Linearization of generalized interval exchange maps

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by  Stefano Marmi

LINEARIZATION OF GENERALIZED INTERVAL EXCHANGE MAPS
STEFANOMARMI,PIERREMOUSSA,ANDJEAN-CHRISTOPHEYOCCOZ
2 ABSTRACT. Astandardintervalexchangemapisaone-to-onemapoftheintervalwhich
1 is locally a translation except at finitely many singularities. We define for such maps,
0 in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical
2 conditioncalledrestrictedRothtypewhichisalmostsurelysatisfiedinparameterspace.
n LetT0beastandardintervalexchangemapofrestrictedRothtype,andletrbeaninteger
a (cid:62)2.Weprovethat,amongstCr+3deformationsofT0whichareCr+3tangenttoT0at
J thesingularities,thosewhichareconjugatedtoT0byaCr diffeomorphismclosetothe
identityformaC1submanifoldofcodimension(g−1)(2r+1)+s.Here,gisthegenus
1
andsisthenumberofmarkedpointsofthetranslationsurfaceobtainedbysuspensionof
1
T0.BothgandscanbecomputedfromthecombinatoricsofT0.

]
S
D
TothememoryofGérardRauzy
.
h
t
a
m
CONTENTS
[

1. Introduction 2
3
1.1. Presentationofthemainresult 2
v
1 1.2. Openproblems 5
9 1.3. Summaryofthepaper 7
1 2. Background 8
1
2.1. Intervalexchangemaps 8
.
3 2.2. TheelementarystepoftheRauzy–Veechalgorithm 9
0 2.3. Rauzydiagrams 9
0
2.4. TheRauzy-Veechalgorithm 9
1
2.5. Suspensionandgenus 10
:
v 2.6. The(discretetime)Kontsevich-Zorichcocycle 11
i
X 3. Thecohomologicalequationrevisited 11
3.1. Theboundaryoperator 11
r
a 3.2. Continuityofthesolutionsofthecohomologicalequation 13
3.3. IntervalexchangemapsofRothType 14
3.4. Thecohomologicalequationinhighersmoothness 16
4. Aconjugacyinvariant 17
4.1. Definitionoftheinvariant 17
4.2. ConjugacyclassesinJr 17
4.3. Invarianceunderconjugacy 18
4.4. Invarianceunderrenormalization 18
Date:November22,2011.
2000MathematicsSubjectClassification. Primary:37C15(Topologicalanddifferentiableequivalence,con-
jugacy,invariants,moduli,classification);Secondary:37E05(mapsoftheinterval),37J40(Perturbations,normal
forms,smalldivisors,KAMtheory,Arnolddiffusion),11J70(Continuedfractionsandgeneralizations).
1
2 S.MARMI,P.MOUSSA,ANDJ.-C.YOCCOZ
4.5. Relationwiththeboundaryoperator 19
5. Themaintheorem: statementandreductiontothesimplecase 19
5.1. Thesetting 19
5.2. StatementoftheTheorem 21
5.3. Simplefamilies 21
6. Proof: Cr-conjugacy,r (cid:62)3 22
6.1. Smoothnessofthecompositionmap 22
6.2. Thecohomologicalequation 23
6.3. RelationbetweenadiffeomorphismanditsSchwarzianderivative 23
6.4. Thefixedpointtheorem 24
6.5. ConditionsforHtobeadiffeomorphism 25
6.6. EquationsfortheconjugacyclassofT 26
0
6.7. EndoftheproofofTheorem5.1forr (cid:62)3 27
7. Proof: C2-conjugacy 30
7.1. Smoothnessofthecompositionmap 30
7.2. Thecohomologicalequation 31
7.3. Relation between a diffeomorphism and the primitive of its Schwarzian
derivative 31
7.4. Thefixedpointtheorem 32
7.5. ConditionsforHtobeadiffeomorphism 32
7.6. EquationsfortheconjugacyclassofT 33
0
7.7. EndoftheproofofTheorem5.1forr =2 34
8. Simpledeformationsoflinearflowsontranslationsurfaces 37
8.1. Translationsurfaces 37
8.2. Theboundaryoperatorandtheconjugacyinvariant 38
8.3. Statementoftheresult 39
8.4. ProofoftheCorollary 40
AppendixA. ThecohomologicalequationwithC1+τ data 43
AppendixB. Thecaseofcirclediffeomorphisms 45
B.1. TheCr-case,r (cid:62)3 45
B.2. TheC2-case 46
AppendixC. Roth-typetranslationsurfaces 47
References 51
1. INTRODUCTION
1.1. Presentationofthemainresult. Manyproblemsofstabilityinthetheoryofdynam-
ical systems face the difficulty of small divisors. The most famous example is probably
givenbyKolmogorov-Arnold-Mosertheoryonthepersistenceofquasi-periodicsolutions
ofHamilton’sequationsforquasi-integrableHamiltoniansystems(bothfiniteandinfinite-
dimensional, like nonlinear wave equations). This is a very natural situation with many
applicationstophysicsandastronomy. Whatallthesedifferentproblemshaveincommon
isroughlyspeakingwhatfollows: onecanassociatesome“frequencies”totheorbitsun-
der investigation and some arithmetical condition is needed to prove their existence and
stability.
Thesimplestexampleofquasiperiodicdynamicsisgivenbyirrationalrotationsofthe
circle. Poincaré asked under which condition a given homeomorphism of the circle is
LINEARIZATIONOFGENERALIZEDINTERVALEXCHANGEMAPS 3
equivalent (in some sense, e.g. topologically or smoothly) to some rotation and proved
that any orientation-preserving homeomorphism of the circle with no periodic orbit is
semi-conjugate to an irrational rotation. Denjoy proved that, when the rotation number
is irrational, adding regularity to a given homeomorphism f (namely requiring f to be
piecewise C1 with Df of bounded variation) is enough to guarantee topological conju-
gacy to a rotation. The step to higher order differentiability for the conjugacy h requires
new techniques and additional hypotheses on the rotation number: a small divisor prob-
lemmustbeovercomeandthiswasfirstachieved(inthecirclecase)byArnoldin[A]:he
provedthatiftherotationnumberverifiesadiophantineconditionandiftheanalyticdif-
feomorphismf iscloseenoughtoarotation,thentheconjugationisanalytic. Atthesame
timeexamplesofanalyticdiffeomorphisms,withirrationalrotationnumber,forwhichthe
conjugation is not even absolutely continuous were given. Later Herman ([He1]) proved
a global result: there exists a full Lebesgue measure set of rotation numbers for which a
C∞ (resp. Cω)diffeomorphismisC∞ (resp. Cω)conjugatedtoarotation. Inthefinitely
differentiablecaseonecanproveasimilarresultbuttheconjugacyislessregularthanthe
diffeomorphism: this phenomenon of loss of differentiability is typical of small divisors
problems.
Thesuspensionofcirclerotationsproduceslinearflowsonthetwo-dimensionaltorus.
When analyzing the recurrence of rotations or the suspended flows, the modular group
GL(2,Z)isoffundamentalimportance,providingtherenormalizationschemeassociated
tothecontinuedfractionoftherotationnumber.
A generalization of the linear flows on the two-dimensional torus is obtained by con-
sidering linear flows on translation surfaces of higher genus (see e.g. [Zo1] for a nice
introductiontothesubject). ByaPoincarésectiontheirdynamicscanbereducedto(stan-
dard)intervalexchangemaps(i.e.m.),whichgeneralizerotationsofthecircle.
A (standard) i.e.m. T on an interval I (of finite length) is a one-to-one map which
islocallyatranslationexceptatafinitenumberofdiscontinuities. ThusT isorientation-
preservingandpreservesLebesguemeasure.ByaskingonlythatT islocallyanorientation-
preservinghomeomorphismoneobtainsthedefinitionofageneralizedi.e.m. Letdbethe
number of intervals of continuity of T. When d = 2, by identifying the endpoints of I,
standard i.e.m.’s correspond to rotations of the circle and generalized i.e.m.’s to homeo-
morphisms of the circle. Standard i.e.m. can be suspended following the construction of
Veech[Ve1]togiverisetotranslationsurfaces.
Typicalstandardi.e.m.’sareminimal([Kea1])butnotethatergodicpropertiesofmini-
malstandardi.e.m.’scandiffersubstantiallyfromthoseofcirclerotations:theyneednotbe
ergodic([Kea2],[KeyNew])butalmosteverystandardi.e.m.(bothinthetopologicalsense
[RK]andinthemeasure-theoreticalsense[Ma,Ve2])isergodic. Moreoverthetypicalnon
rotationalstandardi.e.m.isweaklymixing[AF].
Rauzy and Veech have defined an algorithm that generalizes the classical continued
fractionalgorithm(correspondingtothechoiced = 2)andassociatestoani.e.m.another
i.e.m. which is its first return map to an appropriate subinterval [Ra, Ve2]. Both Rauzy–
Veech"continuedfraction"algorithmanditsacceleratedversionduetoZorich[Zo2]are
ergodic w.r.t. an absolutely continous invariant measure in the space of normalized stan-
dardi.e.m.’s. HoweverinthecaseoftheRauzy–Veechalgorithmthemeasurehasinfinite
masswhereastheinvariantmeasurefortheZorichalgorithmhasfinitemass. Theergodic
propertiesoftheserenormalisationdynamicsinparameterspacehavebeenstudiedindetail
([Ve3],[Ve4],[Zo3],[Zo4],[AvGoYo],[B],[AB],[Y4]).
4 S.MARMI,P.MOUSSA,ANDJ.-C.YOCCOZ
Thepossiblecombinatorialdataforani.e.m.(standardorgeneralized)arethevertices
of Rauzy diagrams; the arrows of these diagrams correspond to the possible transitions
undertheRauzy-Veechalgorithm.
TheRauzy-Veechalgorithm, whichmakessenseforgeneralizedi.e.m.’s, stopsifand
onlyifthei.e.m.hasaconnection,i.e.afiniteorbitwhichstartsandendsatadiscontinuity.
Whenthei.e.m.hasnoconnection,thealgorithmassociatestoitaninfinitepathinaRauzy
diagramthatcanbeviewedasa“rotationnumber”.
Onecancharacterizetheinfinitepathsassociatedtostandardi.e.m.withnoconnections
(∞–completepaths,seesubsection2.3). Onesaysthatageneralizedi.e.m.T isirrational
ifitsassociatedpathis∞–complete;thenT issemi–conjugatedtoanystandardi.e.m.with
thesamerotationnumber[Y2].
ThisgeneralizationofPoincaré’stheoremsuggeststhefollowingverynaturalquestion:
what part of the theory of circle homeomorphisms and diffeomorphisms generalizes to
intervalexchangemaps?
Alltranslationsurfacesobtainedbysuspensionfromstandardi.e.m.withagivenRauzy
diagramhavethesamegenusg,andthesamenumbersofmarkedpoints;thesenumbers
arerelatedtothenumberdofintervalsofcontinuitybytheformulad=2g+s−1.
RegardingDenjoy’stheorem,partialresults([CG],[BHM],[MMY2])gointhenegative
direction,suggestingthattopologicalconjugacytoastandardi.e.m.haspositivecodimen-
sioningenusg ≥2.
Afirststepinthedirectionofextendingsmalldivisorresultsbeyondthetoruscasewas
achievedbyForni’simportantpaper([For1],seealso[For3])onthecohomologicalequa-
tionassociatedtolinearflowsonsurfacesofhighergenus. In[MMY1],weconsideredthe
cohomological equation ψ ◦T −ψ = ϕ for a standard i.e.m. T . We found explicitly
0 0
intermsoftheRauzy-Veechalgorithmafullmeasureclassofstandardi.e.m.(whichwe
calledRothtypei.e.m.) forwhichthecohomologicalequationhasboundedsolutionpro-
videdthatthedatumϕbelongstoafinitecodimensionsubspaceofthespaceoffunctions
having on each continuity interval a continuous derivative with bounded variation. The
improvedlossofregularity(w.r.t.[For1])willbedecisivefortheproofofourmainresult.
ThecohomologicalequationisthelinearizationoftheconjugacyequationT◦h=h◦T
0
forageneralizedi.e.m.T closetothestandardi.e.m.T .
0
We say that a generalized i.e.m. T is a simple deformation of class Cr of a standard
i.e.m.T if
0
• T andT havethesamediscontinuities;
0
• T andT coincideintheneighborhoodoftheendpointsofI andofeachdisconti-
0
nuity;
• T isaCr diffeomorphismoneachcontinuityintervalontoitsimage.
OurmainresultisalocalconjugacytheoremwhichisstatedinfullgeneralityinSection
5. Forsimpledeformationstheresultcanbesummarizedasfollows:
Theorem. For almost all standard i.e.m. T and for any integer r ≥ 2, amongst the
0
Cr+3simpledeformationsofT ,thosewhichareCr-conjugatetoT byadiffeomorphism
0 0
Cr closetotheidentityformaC1submanifoldofcodimensiond∗ =(g−1)(2r+1)+s.
The standard i.e.m. T considered in the theorem are the Roth type i.e.m. for which
0
theLyapunovexponentsoftheKZ-cocycle(seesubsection2.6)arenonzero(wecallthis
restrictedRothtype). TheystillformafullmeasuresetbyForni’stheorem[For2].
ThetangentspaceatT totheC1submanifoldofCr+3simpledeformationswhichare
0
Cr-conjugatetoT isformedofCr+3 functionsϕwhichvanishinaneighborhoodofthe
0
LINEARIZATIONOFGENERALIZEDINTERVALEXCHANGEMAPS 5
singularitiesofT andcanbewrittenas
0
ϕ=ψ◦T −ψ,
0
whereψisaCr functionvanishingatthesingularitiesofT .
0
Toextendthisresulttogeneralizedi.e.m.’sT ofclassCr whicharenotsimpledefor-
mations of a standard i.e.m. T , there are gluing problems of the derivatives of T at the
0
discontinuities. Indeedthereisaconjugacyinvariantwhichisanobstructiontolineariza-
tion(seeSection4).
AnearlierresultispresentedinanunpublishedmanuscriptofDeLaLlaveandGutierrez
[DG],whichwasrecentlycommunicatedtousbyP.Hubert. Theyconsiderstandardi.e.m.
withperiodicpathsfortheRauzy-Veechalgorithm(ford=2,thiscorrespondstorotations
byaquadraticirrational). Theyprovethat,amongstpiecewiseanalyticgeneralizedi.e.m.,
thebi-Lipschitzconjugacyclassofsuchastandardi.e.m.containsasubmanifoldoffinite
codimension. Theyalsoprovethatbi-LipschitzconjugacyimpliesC1-conjugacy.
TheproofofourtheoremisbasedonanadaptationofHerman’sSchwarzianderivative
trick. In [He2] Herman gave simple proofs of local conjugacy theorems for diffeomor-
phisms f of the circle: let ω denote the rotation number, assumed to satisfy a diophan-
tine condition |ω −p/q| ≥ γq−2−τ for some γ > 0, τ < 1, and let R be the corre-
ω
sponding rotation of the circle. Taking Schwarzian derivatives, the conjugacy equation
f ◦h = h◦R becomes(Sh)◦R −Sh = ((Sf)◦h)(Dh)2,alineardifferenceequa-
ω ω
tion in the Schwarzian derivative Sh of the conjugacy (but the r.h.s. depends also on h).
Givenadiffeomorphismh,onecomputesther.h.s.((Sf)◦h)(Dh)2,solvestheequation
ψ◦R −ψ =((Sf)◦h)(Dh)2andthenfindsadiffeomorphismh˜ =Φ(h)assmoothash
ω
withSh˜ =ψ.HermannowusestheSchauder-TychonovtheoremtofindafixedpointofΦ
andthustherequiredconjugacy. Hewasawareofthepossibilityofusingthecontraction
principle(atthecostofonemorederivativeforf)aswedoinourproof.Herman’smethod
ispresentedinmoredetailinAppendixB.1.
InSection8ofthepaper,weexplainhowtoadaptourresulttothesettingofperturba-
tions of linear flows on translation surfaces. Indeed we prove the following corollary of
themaintheorem(wereferthereadertoSection8andtoAppendixCforthedefinitionof
Roth-typetranslationsurfaceandofsimpledeformationofaverticalvectorfield):
Corollary. GivenatranslationsurfaceofrestrictedRothtypeandanyintegerr ≥ 2,
amongst the Cr+3 simple deformations of the vertical vectorfield, those which are Cr-
equivalent to it by a diffeomorphism Cr close to the identity form a C1 submanifold of
codimensiond∗ =(g−1)(2r+1)+s.
1.2. Openproblems.
1. Prove the theorem for r = 1: for almost all standard i.e.m. T , amongst the C4
0
simple deformations of T , those which are C1-conjugate to T by a diffeomorphism C1
0 0
closetotheidentityformaC1submanifoldofcodimensiond∗ =3g−3+s.
A rationale for this conjecture comes from the following argument. Note that d∗ is
equalhereto(d−1)+(g−1). Theintegerd−1isthedimensionofthespaceofstandard
i.e.m.uptoaffineconjugacy.InordertohaveaC1-conjugacybetweenageneralizedi.e.m.
T andastandardi.e.m.T withthesamerotationnumber,anecessaryconditionisthatthe
0
BirkhoffsumsofLogDT (equaltoLogDTn)arebounded. TheintegralofLogDT w.r.t.
theuniqueinvariantmeasureisautomaticallyzero, takingcareofthelargestexponentof
6 S.MARMI,P.MOUSSA,ANDJ.-C.YOCCOZ
theKZ-cocycle;killingthecomponentsw.r.t.theremainingg−1positiveexponentsleads
totheexpectedvalueofd∗.
Ontheotherhand,whenthederivativesoftheiteratesDTn areallowedtogrowexpo-
nentiallyfast,onecouldexpecttohavewanderingintervals(see[MMY2]).
ThissuggeststheexistenceofadichotomybetweenbeingC1-conjugatedtoastandard
i.e.m.andhavingwanderingintervals: onecanthereforeaskwhetherthefollowingistrue:
2. For almost all standard T , any generalized i.e.m. T of class C4 which is a simple
0
deformationofT andistopologicallyconjugatedtoT isalsoC1-conjugatedtoT .
0 0 0
The two conjectures above can be formulated in a slightly more general setting (not
restrictedtosimpledeformations)usingtheconjugacyinvariantintroducedinSection4.
3. ThelocalCr conjugacyclassofastandardi.e.m.T (ofrestrictedRothtype)exhib-
0
ited by our theorem can be considered as a local stable manifold for the renormalization
operatorRdefinedbytheRauzy-Veechinduction(withrescaling)ongeneralizedi.e.m.’s
inasuitablefunctionalspace.Bythestandardtechniquesthislocalstablemanifoldextends
toaglobalstablemanifold
Ws(T )=∪ R−n(Ws (RnT )
0 n≥0 loc 0
whichisthefullCr conjugacyclassofT .
0
Isthisstablemanifold“properlyembedded”inparameterspace?
Moreprecisely,givenasequenceofdiffeomorphismsh inDiffr(I)suchthath →∞,
n n
Is it possible that h ◦ T ◦ h−1 → T in the Cr+3 topology? Is it possible that
n 0 n 0
h ◦T ◦h−1staysboundedintheCr+3topology?
n 0 n
Inthecased=2,theanswertobothquestionsisno. Forthesecondquestion,thisisa
consequenceofHerman’sglobalconjugacytheoremforcirclediffeomorphisms.
4. DescribethesetofgeneralizedCr intervalexchangemapswhicharesemi-conjugate
toagivenstandardi.e.m.T (withnoconnections).
0
In the circle case, for a diophantine rotation number, one has a C∞ submanifold of
codimension1. IntheLiouvillecaseonehasstillatopologicalmanifoldofcodimension1
whichistransversetoall1-parameterstrictlyincreasingfamilies. Onecanthereforedare
toask:
(1) Istheabovesetatopologicalsubmanifoldofcodimensiond−1?
(2) if the answer is positive, does there exist a (smooth) field of "transversal" sub-
spacesofdimensiond−1?
ThequestionsmakesenseforanyT ,buttheanswercoulddependonthediophantine
0
propertiesofT .
0
5. In a generic smooth family of generalized i.e.m.’s, is the rotation number irrational
withpositiveprobability?
In the circle case the answer is affirmative, thanks to Herman’s theorem. This is not
verylikelyinhighergenus.
LINEARIZATIONOFGENERALIZEDINTERVALEXCHANGEMAPS 7
6. Letr ≥ 1. Describeexactly(intermsoftheRauzy-Veechrenormalizationalgorithm)
thesetofrotationnumberssuchthattheCr conjugacyclassofT hasfinitecodimension
0
inthespaceofC∞generalizedi.e.m.’s. Doesthissetdependsonr?
In the circle case, this set is (for any r (cid:62) 1) the set of diophantine rotation numbers
([Y3],[He1]). In higher genus, our theorem (in the stronger form stated in Section 5)
guarantees that this set contains the restricted Roth type rotation numbers and therefore
hasfullmeasure. ItlookslikethatourmethodsextendtoprovethattheunrestrictedRoth
type rotation numbers also belong to this set (but the codimension of the Cr conjugacy
classofT isdifferent). OfcoursethecodimensionoftheCr conjugacyclasswilldepend
0
onrbutthepointhereisthatweonlyrequirethecodimensiontobefinite.
Notethattheanswerisnotknownevenatthelevelofthecohomologicalequation!
Arelatedquestionistheoptimallossofdifferentiability,forinstanceforrestrictedRoth
typerotationnumbers. Acarefulreadingoftheproof(andofAppendixA)willconvince
thereaderthatwemayconsiderCr+2+τ (foranyτ >0)simpledeformationsofT instead
0
of Cr+3 simple deformations and still get the same conclusion. On the other hand, the
cohomological equation suggests that some form of the result could be true for Cr+1+τ
simple deformations of T (for any τ > 0). This is certainly true in genus 1. This is
0
howeverbeyondthereachofourmethod.
1.3. Summaryofthepaper. Inthenextsectionweintroducestandardandgeneralizedin-
tervalexchangemaps. WerecallthedefinitionandthemainpropertiesoftheRauzy-Veech
continuedfractionalgorithm, andexplainhowitallowstodefineinaverynaturalwaya
"rotation number" for certain generalized i.e.m.’s. The algorithm generates a dynamical
systeminparameterspace,equippedwithaveryimportantcocycle,theKontsevich-Zorich
cocycle. The notations and the presentation of this section follow closely the expository
paper[Y1](seealso[Y2],[Y4]).
Section3isdevotedtothestudyofthecohomologicalequation. Weintroduceabound-
aryoperatoronthespaceofpiecewise-continuousfunctionswhichvanishesoncobound-
aries and take care of the neutral component of the KZ-cocycle. We review the results
of[MMY1](Theorem3.10), recallinginparticularthedefinitionofRothtypei.e.m. We
actuallyimproveontheresultsof[MMY1]byshowingthatunderthesameassumptions
onecanobtainacontinuous(insteadofbounded)solution. Wealsoreformulatetheresults
inhighersmoothnessusingtheboundaryoperator.
In Section 4 we introduce, for any integer r (cid:62) 1, an invariant for Cr conjugacy with
valuesintheconjugacyclassesofthegroupJr ofr-jetsoforientation-preservingdiffeo-
morphisms of (R,0). We show that it is also preserved by the renormalization operator
definedbytheRauzy-Veechalgorithm. Weexplaintherelationofthisconjugacyinvariant
withtheboundaryoperator.
Section5containsthepreciseformulationofourmainresult(Theorem5.1):C1param-
eterfamiliesofgeneralizedi.e.m.’sofclassCr+3throughastandardi.e.m.T ofrestricted
0
Rothtypeareconsidered. ItisassumedthattheCr+3-conjugacyinvariantvanishesandan
appropriate transversality hypothesis (related to the cohomological equation) is satisfied.
ThetheoremthenstatesthatthelocalCr-conjugacyclassofT intersectsthefamilyalong
0
asubmanifoldwhosetangentspaceatT isgivenbythecohomologicalequation. Wealso
0
showhowthehypothesisontheconjugacyinvariantallowstoreducetheproofstothecase
ofsimplefamilies.
Section 6 contains the proof of Theorem 5.1 when r (cid:62) 3; following Herman, we use
Schwartzianderivativestoconstructamapwhosefixedpointisacandidatefortheconju-
gatingmap.Inthecirclecase,thisfixedpointisalwaystheconjugatingmap.Inthepresent
8 S.MARMI,P.MOUSSA,ANDJ.-C.YOCCOZ
case,someextraequationsrepresentinggluingconditionshavetobesatisfied;theseequa-
tionsdefinethelocalconjugacyclassinparameterspace.
Section7dealswiththecaser =2ofTheorem5.1. Indeed,aC2-diffeomorphismdoes
not have in general a Schwartzian derivative. We need a little improvement of Herman’s
Schwarzian derivative trick. We show how one can effectively use the primitive of the
Schwarzianderivativetoconstructacontractingmapwhosefixedpointwillturnouttobe
theconjugacy,underappropriategluingconditions.
In Section 8 we explain how to adapt our result to the simple deformations of linear
flowsontranslationsurfaces.Afterabriefintroductiontotranslationsurfaceswestudythe
actionoftheboundaryoperatoratthelevelofthesurfaceandweprovethattheconjugacy
invariant is trivial for simple deformations of the vertical vectorfield. We then introduce
restricted Roth type translation surfaces and we prove the Corollary stated at the end of
subsection1.1.
In Appendix A we show that the main result of [MMY1] (in the improved version of
Theorem3.10)isalsovalidwithdatawhosefirstderivativesareHöldercontinuousinstead
ofhavingboundedvariation.
Appendix B is devoted to the case of circle diffeomorphisms. In subsection B.1, we
deal with Cr-conjugacy, r (cid:62) 3. Herman’s original result (through Schauder-Tychonov
fixedpointtheorem)givesastrongerconclusioninthissetting;however,thesimplevariant
basedonthefixedpointtheoremforcontractingmapsisabetterpreparationforthemore
difficult case of Section 6. In the same way, subsection B.2 introduces the main idea of
Section7inasimplersetting.
FinallyAppendixCisdevotedtothestudyofRoth-typetranslationsurfaces. Proposi-
tionC.1givesseveralequivalentformulationsofcondition(a)inthedefinitionofaRoth-
typei.e.m.(seesubsection3.3).Thisisthenusedinordertoprovethatthei.e.m.’sobtained
asfirstreturnmapsonanopenboundedsegment(ingoodposition)oftheverticalflowon
a(restricted)Roth-typetranslationsurfaceareof(restricted)Roth-type.
Acknowledgements This research has been supported by the following institutions: the
CollègedeFrance,theScuolaNormaleSuperiore,theFrenchANR(grants0863Petitsdi-
viseursetrésonancesengéomtrie,EDPetdynamiqueand0864Dynamiquedansl’espace
de Teichmüller) andthe Italian MURST(PRIN grant 2007B3RBEY Dynamical Systems
and applications). We are also grateful to the two former institutions, to the Centro di
RicercaMatematica“EnnioDeGiorgi”inPisaandtotheMaxPlanckInstitutefürMathe-
matikinBonnforhospitality.Wearegratefultotherefereeforhissuggestionsandremarks
whichledtoaconsiderableimprovementofourpaper.
2. BACKGROUND
2.1. Intervalexchangemaps. LetI beanopenboundedinterval. Ageneralizedinterval
exchange map (g.i.e.m. ) T on I is defined by the following data. Let A be an alphabet
withd (cid:62) 2symbols. Considertwopartitionsmod.0ofI intodopensubintervalsindexed
byA(thetopandbottompartitions):
I =(cid:116)It =(cid:116)Ib .
α α
The map T is defined on (cid:116)It and its restriction to each It is an orientation-preserving
α α
homeomorphismontothecorrespondingIb.
α
LINEARIZATIONOFGENERALIZEDINTERVALEXCHANGEMAPS 9
Theg.i.e.m.T isstandardif|It|=|Ib|foreachα∈AandtherestrictionofT toeach
α α
It isatranslation.
α
Let r be an integer (cid:62) 1 or ∞. The g.i.e.m. T is of class Cr if the restriction of T to
eachIt extendstoaCr-diffeomorphismfromtheclosureofIt ontotheclosureofIb.For
α α α
finiter,itiseasytoseethattheg.i.e.m.’swithfixedAformaBanachmanifold.
Thepointsut < ··· < ut separatingtheIt arecalledthesingularities ofT. The
1 d−1 α
points ub < ··· < ub separating the Ib are called the singularities of T−1. We also
1 d−1 α
writeI =(u ,u ),ut =ub =u ,ut =ub =u .
0 d 0 0 0 d d d
ThecombinatorialdataofT isthepairπ =(π ,π )ofbijectionsfromAonto{1,...,d}
t b
suchthat
It =(ut ,ut ), Ib =(ub ,ub )
α πt(α)−1 πt(α) α πb(α)−1 πb(α)
foreachα∈A.
Wealwaysassumethatthecombinatorialdataareirreducible: for1(cid:54)k <d,wehave
π−1({1,...,k})(cid:54)=π−1({1,...,k}).
t b
2.2. TheelementarystepoftheRauzy–Veechalgorithm. Aconnectionisatriple(ut,ub,m),
i j
wheremisanonnegativeinteger,suchthat
Tm(ub)=ut .
j i
Keanehasproved[Kea1]thatastandardi.e.m.withnoconnectionisminimal.
Let T be a g.i.e.m. with no connection. We have then ut (cid:54)= ub . Set u :=
d−1 d−1 (cid:98)d
max(utd−1,ubd−1), I(cid:98) := (u0,u(cid:98)d), and denote by T(cid:98) the first return map of T in I(cid:98). The
returntimeis1or2.
OnechecksthatT(cid:98)isag.i.e.m.onI(cid:98)whosecombinatorialdataπ(cid:98)arecanonicallylabeled
bythesamealphabetAthanπ (cf.[MMY1]p.829). MoreoverT(cid:98)hasnoconnection; this
allowstoiteratethealgorithm.
WesaythatT(cid:98)isdeducedfromT byanelementarystepoftheRauzy–Veechalgorithm.
Wesaythatthestepisoftop(resp. bottom)typeifut < ub (resp. ut > ub ).
d−1 d−1 d−1 d−1
Onethenwritesπ =R (π)(resp. π =R (π)).
(cid:98) t (cid:98) b
2.3. Rauzydiagrams. ARauzyclassonthealphabetAisanonemptysetofirreducible
combinatorialdatawhichisinvariantunderR ,R andminimalwithrespecttothisprop-
t b
erty. A Rauzy diagram is a graph whose vertices are the elements of a Rauzy class and
whosearrowsconnectavertexπ toitsimagesR (π)andR (π). Eachvertexistherefore
t b
theoriginoftwoarrows. AsR ,R areinvertible,eachvertexisalsotheendpointoftwo
t b
arrows.
An arrow connecting π to R (π) (respectively R (π)) is said to be of top type (resp.
t b
bottomtype). Thewinnerofanarrowoftop(resp.bottom)typestartingatπ = (π ,π )
t b
withπ (α )=π (α )=distheletterα (resp.α )whiletheloserisα (resp.α ).
t t b b t b b t
ApathγinaRauzydiagramiscompleteifeachletterinAisthewinnerofatleastone
arrowinγ;itisk–completeifγ istheconcatenationofkcompletepaths. Aninfinitepath
is∞–completeifitistheconcatenationofinfinitelymanycompletepaths.
2.4. The Rauzy-Veech algorithm. Let T = T(0) be an i.e.m. with no connection. We
denote by A the alphabet for the combinatorial data π(0) of T(0) and by D the Rauzy
diagramonAhavingπ(0)asavertex.
Thei.e.m.T(1),withcombinatorialdataπ(1),deducedfromT(0)bytheelementarystep
of the Rauzy–Veech algorithm has also no connection. It is therefore possible to iterate
this elementary step indefinitely and get a sequence T(n) of i.e.m. with combinatorial
10 S.MARMI,P.MOUSSA,ANDJ.-C.YOCCOZ
dataπ(n),actingonadecreasingsequenceI(n)ofintervalsandasequenceγ(n,n+1)of
arrowsinDfromπ(n) toπ(n+1) associatedtothesuccessivestepsofthealgorithm. For
m < n,wealsowriteγ(m,n)forthepathfromπ(m) toπ(n) madeoftheconcatenation
oftheγ(l,l+1),m(cid:54)l<n.
Wewriteγ(T)fortheinfinitepathstartingfromπ(0)formedbytheγ(n,n+1),n(cid:62)0.
IfT isastandardi.e.m.withnoconnection,thenγ(T)is∞-complete([MMY1],p.832).
Conversely, an ∞-complete path is equal to γ(T) for some standard i.e.m. with no con-
nection. Ontheotherhand,forageneralizedi.e.m.T withnoconnection,thepathγ(T)is
notalways∞-complete.
Definition 2.1. A generalized i.e.m. T is irrational if it has no connection and γ(T) is
∞-complete. Wethencallγ(T)therotationnumberofT.
Inthecirclecased=2,theRauzydiagramhasonevertexandtwoarrows. Iftherota-
tionnumberofacirclehomeomorphismT hasacontinuedfractionexpansion[a ,a ,...],
1 2
the associated ∞-complete path takes a times the first arrow, then a times the second
1 2
arrow,a timesthefirstarrow,....
3
From the definition, a standard i.e.m. is irrational iff it has no connection. Two stan-
dardi.e.m.withnoconnectionaretopologicallyconjugatedifftheyhavethesamerotation
number[Y2]. Moregenerally,ifT isanirrationalg.i.e.m.withthesamerotationnumber
thanastandardi.e.m.T ,thenthereis,asinthecirclecase,asemiconjugacyfromT toT ,
0 0
i.eacontinuousnondecreasingsurjectivemaphfromtheintervalI ofT ontotheinterval
I ofT suchthatT ◦h=h◦T (cf.[Y2]).
0 0 0
2.5. Suspension and genus. Let T be a standard i.e.m. with combinatorial data π =
(π ,π ). Forα∈Alet
t b
λ =|It|=|Ib|, τ =π (α)−π (α), ζ =λ +iτ .
α α α α b t α α α
Inthecomplexplane,drawatop(resp. bottom)polygonallinefromu tou through
0 d
u +ζ ,u +ζ +ζ ,...(resp. u +ζ ,u +ζ +ζ ,...).
0 π−1(1) 0 π−1(1) π−1(2) 0 π−1(1) 0 π−1(1) π−1(2)
t t t b b b
These two polygonal lines bound a polygon. Gluing the ζ bottom and top sides of the
α
polygonproducesatranslationsurfaceM ([Zo]). Theverticesofthepolygonformaset
T
ofmarkedpointsΣonM . ThecardinalitysofΣ,thegenusg ofM andthenumberd
T T
ofintervalsarerelatedby
d=2g+s−1.
Thegenusg canbecomputeddirectlyfromthecombinatorialdataasfollows. Definean
antisymmetricmatrixΩ=Ω(π)by

+1 ifπ (α)<π (β),π (α)>π (β),
 t t b b
Ω = −1 ifπ (α)>π (β),π (α)<π (β),
αβ t t b b
 0 otherwise.
Then the rank of Ω is 2g. Actually ([Y1],[Y4]), if one identifies RA with the relative
homologygroupH (M ,Σ,R)viathebasisdefinedbythesidesζ ofthepolygon, the
1 T α
imageofΩcoincideswiththeabsolutehomologygroupH (M ,R).Anotherwaytocom-
1 T
putes(andthusg)consistsingoingaroundthemarkedpoints,asexplainedinsubsection
3.1.