid,258,278CentralLimitTheorem(CLT),164{167,179,183,184,186,187,249,294,295Circlerotation,3,13,163,303Cones(stableandunstable),62{65,73,74,76,83,86,178,211,212,216,224,252,253,260,263{266,276,277Continuation(ofsingularitylines),89,110,232,241Continuedfractions,55,56,83{85,214,216,217,220{222,265,266Cusps,20,24,25,27,38,67,75,267,269Decayofcorrelations,165{168,176,178,179,249,292,293,302DiusioninLorentzgas,188{190,249Distortionbounds,103,113{115,124,133,139,169,170,197,203,223,235{238,241,242,250Entropy,60{62,104,163,207,266,271,305{307Finitehorizon,26,27,35,37,67,72,88,189Focusingpoint,54{57,62,208,214,220,223,227,232,260,261Globalergodicity,147,151,274,278Grazingcollisions,23,29,30,33,84,86,88{90,92,129,267H-components(homogeneouscomponents),117,118,125{135,137,139,170,172,193{198,200{204H-manifolds(homogeneousmanifolds),109{114,121,136,138,147,148,150,170,172,174,177,178,186,190{193,195{197,200,201,205Hausdormetric,194Holonomymap,118{121,125,144,159,169,201,238,239,242,250,274Homogeneitystrips,108,109,129,130,138,149,169,236Hopfchain,142{144,147,148,156,157Hyperbolicity(nonuniform),43,64,95,141,144,166,220,242,250,270Hyperbolicity(uniform),43,64,65,74,75,95,96,103,106,108,114,115,119{121,123,125,128,139,141,166,213,227,229,231,236{239,242,243,247,248Integrability(offoliations),157,249Involution,38,43,73,86,245,262K-mixing(Kolmogorovmixing),104,153,154,157,158,160,163,278,304,305,307Lindebergcondition,182,295Localergodicity,147,149,151,152,274{278Lorentzgas,17,68,187,189,249Lyapunovexponents,41,43,44,46,48,49,52,59{63,74,143,154,255,270Meanfreepath,36,37,61,62,187,189Mirrorequation,56,106,252Oseledetstheorem,41{43,60,143,270,272p-metric(p-norm),58,59,74{76,97,98,112,218,223,227{229,234309 310INDEXPesinentropyformula,38,61Pinskerpartition(-algebra),104,158,160,305,307,308Propercrossing,194{196Properfamily,172,173,175{177,193{195,197,198,200,203Quadraticforms,64,254{257,260,270,274,277Rectangle,173Rectangle(Cantor),190{196,199,200Rectangle(solid),190,191,195Regularcoverings,141,250,251,274,275Stadium,210,219,224,227,229,230,232,233,235,236,238,242{245,247{249,251Standardpair,169{173,175,176,194{197,199,201Suspension\row,31,61,154,156,157,186Timereversibility,73,76,85,95,104,110,122,136{138,155,220,232,240u-SRBmeasuresanddensities,104{108,110,113,170,172,177,223,235WeakInvariancePrinciple(WIP),184,185Weaklystable(unstable)manifolds,156,157 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