Table Of ContentTexts 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
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ApoorvaKhare,IndianInstituteofSciences,Bangalore,India
T.R.Ramadas,ChennaiMathematicalInstitute,Chennai,India
V.Srinivas,TataInstituteofFundamentalResearch,Mumbai,India
TechnicalEditor
P.Vanchinathan,VelloreInstituteofTechnology,Chennai,India
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· ·
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
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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
