Lecture notes on the dynamics of the Weil-Petersson flow

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by  Carlos Matheus

LECTURE NOTES ON THE DYNAMICS OF THE WEIL-PETERSSON FLOW
CARLOSMATHEUS
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CONTENTS
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4 1. Introduction 2

1.1. Somewordsontheoriginofthesenotes 2
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S 1.2. AnoverviewofthedynamicsofWPflow 2
D 1.3. ErgodicityofWPflow:outlineofproof 5
. 1.4. RatesofmixingofWPflow 11
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t 1.5. Organizationofthetext 12
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m 2. ModulispacesofRiemannsurfacesandtheWeil-Peterssonmetric 12
2.1. Definitionandexamplesofmodulispaces 13
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2.2. Teichmüllermetric 14

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2.3. Teichmüllerspacesandmappingclassgroups 15
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2.4. Fenchel-Nielsencoordinates 16
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9 2.5. CotangentbundletomodulispacesofRiemannsurfaces 18
6 2.6. Integrablequadraticdifferentials 20
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2.7. TeichmüllerandWeil-Peterssonmetrics 21
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. 2.8. ErgodicityofWPflow:outlineofproofrevisited 24
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3. GeometryoftheWeil-Peterssonmetric 26
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6 3.1. Items(I)and(II)ofTheorem1.5forWPmetric 26
1 3.2. Item(III)ofTheorem1.5forWPmetric 29
:
v 3.3. Item(IV)ofTheorem1.5forWPmetric 30
Xi 3.4. Item(V)ofTheorem1.5forWPmetric 42
3.5. Item(VI)ofTheorem1.5forWPflow 43
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a 4. DecayofcorrelationsfortheWeil-Peterssongeodesicflow 48
4.1. RatesofmixingoftheWPflowonT1M I:ProofofTheorem4.1 50
g,n
4.2. RatesofmixingoftheWPflowonT1M II:ProofofTheorem4.2 55
g,n
References 70
2010MathematicsSubjectClassification: Primary:37D20;Secondary:57N10.
Keywordsandphrases: Riemannsurfaces,modulispaces,Teichmüllerspaces,Weil-Petersson
metric,Weil-Peterssongeodesicflow,ergodicity,mixing,ratesofmixing.
ThisworkwaspartiallysupportedbytheFrenchANRgrant“GeoDyM”(ANR-11-BS01-0004)
andtheBalzanResearchProjectofJ.Palis.
1 ©2016THEAUTHOR
2 CARLOSMATHEUS
1. INTRODUCTION
1.1. Some words on the origin of these notes. This text is an expanded ver-
sion of some lecture notes prepared by the author in the occasion of a series
ofthreelecturesduringtheworkshopYoungmathematiciansindynamicalsys-
temsorganizedbyFrançoiseDal’bo,LouisFunar,BorisHasselblattandBarbara
Schapira in November 2013 at Centre International de Rencontres Mathéma-
tiques(CIRM),Marseille,France.
AsitisexplainedintheintroductionofHasselblatt’stext[29]inthisvolume,
the three lectures at the origin of this text were part of a minicourse by Keith
Burns, Boris Hasselblatt and the author around the recent theorem of Burns-
Masur-Wilkinson[15]ontheergodicityoftheWeil-Petersson(WP)geodesicflow.
Of course, the goal of these notes is the same of the author’s lectures: we
wanttocoversomeoftheaspectsrelatedtomodulispacesofRiemannsurfaces
(andTeichmüllertheory)intheproofsoftheergodicityofWPflow[15](seealso
Theorem1.1below)andtherecentresultsofBurns,Masur,Wilkinsonandthe
author[14]ontheratesofmixingofWPflow(seealsoTheorem1.2below).
1.2. AnoverviewofthedynamicsofWPflow. Beforegivingprecisedefinitions
ofthe terms introducedabove (e.g.,moduli spaces ofRiemann surfaces,Weil-
Peterssongeodesicflow,etc.),letuslistandcomparesomepropertiesoftheWP
flow and its close cousin the Teichmüller (geodesic) flow (see [69]) in order to
getaflavoroftheirdynamicalbehaviors.
Teichmüllerflow WPflow
(a) comesfromaFinslermetric comesfromaRiemannianmetric
(b) complete incomplete
(c) ispartofaSL(2,(cid:82))-action isnotpartofaSL(2,(cid:82))-action
(d) non-uniformlyhyperbolic singularhyperbolic
(e) related to flat geometry of Rie- related to hyperbolic geometry of
mannsurfaces Riemannsurfaces
(f) transitive transitive
(g) periodicorbitsaredense periodicorbitsaredense
(h) finitetopologicalentropy infinitetopologicalentropy
(i) ergodic for the Liouville measure ergodic for the Liouville measure
µ µ
T WP
(j) metricentropy0<h(µ )<∞ metricentropy0<h(µ )<∞
T WP
(k) exponentialrateofmixing mixingatmostpolynomial(ingen-
eral)
Letusmakesomecommentsonboththecommonfeaturesandthesignifi-
cantdifferencesbetweentheTeichmüllerandWPflowshighlightedintheitems
above.
LECTURENOTESONTHEDYNAMICSOFTHEWEIL-PETERSSONFLOW 3
TheTeichmüllerflowisassociatedtoaFinslermetric(i.e.,acontinuousfamily
ofnorms)onthefibersofthecotangentbundleofthemodulispaces1,whilethe
WPflowisassociatedtoaRiemannian(and,actually,Kähler)metriccalledWeil-
Petersson (WP) metric. In particular,the item (a) says that the WP flow comes
fromametricthatissmoother thanthemetricgeneratingtheTeichmüllerflow.
WewillcomebacktothispointlaterwhendefiningtheWPmetric.
Ontheotherhand,theitem(b)saysthatthedynamicsofWPflowisnotso
nicebecauseitisincomplete,thatis,therearecertainWPgeodesicsthat“goto
infinity”infinitetime.Inparticular,theWPflowisnot definedforalltimet∈(cid:82)
when we start from certain initial data. We will make more comments on this
later.Nevertheless,Wolpert[62]showedthattheWPflowisdefinedforalltime
t∈(cid:82)foralmostevery initialdatawithrespecttotheLiouville(volume)measure
inducedbyWPmetric,and,thus,theWPflowisalegitimeflowfromthepoint
ofviewofErgodicTheory.
The item (c) says that WP flow is less algebraic than Teichmüller flow be-
cause the former is not part of a SL(2,(cid:82))-action while the latter corresponds
tothediagonalsubgroup g =diag(et,e−t)ofSL(2,(cid:82))acting(inanaturalway)
t
ontheunitcotangentbundleofthemodulispacesofRiemannsurfaces.Here,
it is worth to mention that the mere fact that the Teichmüller flow is part of
a SL(2,(cid:82))-action makes its dynamics very rich: for instance, once one shows
that the Teichmüller flow is ergodic (with respect to some SL(2,(cid:82))-invariant
probabilitymeasure),itispossibletoapplyHowe-Moore’stheorem(orvariants
of it) to improve ergodicity into mixing (and, actually, exponential mixing) of
Teichmüllerflow(see,e.g.,[2]and[3]formoredetails).
Theitem(d)saysthatWPandTeichmüllerflows(morally)arenon-uniformly
hyperbolicinthesenseofPesintheory[44],buttheyaresofordistinct reasons.
The non-uniform hyperbolicity of the Teichmüller flow was shown by Veech
[58](for“volume”/Masur-Veechmeasure)andForni[26](forarbitraryinvariant
probabilitymeasures)anditfollowsfromuniformestimatesforthederivative
oftheTeichmüllerflowoncompactsets.Ontheotherhand,thenon-uniform
hyperbolicityoftheWPflowrequiresaslightlydifferentargumentbecausesome
sectionalcurvaturesofWPmetricapproach−∞or0atcertainplacesnearthe
“boundary”ofthemodulispaces.Wewillreturntothispointinthefuture.
Theitem(e)partlyexplainstheinterestofseveralauthorsinTeichmüllerand
WPflows.Indeed,sincetheirintroductionbyBernardRiemannin1851(inhis
PhD thesis), the study of Riemann surfaces and their moduli spaces became
an important topic of research in both Mathematics and Physics (for reasons
whoseexplanationsarebeyondthescopeofthesenotes).Inparticular,thefact
that the properties of the Teichmüller and WP flows on moduli spaces allows
torecovergeometricalinformationaboutRiemannsurfacesmotivatedpartof
theliteratureonthedynamicsoftheseflows.Concerningapplicationsofthese
1Actually,theFinslermetriccorrespondingtoTeichmüllerflowisaC1butnotC2familyof
norms:see,e.g.,pages308and309ofHubbard’sbook[31].
4 CARLOSMATHEUS
flowstotheinvestigationofRiemannsurfaces,itisnaturaltostudytheTeich-
müller flow whenever one is interested in the properties of flat metrics with
conical singularities on Riemann surfaces (cf. Zorich’s survey [69]), while it is
more natural to study the WP metric/flow whenever one is interested in the
propertiesofhyperbolicmetricsonRiemannsurfaces:forinstance,Wolpert[63]
showedthatthehyperboliclengthofaclosedgeodesicinafixedfreehomotopy
classisaconvexfunction alongorbitsoftheWPflow,Mirzakhani [41]proved
thatthegrowthofthehyperboliclengthsofsimplegeodesicsonhyperbolicsur-
faces is relatedto the WP volume ofthe moduli space,and,afterthe works of
Bridgeman[8],McMullen[38]andmorerecentlyBridgeman-Canary-Labourie-
Sambarino[9](amongotherauthors),weknowthattheWeil-Peterssonmetric
isintimatelyrelatedtothermodynamicalinvariants(entropy,pressure,etc.)of
thegeodesicflowonhyperbolicsurfaces.
Concerning items (f) to (h),Pollicott-Weiss-Wolpert [46] showed the transi-
tivityanddensenessofperiodicorbitsoftheWPflowintheparticularcaseof
the unit cotangent bundle of the moduli space M (of once-punctured tori).
1,1
In general,the transitivity,the denseness of periodic orbits and the infinitude
ofthetopologicalentropyoftheWPflowontheunitcotangentbundleofthe
moduli space M of genus g Riemann surfaces with n marked points (for
g,n
any g ≥1,n≥1) were shown by Brock-Masur-Minsky [10]. Moreover,Hamen-
städt[27]provedtheergodicversionofthedensenessofperiodicorbits,i.e.,the
densenessofthesubsetofergodicprobabilitymeasuressupportedonperiodic
orbitsinthesetofallergodicWPflowinvariantprobabilitymeasures.
TheergodicityofWPflow(mentionedinitem(i))wasfirststudiedbyPollicott-
Weiss [45] in the particular case of the unit cotangent bundle T1M of the
1,1
moduli space M of once-punctured tori: they showed that if the first two
1,1
derivatives of the WP flow on T1M are suitably bounded, then this flow is
1,1
ergodic.Morerecently,Burns-Masur-Wilkinson[15]wereabletocontrolingen-
eral the first derivatives of WP flow and they used theirestimates to show the
followingtheorem:
THEOREM 1.1 (Burns-Masur-Wilkinson). The WP flow on the unit cotangent
bundleT1M ofthe moduli space M ofRiemann surfacesofgenus g with
g,n g,n
n markedpointsisergodicwithrespecttotheLiouvillemeasureµ oftheWP
WP
metric whenever 3g −3+n ≥ 1. Actually, it is Bernoulli (i.e., it is measurably
isomorphictoa Bernoulli shift)and,afortiori,mixing. Furthermore,itsmetric
entropyh(µ )ispositiveandfinite.
WP
TheTeichmüller-theoreticalaspectsofthistheoremwilloccupythenexttwo
sections of this text. For now, we will just try to describe the general lines of
Burns-Masur-WilkinsonargumentsinSubsection1.3below.
However, before passing to this topic, let us make some comments about
item(k)aboveontherateofmixingofTeichmüllerandWPflows.
Generallyspeaking,itisexpectedthattherateofmixingofasystem(diffeo-
morphism or flow) displaying a “reasonable” amount of hyperbolicity is expo-
nential:forexample,thepropertyofexponentialrateofmixingwasshownby
LECTURENOTESONTHEDYNAMICSOFTHEWEIL-PETERSSONFLOW 5
Dolgopyat[24](seealsothisarticleofLiverani[34])foralargeclassofcontact
Anosov flows2, and by Avila-Gouëzel-Yoccoz [3] and Avila-Gouëzel [2] for the
Teichmüllerflowequippedwith“nice”measures.
Here,werecallthattherateofmixing/decayofcorrelationsofamixingflow
ψt isthespeedofconvergencetozeroofthecorrelationsfunctionsC (f,g):=
t
(cid:82) f ·g◦ψt−(cid:161)(cid:82) f(cid:162)(cid:161)(cid:82) g(cid:162)ast→∞(forchoicesof“sufficientlysmooth”observables
f and g). Intuitively,the rate of mixing is a quantitative measurement of how
fasttheflowψt mixdistinctregionsofthephasespace(suchasthesupportsof
theobservables f andg).See,e.g.,Subsection6.16ofHasselblatt’slecturenotes
[29]formorecomments.
In this context, given the ergodicity and mixing theorem of Burns-Masur-
Wilkinsonstatedabove,itisnaturaltotryto“determine”therateofmixingof
WPflow.Inthisdirection,weobtainedthefollowingresult(cf.[14]):
THEOREM 1.2(Burns-Masur-M.-Wilkinson). The rate of mixing of WP flow on
T1M (for“reasonablysmooth”observables)is
g,n
• atmostpolynomialfor3g−3+n>1and
• rapid(super-polynomial)for3g−3+n=1.
Wewillpresentasketchofproofofthisresultinthelastsectionofthistext.
Fornow,wewillcontentourselveswithavaguedescriptionofthegeometrical
reasonforthedifferenceintherateofmixingoftheTeichmüllerandWPflows
inSubsection1.4below.
1.3. Ergodicity of WP flow: outline of proof. The initial idea to prove Burns-
Masur-Wilkinsontheoremisthe“usual”argumentfortheproofofergodicityof
asystemexhibitingsomehyperbolicity,namely,Hopf’sargument.
1.3.1. A quick review of Hopf’s argument. Traditionally,Hopf’s argument runs
asfollows(cf.Subsection4.3ofHasselblatt’slecturenotes[29]).Givenasmooth
flow(ψt)t∈(cid:82):X →X onacompactRiemannianmanifold(X,d)preservingthe
corresponding volume measure µ and a continuous observable f :X →(cid:82),we
considerthefutureandpastBirkhoffaverages:
f+(x):= lim 1 (cid:90) T f(ψs(x))ds and f−(x):= lim 1 (cid:90) T f(ψs(x))ds
T→+∞T 0 T→−∞T 0
By Birkhoff’s ergodic theorem (cf. Subsection 6.3 of [29]), for µ-almost every
x∈X,thequantities f+(x)and f−(x)existand,actually,theycoincide f+(x)=
f−(x):=f(cid:101)(x).Intheliterature,apointxsuchthat f+(x), f−(x)existand f+(x)=
f−(x)=f(cid:101)(x)iscalledaBirkhoffgeneric point(withrespecttoµ).
Bydefinition,theergodicityofψt (withrespecttoµ)isequivalenttothefact
thatthefunctions f+ and f− areconstant atµ-almosteverypoint.
Inordertoshowtheergodicityofaflowψt withsomehyperbolicity,Hopf[30]
+ −
observesthatthefunction f ,resp. f ,isconstantalongstable,resp.unstable,
2IncludingcertaingeodesicflowsoncompactRiemannianmanifoldswithnegativecurvature.
6 CARLOSMATHEUS
sets
Ws(x):={y: lim d(ψt(y),ψt(x))=0},resp.Wu(x)={y: lim d(ψt(y),ψt(x))=0},
t→+∞ t→−∞
i.e., f+(x)=f+(y)whenever y∈Ws(x),resp. f−(x)=f−(z)wheneverz∈Wu(x).
Weleavetheverificationofthisfactasanexercisetothereader.
InthecaseofanAnosovflow ψt on X,weknowthatthestableandunstable
setsareimmersedsubmanifolds(cf.Subsection5.5ofHasselblatt’snotes[29]).
Moreover,ifoneforgetsabouttheflowdirection,thestableandunstablemani-
foldshavecomplementarydimensionsandintersecttransversely.Hence,given
twopoints p,q∈X (lyingindistinctorbitsofψt),wecanconnectthemusing
piecesofstableandunstablemanifoldsasshowninthefigurebelow:
q
p
FIGURE 1. Connectingp and q withpiecesofstableandunsta-
blemanifolds.
In particular, this indicates that a volume-preserving Anosov flow ψt is er-
+ −
godic because the functions f and f are constant along stable and unsta-
ble manifolds, they coincide almost everywhere and any pair of points can
be connected via pieces of stable and unstable manifolds. However,this argu-
menttowardsergodicityofψt isnot completeyet:indeed,oneneedstoknow
that the intersection points z ,...,z between the pieces of stable and unsta-
1 n
blemanifoldsconnectingp andq areBirkhoffgenericinordertoconludethat
f(cid:101)(p)=f(cid:101)(z1)=···=f(cid:101)(zn)=f(cid:101)(q).
Intheoriginalcontextofhisarticle,Hopf[30]studiesageodesicflowψt of
acompactsurfaceofconstant negativecurvature,andheusesthefactthatthe
stableandunstablemanifoldsformC1foliationstodeducethattheintersection
points z ,...,z can be taken to be Birkhoff generic points. Indeed, since the
1 n
invariantfoliations areC1 in his context,Hopfapplies Fubini’s theorem to the
setB offullµ-volumeconsistingofBirkhoffgenericpointsinordertoensure
thatalmostallstableandunstablemanifoldsWs(x)andWu(x)intersectB in
asubsetoftotallengthmeasureofWs(x)andWu(x)(comparewiththeproof
ofProposition4.10of[29]).
LECTURENOTESONTHEDYNAMICSOFTHEWEIL-PETERSSONFLOW 7
On the other hand, it is known that the stable and unstable manifolds of a
general Anosovflow(suchasgeodesicflowsoncompactmanifoldsofvariable
negativecurvature)donot formnecessarilyaC1-foliation,butonlyHöldercon-
tinuousfoliations(seee.g.thepapersofAnosov[1]and/orHasselblatt[28]for
concreteexamples).Inparticular,thisisanobstacletotheargumentàlaFubini
ofthepreviousparagraph.Nevertheless,Anosov[1]showedthatthestableand
unstable foliations of a smooth Anosov flow are always absolutely continuous,
so that one can still apply Fubini’s theorem to conclude ergodicity along the
linesofHopf’sargumentpresented.
In summary, we know that a smooth (C2) volume-preserving Anosov flow
onacompactmanifoldisergodicthankstoHopf’sargumentandtheabsolute
continuityofstableandunstablefoliations.
REMARK 1.3. Robinson-Young [51] showed that the stable and unstable folia-
tionsofaC1 Anosovsystemarenotnecessarilyabsolutelycontinuous. Inpar-
ticular,thesmoothness(C2)assumptionontheAnosovflowisnecessaryforthe
ergodicityargumentdescribedabove.
REMARK 1.4. The absolute continuity of a foliation invariant under some sys-
tem depends on some hyperbolicity. In fact,Shub-Wilkinson [55] constructed
examplesofinvariantcentral(alongwhichthedynamicsisneutral)foliationsof
certainpartiallyhyperbolicdiffeomorphismsfailingtosatisfyFubini’stheorem:
eachleafofthesecentralfoliationsintersectsasetoffullvolumeexactlyatone
point!ThisphenomenonissometimesreferredtoasFubini’snightmareinthe
literature(see,e.g.,thisarticleofMilnor[40])andsometimesafoliation“failing”
Fubini’stheoremiscalledapathologicalfoliation.
AfterthisbriefsketchofHopf’sargumentfortheergodicityofsmoothvolume-
preservingAnosovflowsoncompactmanifolds,letusexplainthedifficultiesof
extendingthisargumenttothesettingofWPflow.
1.3.2. Hopf’sargumentinthecontextofWPflow. Aswealreadymentioned(cf.
item(d)ofthetableabove),theWPflowissingularhyperbolic.Inanutshell,this
means that,even thoughWP flow is notAnosov,itis (morally) non-uniformly
hyperbolicinthesenseofPesintheoryanditsatisfiessomehyperbolicityesti-
matesalongpiecesoforbitsstayingincompactpartsofmodulispace.
Inparticular,thanksto(Katok-Strelcyn[33]versionof)Pesin’sstablemanifold
theorem[44],thestableandunstablesetsofalmosteverypointareimmersed
submanifolds,and,ifwe forgetaboutthe flowdirection,the stable andunsta-
ble manifolds have complementary dimensions. Furthermore, the stable and
unstable manifolds are part of absolutely continuous laminations. Here, it is
importantthatthedynamicsissufficientlysmooth(see,e.g.,thispaperofPugh
[47],andthispreprintofBonatti-Crovisier-Shinohara[7]).
Thus, this gives hopes that Hopf’s argument could be applied to show the
ergodicityofvolume-preservingnon-uniformlyhyperbolicsystems.
8 CARLOSMATHEUS
However,byinspectingthefigure1above,weseethatHopf’sargumentrelies
onthefactthatstableandunstablemanifoldsofAnosovflowshaveanice,well-
controlled,geometry.
Forinstance,ifwestartwithapointp andwewanttoconnectitwithpieces
of stable and unstable manifolds to a point q at a large distance, we have to
make sure that the pieces of stable and unstable manifolds used in figure 1
are“uniform”,e.g.,theyaregraphsofdefinitesizeandboundedcurvaturewith
respecttothesplittingintostableandunstabledirections,and,moreover,the
anglesbetweenthestableandunstabledirectionsareuniformlyboundedaway
fromzero.
Indeed,ifthepiecesofstableandunstablemanifoldsgetshorterandshorter,
and/orifthey“curve”alot,and/ortheanglesbetweenstableandunstabledi-
rectionsarenotboundedawayfromzero,onemightnotbeabletoreach/access
q fromp withstableandunstablemanifolds:
p
FIGURE 2. Pesinstableandunstablemanifoldswith“bad”geometry.
Asitturnsout,whilethesekindsofnon-uniformitydonotoccurforAnosov
flows, they can actually occur for certain non-uniformly hyperbolic systems.
Moreprecisely,thesizesandcurvaturesofstableandunstablemanifolds,and
the angles between stable andunstable directions ofa generalnon-uniformly
hyperbolicsystemvaryonlymeasurably frompointtopoint.
Inparticular,thisexcludesapriorianaivegeneralizationofHopf’sergodicity
argumentfornon-uniformlyhyperbolicsystems,and,infact,thereareconcrete
examples3 byDolgopyat-Hu-Pesin[5]ofvolume-preservingnon-uniformlyhy-
perbolicsystemswithcountablymanyergodiccomponentsconsistingofinvari-
antsetsofpositivevolumesthatareessentiallyopen.
Insummary,theergodicityofanon-uniformlyhyperbolicsystemdependson
theparticulardynamicalfeaturesofthegivensystem.
3As a matter of fact,these examples are “sharp”: Pugh-Shub [48] showed that a volume-
preservingnon-uniformlyhyperbolicsystemhasatmostcountablymanyergodiccomponents.
LECTURENOTESONTHEDYNAMICSOFTHEWEIL-PETERSSONFLOW 9
In this direction,there is an important literature dedicated to the construc-
tion of large classes of ergodic non-uniformly hyperbolic systems: for exam-
ple,theergodicityofseveralclassesofbilliards wasshownbySinai[56],Buni-
movich[11],Bunimovich-Chernov-Sinai[12]amongothers(seealsoChernov-
Markarian’sbook[18])andtheergodicityofnon-uniformlyhyperbolicsystems
exhibiting partial hyperbolicity (or dominated splitting) was shown by Pugh-
Shub[49],Rodriguez-Hertz[52],Tahzibi[57],Burns-Wilkinson[16],Rodriguez-
Hertz–Rodriguez-Hertz–Ures[53]amongothers.
FortheproofoftheirergodicityresultfortheWPflow,Burns-Masur-Wilkinson
takepartoftheirinspirationfromtheworkofKatok-Strelcyn[33]wherePesin’s
theory [44] (of existence and absolute continuity of stable manifolds) is ex-
tendedtosingularhyperbolicsystems.
Inanutshell,thebasicphilosophybehindKatok-Strelcyn’sworkisthefollow-
ing.Givenanon-uniformlyhyperbolicsystemwithsomenon-trivialsingularset,
alldynamicalfeaturespredictedbyPesintheoryinvirtueofthe(non-uniform)
exponential contraction and expansion are not affected if the loss of control
on the system is at most polynomial as one approaches the singular set. In
other terms,the exponential (hyperbolic) behavior of a singular system is not
disturbedbythepresenceofasingularsetwherethefirsttwoderivativesofthe
systemlosecontrolinapolynomial way.Inparticular,thishintsthatHopf’sar-
gumentcanbeextendedtosingularhyperbolicsystemswithpolynomiallybad
singularsets.
Inthiscontext,Burns-Masur-Wilkinsonshowsthefollowingergodicitycrite-
rionforsingularhyperbolicgeodesicflows(cf.Theorem3.1of[15]).
LetN bethequotientN =M/Γofacontractible,negativelycurved,possibly
incomplete,RiemannianmanifoldM byasubgroupΓofisometriesofM acting
freelyandproperlydiscontinuously.Byslightlyabusingnotation,wedenoteby
d themetricsonN andM inducedbytheRiemannianmetricofM.
We consider N the (Cauchy) metric completion of the metric space (N,d),
i.e.,the(complete)metricspaceconsistingofallequivalenceclassesofCauchy
sequences{x }⊂N undertherelation{x }∼{y }ifandonlyif lim d(x ,y )=0
n n n n→∞ n n
equippedwiththemetricd({x },{z })= lim d(x ,z ),andwedefinethe(Cauchy)
n n n→∞ n n
boundary ∂N :=N−N.
THEOREM1.5(Burns-Masur-Wilkinsonergodicitycriterionforgeodesicflows).
Let N =M/Γbeamanifoldasabove.Supposethat:
(I) the universal cover M of N is geodesically convex, i.e., for every p,q ∈M,
thereexistsanuniquegeodesicsegmentinM connecting p and q.
(II) themetriccompletionN of(N,d)iscompact.
(III) theboundary∂N isvolumetricallycusplike,i.e.,forsomeconstantsC >1
andν>0,thevolumeofaρ-neighborhoodoftheboundarysatisfies
Vol({x∈N :d(x,∂N)<ρ})≤Cρ2+ν
foreveryρ>0.
10 CARLOSMATHEUS
(IV) N has polynomially controlled curvature, i.e., there are constants C >1
andβ>0suchthatthecurvaturetensorR ofN anditsfirsttwoderivatives
satisfythefollowingpolynomialbound
max{(cid:107)R(x)(cid:107),(cid:107)∇R(x)(cid:107),(cid:107)∇2R(x)(cid:107)}≤Cd(x,∂N)−β
foreveryx∈N.
(V) N haspolynomiallycontrolledinjectivityradius,i.e.,thereareconstants
C >1andβ>0suchthat
inj(x)≥(1/C)d(x,∂N)β
foreveryx∈N (whereinj(x)denotestheinjectivityradiusat x).
(VI) Thefirstderivativeofthegeodesicflowϕ ispolynomiallycontrolled,i.e.,
t
thereareconstantsC >1andβ>0suchthat,foreveryinfinitegeodesicγ
on N andevery t∈[0,1]:
(cid:107)Dγ.(0)ϕt(cid:107)≤Cd(γ([−t,t]),∂N)β
Then, the Liouville (volume) measure m of N is finite, the geodesic flow ϕ
t
on the unit cotangent bundle T1N of N is defined at m-almost every point for
alltimet,andthegeodesicflowϕ isnon-uniformlyhyperbolic(inthesenseof
t
Pesin’stheory)andergodic.
Actually,thegeodesicflowϕ isBernoulliand,furthermore,itsmetricentropy
t
h(ϕ )ispositive,finiteandh(ϕ )isgivenbyPesin’sentropyformula(i.e.,h(ϕ )
t t t
isthesumofpositiveLyapunovexponentsofϕ countedwithmultiplicities).
t
Theproofofthisergodicitycriterionforgeodesicflowswasoneofthemain
motivationsofBurns’lectures(see[13])and,forthisreason,wewillnotdiscuss
it here. Instead, we will always assume Theorem 1.5 in the sequel, so that the
proof of Theorem 1.1 (ergodicity of the WP flow) will be complete4 once we
showthatthemodulispaceofRiemannsurfacesequippedwiththeWPmetric
satisfiesthesixitems(I)to(VI)above.
1.3.3. AbriefcommentontheverificationoftheergodicitycriterionforWPflow.
Incomparisonwithpreviouslyknownresultsintheliterature,someofthemain
noveltiesinBurns-Masur-Wilkinsonwork[15]concerntheverificationofitems
(IV) and (VI) for the WP metric: in fact, those items are the most delicate to
checkandtheirverificationsarestronglybasedonimportantpreviousworksof
McMullen[37]andWolpert[62],[63],[64],[66].
Inanycase,thiscompletesouroutlineoftheproofofBurns-Masur-Wilkinson
theoremontheergodicityofWPflow.
4Actually,thereisasubtlepointinthereductionofTheorem1.1toTheorem1.5relatedtothe
orbifoldicnatureofmodulispaces.WewilldiscussthislaterinSubsection2.8.