Table Of ContentLecture Notes in Mathematics 2164
Boris Hasselblatt Editor
Ergodic Theory
and Negative
Curvature
CIRM Jean-Morlet Chair, Fall 2013
Lecture Notes in Mathematics 2164
Editors-in-Chief:
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BernardTeissier,Paris
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AlessioFigalli,Zurich
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MarkPodolskij,Aarhus
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Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Boris Hasselblatt
Editor
Ergodic Theory and Negative
Curvature
CIRM Jean-Morlet Chair, Fall 2013
123
Editor
BorisHasselblatt
DepartmentofMathematics
TuftsUniversity
Medford,Massachusetts
USA
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-43058-4 ISBN978-3-319-43059-1 (eBook)
DOI10.1007/978-3-319-43059-1
LibraryofCongressControlNumber:2017953454
MathematicsSubjectClassification(2010):37C40,37D40
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Preface
Thisvolumeconsistsofnotesfromminicoursesgiveninworkshopsheldunderthe
auspicesoftheJean-MorletChairatCIRMin2013and2014andwithsubstantial
coresupportandfundingbyCIRM.
Mostofthesecoursesweregivenduringtheworkshop“YoungMathematicians
in Dynamical Systems” organized by the GDR Platon under the direction of
Françoise Dal’Bo of Université Rennes 1; her co-organizers were Louis Funar
(Université Grenoble 1), Boris Hasselblatt (Tufts University, CIRM, and Aix-
MarseilleUniversité),andBarbaraSchapira(UniversitédePicardieJulesVerne).1
The event was supported by several ANRs such as GEODE, directed by Barbara
Schapira, and by the LABEX Archimede. The centerpieces were minicourses
by Keith Burns (Northwestern University), Carlos Matheus (Instituto Nacional
de Matemática Pura e Aplicada in Brazil and Université Paris 13), and Boris
Hasselblatt.ThescientificfocuswastheergodicityoftheWeil–Peterssongeodesic
flow on Teichmüller space. The course by Burns presented geodesic flows and
methodsforprovingtheirergodicity,thatbyMatheusintroducedTeichmüllerspace
and the Weil–Petersson metric, and that by Hasselblatt provided an introduction
to hyperbolic dynamical systems and ergodic theory. Barbara Schapira presented
in a unified way the classical dynamical and ergodic properties of the horocy-
cle flow in the Spring 2014 School on Geometry and Dynamics,2 which was
organizedbyNicolasBedaride(Aix-MarseilleUniversité),AlexanderBufetov(Aix-
Marseille Université), Moon Duchin (Tufts University), Boris Hasselblatt, Pascal
Hubert (Aix-Marseille Université), and Federico Rodriguez Hertz (Pennsylvania
State University) with the scientific committee consisting of Giovanni Forni (the
University of Maryland), Boris Hasselblatt, Howard Masur (the University of
IllinoisatChicago andUniversityof Chicago),and GrigoriOlshanski(Dobrushin
1The participant list is at http://www.cirm-math.fr/Archives/?EX=liste_participants&annee=
2013&id_renc=1097&num_semaine=0.
2The participant list is at http://www.cirm-math.fr/Archives/?EX=liste_participants&annee=
2014&id_renc=1129&num_semaine=0.
v
vi Preface
Mathematics Laboratory at the Institute for Information Transmission Problems
of the Russian Academyof Sciences, the IndependentUniversityof Moscow,and
the NationalResearch University–HigherSchoolofEconomics).Itwas supported
amongothersbytheNationalScienceFoundation,A*Midex,andtheGDRPlaton.
Thecontentsofthisvolumeareasfollows.Itbeginswithfoundationalmaterial:
an introduction to hyperbolic dynamics and ergodic theory (plus a translation of
Hadamard’sproofofthestable-manifoldtheorem)byBorisHasselblattfollowedby
onetothedynamicsofgeodesicandhorocyclicflowsbyBarbaraSchapira.
The nextthree chapterscover the motivating mathematicsfor the workshopas
follows:AdetailedsummaryoftheBurns–Masur–Wilkinsonpaper“Ergodicityof
theWeil–PeterssonGeodesicFlow”laysouttheagendaandcoreideas.Atextbased
onthelecturesbyBurns(“ErgodicityofGeodesicFlowsonIncompleteNegatively
CurvedManifolds”)thenprovidesalltheinsightsintotheergodicityproof.Finally,
thelecturenotesbyMatheus(“TheDynamicsoftheWeil–PeterssonFlow”)provide
comprehensive background on the Teichmüller theory at the base of the whole
problem.
We also includeda surveyof somearithmeticapplicationsof ergodictheoryin
negativecurvature,whichroundsoutthevolumebyaddinganothertopiccentered
onergodicityofgeodesicflowsinnegativecurvature.
Thereisnoneedtoreadthisbooksequentially.Thechaptersmakeitaseasyas
possible to read any one of them independently while keeping it easy to refer to
otherchaptersforanyneededcontext.
TheeditorgratefullyacknowledgespartialsupportbytheCommitteeonFaculty
Research Awards of Tufts University and will forever be indebted to CIRM for 6
monthsoftrulyexceptionalworkingconditions.
Medford,MA,USA BorisHasselblatt
April2017
Contents
1 IntroductiontoHyperbolicDynamicsandErgodicTheory ............ 1
BorisHasselblatt
2 OnIterationandAsymptoticSolutionsofDifferentialEquations
byJacquesHadamard....................................................... 125
BorisHasselblatt(Translator)
3 DynamicsofGeodesicandHorocyclicFlows ............................. 129
BarbaraSchapira
4 ErgodicityoftheWeil–PeterssonGeodesicFlow......................... 157
KeithBurns,HowardMasur,andAmieWilkinson
5 ErgodicityofGeodesicFlowsonIncompleteNegativelyCurved
Manifolds ..................................................................... 175
KeithBurns,HowardMasur,CarlosMatheus,andAmieWilkinson
6 TheDynamicsoftheWeil–PeterssonFlow................................ 209
CarlosMatheus
7 ASurveyofSomeArithmeticApplicationsofErgodicTheory
inNegativeCurvature....................................................... 293
JouniParkkonenandFrédéricPaulin
vii
Chapter 1
Introduction to Hyperbolic Dynamics
and Ergodic Theory
BorisHasselblatt
1.1 Introduction
1.1.1 GuidedTour
These are notes based on a minicourses given at the Centre International de
RencontresMathématiques,MarseilleinNovember2013,attheUniversityofTokyo
inJune2014,andthe2015HoustonSummerSchoolonDynamicalSystems.They
owe much to these experiences,and I would like to thank the organizersof these
schools for their invitation and hospitality as well as the participants for their
engagementandtheirattentivecommentsonthesenotes.
Whiletherearemanygoodintroductionstohyperbolicdynamicalsystems,1two
aspectsofthisoneareofinterest.Ononehand,weimplementanunderappreciated
approachduetoBowen,AnosovandKatok[Bo75,Bo78,Ka81]toobtainthebasic
topologicaldynamicsofuniformlyhyperbolicdynamicalsystemsfromshadowing
and expansivity; this is the content of Sect.1.3.3. On the other hand, we use the
Hopfargumenttoobtainmultiplemixingofhyperbolicdynamicalsystems—which
meansthatwecandosowithoutanyreferencetoentropytheoryorresultsthatuse
it.Thus,pages24–44canberegardedastheprincipalnoveltyofthesenotes.
The lectures gave an introduction to some features of the topological and
measurable dynamics of hyperbolic systems, mainly in discrete time, and this is
an essentially self-contained account of these basics. The first half is devoted to
1Notably[Yo95],whichinspiredSect.1.6,whereweprovetheStable/UnstableManifoldTheorem
usingthePerron–Irwinmethod.Altogetherthesenotesowemuchto[Co07,KaHa95,Yo95].
B.Hasselblatt((cid:2))
DepartmentofMathematics,TuftsUniversity,Medford,MA02155,USA
e-mail:[email protected]
©SpringerInternationalPublishingSwitzerland2017 1
B.Hasselblatt(ed.),ErgodicTheoryandNegativeCurvature,
LectureNotesinMathematics2164,DOI10.1007/978-3-319-43059-1_1
2 B.Hasselblatt
hyperbolic dynamical systems, and the second half (Sect.1.7) introduces ergodic
theory. While a central point is that these subjects interact deeply, the halves are
essentially independent,to the point of having some duplication(mainly between
Sects.1.5and1.7).Acommonfeatureisthatentropytheoryisabsent,andanovelty
isthatmultiplemixingpropertiesareobtainedwithoutit.
The sections on hyperbolic dynamics are modular and can be read largely
independently.That is, each part can largely be read on its own or omitted on its
own.ThisismostevidentlysoforthissectionandthehistoricalsketchinSect.1.2.
Theremaining“hyperbolic”sectionsarerelatedasfollows.
Section 1.3 forms the centerpiece: basic and iconic features of a hyperbolic
dynamicalsystemarederivedfromwhattherebyappearsastheverycorefeaturesof
hyperbolicdynamics:Expansivityandshadowing.Thelatteristhatinahyperbolic
systemanythingonecanimagineapproximatelyhappeningis,togoodapproxima-
tion, actually happening in the system. Sections 1.3.3–1.3.6 (pages 24–31) show
that the Shadowing Lemma produces the essential richness and rigidity of the
orbitstructureofahyperbolicdynamicalsystem(expansivity,theAnosovClosing
Lemma, specification, spectral decomposition,topologicalstability). The stronger
Anosov Shadowing Theorem more easily yields structural stability and symbolic
descriptions.ThiswholedevelopmentusestheContraction-MappingPrinciple(Pro-
position1.6.3)butnotthetechnicalsectionsoninvariantmanifolds(Sects.1.6.4and
1.6.5),yetevenwithcompleteproofsitonlyoccupiespages24–37.
Section 1.5 can also be read independently of the preceding material, though
the examples in the present section will help. Based on ideas originally due to
BabillotandCoudène,itshowstheHopfargumenttomaximumadvantage,which
is thereforepresentedby stating explicitly what is neededfor the argumentrather
thanrelyingoncontextandbackground.WhiletheHopfargumentwasdevelopedto
showergodicity,weshowthatitcanbeusedeffectivelytoestablishmixingwithno
addedeffort,andintherightcircumstancesmultiplemixingforevenlesseffort.A
notableingredientofindependentinterestistheergodicityofthestable(orunstable)
foliation,whichisrarelyfeaturedinintroductions,andwhichresultsfromasimple
argument.2
Section 1.6 provides the Contraction-Mapping Principle and the Hadamard–
PerronStable/UnstableManifoldTheorem.TheformerisinvokedinSect.1.3,and
weillustratethemajorimportanceofthelatterinthesubjectbeyondthearguments
in Sect.1.5 by using these invariant foliations to further the understanding of the
topological dynamics of a hyperbolic set. We prove this using the Perron–Irwin
method and provide the Hadamard method in a separate chapter in the form of
Hadamard’soriginalpresentation[Ha01],translatedhereintoEnglish,presumably
forthefirsttime.
2Self-containedsaveforinvokingabsolutecontinuity,whichcanbefoundin[Br02,Chap.6].
