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Texts and Readings in Mathematics 79 Anima Nagar Riddhi Shah Shrihari Sridharan Editors Elements of Dynamical Systems Texts and Readings in Mathematics Volume 79 AdvisoryEditor C.S.Seshadri,ChennaiMathematicalInstitute,Chennai,India ManagingEditor RajendraBhatia,AshokaUniversity,Sonepat,India Editors ManindraAgrawal,IndianInstituteofTechnology,Kanpur,India V.Balaji,ChennaiMathematicalInstitute,Chennai,India R.B.Bapat,IndianStatisticalInstitute,NewDelhi,India V.S.Borkar,IndianInstituteofTechnology,Mumbai,India ApoorvaKhare,IndianInstituteofSciences,Bangalore,India T.R.Ramadas,ChennaiMathematicalInstitute,Chennai,India V.Srinivas,TataInstituteofFundamentalResearch,Mumbai,India TechnicalEditor P.Vanchinathan,VelloreInstituteofTechnology,Chennai,India The Texts and Readings in Mathematics series publishes high-quality textbooks, research-level monographs, lecture notes and contributed volumes. Undergraduate and graduate students of mathematics, research scholars and teachers would find thisbookseriesuseful.Thevolumesarecarefullywrittenasteachingaidsandhigh- lightcharacteristicfeaturesofthetheory.Booksinthisseriesareco-publishedwith HindustanBookAgency,NewDelhi,India. · · Anima Nagar Riddhi Shah Shrihari Sridharan Editors Elements of Dynamical Systems Lecture Notes from NCM School Editors AnimaNagar RiddhiShah DepartmentofMathematics SchoolofPhysicalSciences IndianInstituteofTechnologyDelhi JawaharlalNehruUniversity NewDelhi,India NewDelhi,India ShrihariSridharan SchoolofMathematics IndianInstituteofScienceEducation andResearch-Thiruvananthapuram Thiruvananthapuram,India ISSN2366-8717 ISSN2366-8725 (electronic) TextsandReadingsinMathematics ISBN978-981-16-7962-9 (eBook) https://doi.org/10.1007/978-981-16-7962-9 Thisworkisaco-publicationwithHindustanBookAgency,NewDelhi,licensedforsaleinallcountries inelectronicformonly.SoldanddistributedinprintacrosstheworldbyHindustanBookAgency,P-19 GreenParkExtension,NewDelhi110016,India. JointlypublishedwithHindustanBookAgencyISBNoftheHindustanBookAgencyedition:978-93- 86279-83-5 MathematicsSubjectClassification:37-XX,11-XX,37Axx,37Bxx,37B10 ©HindustanBookAgency2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface VariousAdvancedTraininginMathematics(ATM)schools,originallylaunchedby theNationalBoardforHigherMathematics(NBHM),arethemostsuccessfulwork- shopswhichhavehelpedstudents,teachersandresearcherstoenhancetheirschol- arship and improve research. The efforts of the National Centre of Mathematics (NCM) and its apex committee in continuing this yeoman service by conducting severalAnnualFoundationSchools(AFS),AdvancedInstructionalSchools(AIS), NCMWorkshops(NCMW),InstructionalSchoolsforTeachers(IST)andTeachers’ EnrichmentWorkshops(TEW)throughouttheyeararepraiseworthy.Organisedby theNCM,withsupportfromNBHM,DepartmentofAtomicEnergy(DAE),Govern- mentofIndia,theseworkshopshaveattainedastatusoftheirownamidsttheMath community,withinthecountryandglobally. Theselecturenotesgrewoutofathree-weekAISon“ErgodicTheoryandDynam- icalSystems”(https://www.atmschools.org/2017/ais/etds)thatwasconductedatthe IndianInstituteofTechnologyDelhi(IITD),during4th–23rdDecember2017,organ- isedbyNCM,withthesupportofNBHM,DAE,GovernmentofIndia.Thespeakers at this school were C. S. Aravinda, Siddhartha Bhattacharya, S. G. Dani, Anish Ghosh,V.Kannan,AnimaNagar,C.R.E.RajaandKaushalVerma.Theirlectures wereaidedbythehugesupportofthetutorsoftheprogramme,NikitaAgarwal,P. Chiranjeevi,ManojChoudhuri,RajkumarKrishnan,ShrihariSridharanandPuneet Sharma.Wearethankfultothecontributionsofallthelecturersandthetutors. Dynamicsisthestudyoftheevolutionofanygivensystemwithtime,governed by some physical law. Different laws imposed on the system could give rise to a varietyofdynamicalsystems.Thelawsmayariseinavarietyofways;somewith respecttothestructureoftheunderlyingspacewherethesystemismanifested,some withrespecttonatureoftheactiononthespace,somewithrespecttoournotionof observationoftheevolutionetc.Thetopicofdynamicalsystemsisthusveryrich; withdifferentresearchersfocussingondifferentaspects. These lecture notes are intended to help a new researcher understand various aspectsofdynamicalsystems.Inkeepingwiththetruespiritsoftheavailabilityof a variety of means to study dynamical systems, this book begins with chapters on various kinds of dynamics; real dynamics, topological dynamics, ergodic theory, v vi Preface symbolicdynamics,complexdynamics.Further,asisnaturalforatopicthatspans avarietyofinterestsacrossareas,thetheoryofdynamicalsystemshasusefulappli- cationsacrossabroadspectrumofareasinmathematics,suchastopology,complex analysis,numbertheoryandrepresentationtheory.Inthisbook,welaterprovidea glimpseofsuchapplicationstonumbertheoryandgametheory. Every chapter of thisbook specialises inone aspect of dynamical systems;and thus begins at an elementary level and goes on to cover fairly advanced materials. Even though the lectures were delivered to a slightly mature audience comprising of graduate students from across the country, the chapters have been written by the respective authors so articulately that a beginner can read and understand the materialscovered,withabitofaneffort. Inthefirstchapter,westudydynamicsofmapsonthereallineoronaninterval there.Mostofthetheoremsprovedinthischapterarespecialtotherealline;their analogues do not hold in general dynamical systems. Section one starts with the definitions of such terms as fixed point, periodic point, eventually periodic point, recurrent point and non-wandering point. Their inter-relations are observed. Next, elementaryexamplesofmapslikecontractionmap,identitymap,squaringmap,tent map,logisticmapandshiftmapareintroduced.Ineachoftheseexamples,periodic points,recurrentpoints,etc.areexplicitlycalculated.Whichkindsofsubsetscanarise asthesetFix(f)ofallfixedpoints?Thisandfourofitsanaloguesareanswered.Three morenotions,namelyinvariantsets,omega-limitsetsandcyclesareintroduced. Insectiontwo,notionsofattractingandrepellingcyclesarestudied.Itstartswith theclassicaltheoremofBanachknownasthecontractionmappingtheorem.There arevariouswaysofunderstandingtheattractingnatureofafixedpoint,fromtheview pointsofcalculus,topology,metric,etc.Wediscussmutualimplicationsamongthem. Severalcounterexamplesareprovidedtodisprovesomeoftheimplications. In section three, topological transitivity is studied through various equivalent formulations.Fivedifferentproofsareincludedforthefactthatthetentmapistopo- logicallytransitive.Theseproofsleadtofivedifferentgeneraltheoremsthatopenup fivesignificantdirectionsofstudy.Incidentally,somemoreconceptssuchastopo- logicalconjugacy,Markovmapsandexpandingmapsarealsointroduced.Insection four,Devaney’sdefinitionofchaosisintroducedthroughthreeingredientproperties. Theindependenceofthesethreeisestablishedbyasetofeightcounterexamples. While doing so, about a dozen propositions involving transitivity, sensitivity and denseperiodicityareproved.Insectionfive,itisseenthattheindependenceresults obtainedintheprevioussectionarenotvalidwhentheunderlyingspaceisrestricted. Forexample,weprovethatontherealline,everytransitivemapisnecessarilychaotic. SectionsixismainlydevotedtoatheoremofSarkovskiioncyclelengthsavailable for real maps and the forcing relation among them. Here the proofs are merely outlined.NextcomesashortsectioninwhichBaireCategorytheoremandanother theorem (that have been used earlier) are proved. The chapter ends with a short sectionconsistingofnotesandexercises. In chapter two, we introduce G-systems and describe basic notions such as recurrence,minimalityandenvelopingsemigroups.WeprovideaproofofVander Waerden’stheorem.Wealsodiscussproximalanddistalnotionsanditsrelationwith Preface vii enveloping semigroup. Topological dynamics is inspired by the qualitative study ofdifferentialequations,initiatedbytheapproachofHenriPoincare,andfollowed largelybythecontributionofG.D.Birkhoff. G-systemsarejointlycontinuousactionsofatopologicalgrouponaHausdorff space.ThisabstractapproachwasinitiatedbyW.H.GottschalkandG.A.Hedlund. Weadopttheirapproachtostudythebasicnotionsofrecurrenceinthefirstsection. The second section is devoted to minimal systems which is fundamental to many recurrence theorems. Such phenomena have a wide range of applications and we provideonesuchapplicationinthefieldofnumbertheory.Wediscussthefamous proofofthecelebratedVanderWaerden’stheoremgivenbyH.FurstenbergandB. Weissinthethirdsection.Inthefourthsection,wediscussthealgebraictheoryof envelopingsemigroupsthatformafundamentaltooltostudytopologicaldynamics. Thenotionsofproximalanddistalsystemsareimportantaspects,whichwediscuss inthefifthsection.Thesixthsectionisbasicallydedicatedtotheevergreennotion oftopologicaltransitivityanditsvariousforms. Chapter three is a gently paced introduction to some of the key ideas in the generaltopicofErgodictheory,providingessentialbackgroundtodiscusssomeof thecornerstoneresultsinthefield. Thefirstcoupleofdecadesofthetwentiethcenturywitnessedadefinitive,neat andclearunderstandingoftheallimportantnotionsofmeasureinageneralcontext. Apartfromservingasawarmupontherudimentsofmeasure,thesecondsection ofthischapterintendstoparticularlyhighlighttheworkofC.Carathéodaryinthis context,andpointoutthepossiblelogicbehindtheintroductionoftheCarathéodory criterion for a set to be measurable. The section ends with a quick description of HausdorffmeasuresandHausdorffdimension. Recallingamotivationfromcertainquestionsinstatisticalmechanics,themain aimofthethirdsectionistogiveaproofofthecelebratedBirkhoffergodictheorem. Alsoknownasthepointwiseergodictheorem,firstprovedin1931byG.D.Birkhoff, this lofty result brought in much clarity on the notion of ergodicity, and triggered significantprogressinthemathematicalaspectsofthetheory. Building further on the discussion in the previous sections, the fourth and final sectionsketchestheproofofergodicityofoneoftheearliestinterestingexamplesof anergodicdynamicalsystem—thegeodesicflowontheunittangentbundleofclosed surfaceofconstantnegativecurvature.Firstprovenintheyear1934byG.Hedlund, the proof sketched here is the one due to E. Hopf which has inspired monumental later work in hyperbolic dynamics. The first subsection to section four may also serveasanintroductiontohyperbolicgeometry.Thus,thethirdchapteressentially captures the spirit of the remarkable development heralding the beginnings of this importantareaofresearchduringthefirstfourdecadesofthetwentiethcentury. Symbolic dynamics is the study of shift spaces, which consist of infinite or bi- infinitesequencesonapre-determinedalphabetset.Thesesequencesalmostcapture theessenceofabstractsystemsandprovideasimplifiedmodelofstudy.Codingsgive mappingsbetweentwosuchshiftspaces.Further,aidedbythecombinatorial,alge- braic,topologicalandmeasure-theoreticinvariants,codingsgiveasubtledescription ofmanydynamicalproperties,aswell. viii Preface Afterintroducingthesetupofsymbolicdynamicsinthefirstsectionofchapter four,wediscusssomebasicpropertiesinthesecondsection.Theconceptofentropy isdefinedinthethirdsection,andthefourthsectiondealswithmethodstocompute suchentropy.Inthefifthsection,aclassofsymbolicdynamicalsystemsrelatedto tilingspacesisdefinedandaprofoundresultduetoM.Szegedyisproved.Thelast sectionisdevotedtoanalgebraicdynamicalsystemknownas3-dotsystem,which isusedtostudysymbolicsystemsthatcanexhibitstrongrigidityproperty. Thepurposeofchapterfiveistopresentsomebasicideasandtoolsincomplex dynamics.Startingwithsomeelementaryobservationsthatmotivateustostudythis topicindetail,wemakeuseofthevariousversionsofMontel’stheoremthatdescribes normalityinafamilyofholomorphicfunctionsdefinedonadomainintheRiemann sphere,P1 =C∪{∞}.DichotomisingtheRiemannsphereusingMontel’snormality criterion on the family of iterates of a rational map, we obtain the Fatou and Julia sets of the considered rational map. Various properties of these two sets are then investigated;oneimportantpropertybeingthenon-vacuousnessoftheJuliasetfor rationalmapsofdegree atleast2.Answeringournaturalcuriosityaboutasimilar propertyfortheFatousetofarationalmap,weconstructafamilyofrationalmaps forwhichtheJuliasetisallofP1;whichimpliesthattheFatousetisempty,inthis case.Lattès’exampleisasimplecaseofthisconstruction. The authors then focus on some statements that characterise the Julia set of a rationalmap,alternativelyusingvariousresultsfromcomplexanalysis.Thesestate- mentsaremoreusefulindeterminingtheJuliasetscomputationally.Then,thefocus shiftstostudyinglocalnormalformsnearfixedpointsandtheclassificationofFatou componentsforrationalmaps. Oneimportantresultinthisfieldpertainstotherelationbetweenthedensesetof pre-imagesofanygenericpointintheJuliasetandtheequilibriummeasureofany compactsubsetofP1,usingtheenergyintegral,asencapsulatedbyaresultdueto H. Brolin. The authors build their case for the Brolin’s theorem in P1 and discuss analogousresultsinhigherdimensions. Recent decades have seen dramatic progress in the study of ergodic aspects of groupactionsonhomogeneousspacesofLiegroups.Muchofthisprogress,begin- ning with Margulis’ famous proof of Oppenheim’s conjecture, has been closely associatedtoDiophantineanalysis.Another,morerecentexampleistheimportant workofEinsiedler,KatokandLindenstrausstowardsLittlewood’sconjecture.The aimofchaptersixistopresentsometopicsattheinterfaceofhomogeneousdynamics andnumbertheorywiththeaimofgivingthereaderaglimpseoftherichconnec- tionsbetweenthetwosubjects.Thegoalistowhettheappetiteofthereader.The interestedreadercanthenmoveontoamoresystematicanddetailedsourcelikethe bookbyEinsiedlerandWard.Thisissuitablefortalentedundergraduateswithsome backgroundinLiegroups,forgraduatestudents,aswellasformathematicianswho wishtogetacquaintedwiththearea. Theaimofchaptersevenistogiveanintroductiontoanotionof“largesubsets” ofEuclideanandothersimilarspaces,thathasattractedmuchattentionintherecent decades, in the theory of Diophantine approximation, geometry, and dynamics of flowsonhomogeneousspaces.Thesetsaredefinedintermsofexistenceofwinning Preface ix strategies for various two-player infinite games. Their origin goes back to a 1966 paperofW.M.Schmidt,whichmadeinterestingobservationsaboutthesetofbadly approximablerealnumbershavingcertainunusuallargenessproperties,whichhas nowfoundgeneralisationsinavarietyofcontexts. Asaspecialistmayhaveobserved,thetopicsdealtwithineachofthesechapters isanareaofresearchinitsownright,howeverthatdependsontheotherareasalso describedintheotherchapters.Onemayreligiouslycoverallmaterialsinthisbook, if one is interested to give a year-long course on various elements of dynamical systems, as the title of the book suggests. However, within the book lies various ideasforaone-semestercourse;eachofthecombinationbelowdescribingonesuch. • chapters1–3and5; • chapters1,2,4and5; • chapters2–4and5; • chapters3,6and7;etc. We are grateful to the participants Mahboob Alam, G. K. Chaitanya, Haritha Cheriyath,PramodDas,ShreyasiDatta,MuktaGarg,DileepKumar,DineshKumar, Pabitra Narayan Mandal, Manoj B. Prajapati, Yogesh Prajapaty, Manish Rajput, Manpreet Singh, Pradeep Singh, Sharvari Neetin Tikekar and Atma Ram Tiwari, fortheirspecialeffortsintakingnoteswhichhelpedtheauthorsinpreparingtheir lecture notes. It is a pleasure to thank the students for their contributions to these lecturenotes. Lastly,wethanktheIndianInstituteofTechnologyDelhi(IITD)fortheirexcellent hospitality. NewDelhi,India AnimaNagar NewDelhi,India RiddhiShah Thiruvananthapuram,India ShrihariSridharan