Cocycles over interval exchange transformations and multivalued Hamiltonian flows

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multivalued Hamiltonian flows

Jean-Pierre Conze, Krzysztof Fraczek

To cite this version:

Jean-Pierre Conze, Krzysztof Fraczek. Cocycles over interval exchange transformations and mul- tivalued Hamiltonian flows. Advances in Mathematics, Elsevier, 2011, 226 (5), pp.4373-4428.

�10.1016/j.aim.2010.11.014�. �hal-00462646�

FLOWS

JEAN-PIERRECONZEANDKRZYSZTOFFRCZEK

Abstrat. Weonsider interval exhange transformations of periodi type

andonstrutdierentlassesofreurrentergodioylesofdimension≥1

over thisspeiallassof IETs. Thenusing Poinaré setions we applythis

onstrution to obtain reurrene and ergodiityfor somesmooth ows on

non-ompatmanifoldswhihareextensionsofmultivaluedHamiltonianows

onompatsurfaes.

Contents

1. Introdution 2

2. Preliminaries 3

2.1. Intervalexhangetransformations 3

2.2. IETsofperioditype 4

2.3. GrowthofBVoyles 5

2.4. Reurrene,essentialvalues,andergodiityofoyles 6

3. Ergodiityofpieewiselinearoyles 8

3.1. Pieewiselinearoyles 9

3.2. Produtoyles 10

4. Ergodiityofertainstepoyles 15

4.1. Stepoyles 15

4.2. Ergodioylesinaseκ >1 ¹⁷

5. Ergodiityoforretedoyles 18

5.1. Rauzy-Veehindution foroyles 18

5.2. Corretionoffuntions ofbounded variation 19

5.3. Ergodiityoforretedstepfuntions 20

6. ReurreneandergodiityofextensionsofmultivaluedHamiltonians 22

6.1. Speialows 22

6.2. BasipropertiesofmultivaluedHamiltonianows 22

6.3. ExtensionsofmultivaluedHamiltonianows 24

7. ExamplesofergodiextensionsofmultivaluedHamiltonianows 30

7.1. ConstrutionofmultivaluedHamiltonians 30

7.2. Examples 33

AppendixA. Deviationofoyles: proofs 35

2000MathematisSubjetClassiation. 37A40,37C40.

Keywordsand phrases. intervalexhangetransformation,oyle,multivaluedHamiltonian

ow,inniteinvariantmeasure,ergodiity.

ResearhpartiallysupportedbyMNiSzWgrantNN201384834andMarieCurie"Transferof

Knowledge"program,projetMTKD-CT-2005-030042(TODEQ).

AppendixB. Possiblevaluesofθ2/θ1 ³⁶

AppendixC. Deviation oforretedfuntions 37

AppendixD. Exampleofnon-regularstepoyle 43

Referenes 44

1. Introdution

LetT : (X,B, µ)→(X,B, µ)^{bean ergodi}automorphism ofastandardBorel probabilityspaeandG^{bealoally ompatabeliangroupwithidentityelement}

denotedby0^{. Wewillonsider} essentiallytheaseG=R^ℓ,forℓ≥1^.

Eah measurable funtion ϕ: X → G^{determines aoyle}ϕ^(·) : Z×X →G

forT ^bytheformula ϕ⁽ⁿ⁾(x) =





ϕ(x) +ϕ(T x) +. . .+ϕ(Tⁿ⁻¹x), ^if n >0

0, ^if n= 0,

−(ϕ(Tⁿx) +ϕ(Tⁿ⁺¹x) +. . .+ϕ(T⁻¹x)), ^if n <0.

Weonsidertheassoiatedskewprodut

Tϕ: (X×G,B × B^G, µ×mG) → (X×G,B × B^G, µ×mG), Tϕ(x, g) = (T x, g+ϕ(x)),

(1.1)

whereB^{G denotesthe}σ^{-algebraofBorelsubsetsand}mG ^{theHaarmeasureof}G^.

Theoyle(ϕ^(·))^{anbeviewedasa}"stationary"walkinG^{overthedynamial}

system(X, µ, T)^{. Wesaythatitisreurrentif}(ϕ⁽ⁿ⁾(x))^{returnsfora.e.} x^innitely

oftenin anyneighborhoodoftheidentityelement. ThetransformationTϕ ^isthen

onservative for the invariant σ^{-nite measure} µ×mG^{. If moreoverthe system}

(X×G, µ×mG, Tϕ)^{isergodi,wesaythattheoyle}ϕ^{(·)isergodi. Forsimpliity,}

the expression "oyle ϕ^{" refers to the oyle} (ϕ^(·)) ^{generated by} ϕ ^{over the}

dynamialsystem(X,B, µ, T)^.

Aproblemistheonstrutionofreurrentergodioylesdened overagiven

dynamialsystembyregularfuntionsϕ^withvaluesin R^{ℓ. Thereisanimportant}

literature on skew produts overan irrational rotation on the irle, and several

lassesofergodioyleswithvaluesinR^orR^{ℓareknowninthatase(see[23℄,[25℄}

and[26℄forsomelassesofergodipieewiseabsolutelyontinuousnon-ontinuous

R^{-oyles, [16℄for examplesof ergodioyleswithvaluesin anilpotent group,}

[7℄for ergodi oyles in Z^{2 assoiated to speial diretional retangularbilliard}

owsintheplane).

Skew produtsappearin anaturalway in thestudy of thebilliardowin the

plane withZ² periodiallydistributed obstales. Forinstane whenthe obstales areretangles,theyanbemodeledasskewprodutsoverintervalexhangetrans-

formations(abbreviated asIETs). Reurreneand ergodiityof these models are

mainly open questions. Nevertheless a rst step is the onstrution of reurrent

ergodi oylesoversome lasses of IETs (see also areent paper byP. Hubert

For the rotations on the irle, a speial lass onsists in the rotations with

boundedpartialquotients. ForIETs,itisnaturaltoonsidertheso-alledinterval

exhange transformationsof periodi type. Theaim of this paper isto onstrut

dierentlassesof reurrentergodioylesoverIETsin thisspeial lass.

This is donein Setions 3, 4, and 5. In Setion 2we reall basi fats about

IETs of periodi type, as well as from the ergoditheory of oyles. Inthe ap-

pendixproofsoftheneededresultsonthegrowthofoylesofboundedvariation

(abbreviatedasBVoyles)aregiven,mainlyadapted from[24℄.

In Setions 6 and 7 we present smooth models for reurrent and ergodi sys-

tems based on the previous setions. We deal with a lass of smooth ows on

non-ompatmanifolds whih areextensions ofmultivaluedHamiltonian owson

ompat surfaes of higher genus. These ows have Poinaré setions for whih

therstreurrenemap isisomorphito askew produtof anIET andaBVo-

yle. This allowsus to proveasuientonditionfor reurreneand ergodiity

(seeSetion6)whenevertheIETisofperioditype. InSetion7weshowhowto

onstrutexpliitnon-ompatergodiextensionsofsomeHamiltonianows.

2. Preliminaries

2.1. Interval exhange transformations.

In this subsetion, we reall standard fats on IET's, with the presentation and

notationsfrom [32℄ and [33℄. Let A^bead^{-elementalphabet andlet}π = (π0, π1)

beapairofbijetionsπε:A → {1, . . . , d}^forε= 0,1^{. Denoteby}SA^{0 thesubsetof}

irreduiblepairs, i.e.suhthat π1◦π₀⁻¹{1, . . . , k} 6={1, . . . , k} ^for1 ≤k < d^{. We}

willdenotebyπ^sym_d ^anypair(π0, π1)^suhthatπ1◦π₀⁻¹(j) =d+ 1−j^for1≤j≤d^.

Letusonsiderλ= (λα)α∈A∈R^A₊^,whereR₊= (0,+∞)^{. Set}

|λ|= X

α∈A

λα, I= [0,|λ|)

and

Iα= [lα, rα), ^where lα= X

π0(β)<π0(α)

λβ, rα= X

π0(β)≤π0(α)

λβ.

Then|Iα|=λα^{. Denote by}Ωπ ^thematrix[Ωα β]α,β∈A ^givenby

Ωα β=





+1 ^ifπ1(α)> π1(β)^andπ0(α)< π0(β),

−1 ^ifπ1(α)< π1(β)^andπ0(α)> π0(β), 0 ^{in allotherases.}

Given(π, λ)∈ SA⁰×R^A₊^,letT(π,λ): [0,|λ|)→[0,|λ|)^{standfortheintervalexhange}

transformation(IET)ond^intervalsIα^, α∈ A^{, whih arerearrangedaordingto}

thepermutationπ⁻¹₁ ◦π0^,i.e.T(π,λ)x=x+wα^forx∈Iα^,wherew= Ωπλ^.

Notethatforeveryα∈ A^withπ0(α)6= 1^thereexistsβ∈ A^suhthatπ0(β)6=d

andlα=rβ^{. Itfollowsthat}

(2.1) {lα:α∈ A, π0(α)6= 1}={rα:α∈ A, π0(α)6=d}.

By Tb(π,λ) : (0,|I|] → (0,|I|] ^{denote the exhange of the intervals} Ibα = (lα, rα]^, α∈ A^,i.e.T(π,λ)x=x+wα^forx∈Ibα^{. Notethatforevery}α∈ A^withπ1(α)6= 1

thereexistsβ ∈ A^{suh that}π1(β)6=d^andT(π,λ)lα=Tb(π,λ)rβ^{. Itfollowsthat}

(2.2) {T(π,λ)lα:α∈ A, π1(α)6= 1}={Tb(π,λ)rα:α∈ A, π1(α)6=d}.

Apair(π, λ)^{satises theKeane onditionif}T_(π,λ)^m lα6=lβ ^forallm≥1 ^andfor

allα, β∈ A^withπ0(β)6= 1^.

LetT =T(π,λ)^,(π, λ)∈ SA⁰ ×R^A₊^{,beanIETsatisfyingKeane'sondition. Then} λ_π⁻¹

0 (d)6=λ_π⁻¹

1 (d)^{. Let}

I˜=h

0,max l_π⁻¹

0 (d), l_π⁻¹

1 (d)

anddenotebyR(T) = ˜T : ˜I→I˜^{therstreturnmapof}T ^{to theinterval}I˜^{. Set} ε(π, λ) =

( 0 ^if λ_π−1

0 (d)> λ_π−1 1 (d), 1 ^if λ_π⁻¹

0 (d)< λ_π⁻¹

1 (d).

Letusonsiderapairπ˜ = (˜π0,π˜1)∈ SA^0,where

˜

πε(α) = πε(α)^forallα∈ A^and

˜

π1−ε(α) =





π1−ε(α) ^if π1−ε(α)≤π1−ε◦π⁻¹_ε (d), π1−ε(α) + 1 ^if π1−ε◦π⁻¹_ε (d)< π1−ε(α)< d, π1−επ_ε⁻¹(d) + 1 ^if π1−ε(α) =d.

AsitwasshownbyRauzyin [27℄,T˜ ^{isalsoanIET on}d^-intervals T˜=T_(˜π,λ)˜ ^with˜λ= Θ⁻¹(π, λ)λ,

where

Θ(T) = Θ(π, λ) =I+E_π⁻¹_{ε (d)π}⁻¹

1−ε(d)∈SL(Z^A).

Moreover,

(2.3) Θ^t(π, λ)ΩπΘ(π, λ) = Ωπ˜.

It followsthat ker Ωπ = Θ(π, λ) ker Ωπ˜^{. Wehavealso}Ω^t_π =−Ωπ^{. Thus taking}

Hπ = Ωπ(R^A) = ker Ω^⊥_π^{, we get} Hπ˜ = Θ^t(π, λ)Hπ^{. Moreover,}dimHπ = 2g ^and dim ker Ωπ=κ−1^,whereκ^{isthenumberof}singularitiesandg^{isthegenusofthe}

translationsurfaeassoiatedtoπ^{. Formoredetails wereferthereaderto[33℄.}

The IET T˜ ^{fullls the Keane ondition as well. Therefore we aniterate the}

renormalizationproedureandgenerateasequeneofIETs(T⁽ⁿ⁾)n≥0^,whereT⁽ⁿ⁾= Rⁿ(T) ^for n ≥ 0^{. Denote by} π⁽ⁿ⁾ = (π₀⁽ⁿ⁾, π⁽ⁿ⁾₁ ) ∈ SA^{0 the pair and by} λ⁽ⁿ⁾ = (λ⁽ⁿ⁾α )α∈A ^{thevetorwhih determines}T^{(n). Then} T^{(n)is therstreturnmap of} T ^{totheinterval}I⁽ⁿ⁾= [0,|λ⁽ⁿ⁾|)^and

λ= Θ⁽ⁿ⁾(T)λ^(n)withΘ⁽ⁿ⁾(T) = Θ(T)·Θ(T⁽¹⁾)·. . .·Θ(T⁽ⁿ⁻¹⁾).

2.2. IETs ofperiodi type.

Denition (see [29℄). An IET T ^{is of periodi type if there exists} p > 0 ^{(alled a}

period of T^{) suh that} Θ(T^(n+p)) = Θ(T⁽ⁿ⁾)^{for every}n ≥0 ^and Θ^(p)(T) ^(alled

aperiodi matrix of T ^{anddenoted by} A^{in allthat follows) has stritly positive}

Remark 2.1. SupposethatT =T(π,λ) ^{isofperioditype. It followsthat}

λ= Θ^(pn)(T)λ^(pn)= Θ^(p)(T)ⁿλ^(pn)∈Θ^(p)(T)ⁿR^A,

andheneλ^belongsto T

n≥0Θ^(p)(T)ⁿR^{A whihis a}one-dimensionalonvexone (see[30℄). Thereforeλ^{isapositiveright}Perron-Frobeniuseigenvetorofthematrix

Θ^(p)(T)^{. Sine the set} SA^{0 is nite,} multiplyingthe period p ^{ifneessary, we an}

assume that π^(p) = π^{. It follows that} (π^(p), λ^(p)/|λ^(p)|) = (π, λ/|λ|) ^and ρ :=

|λ|/|λ^(p)| ^{is the} Perron-Frobenius eigenvetor of the matrix Θ^(p)(T)^{. Reall that}

similarargumentstothoseaboveshowthat everyIETofperioditypeisuniquely

ergodi.

A proeduregivinganexpliitonstrutionof IETsof periodi typewasintro-

dued in [29℄. The onstrution is based on hoosing losed paths on the graph

givingtheRauzylasses. EveryIETofperioditypeanbeobtainedthisway.

LetT =T(π,λ)^{beanIETofperioditypeand}p^{beaperiodsuhthat}π^(p)=π^.

LetA= Θ^(p)(T)^{. By(2.3),}

A^tΩπA= Ωπ ^andhene ker Ωπ=Aker Ωπ ^andHπ=A^tHπ.

Multiplying theperiod p^{ifneessary,weanassumethat} A|^{ker Ωπ} =Id^(seeAp-

pendix C for details). Denote by Sp(A) ^{the olletionof omplex}eigenvalues of

A^{, inluding} multipliities. Letus onsider the olletionof Lyapunov exponents

log|ρ|^,ρ∈Sp(A)^{. Itonsistsofthenumbers}

θ1> θ2≥θ3≥. . .≥θg≥0 =. . .= 0≥ −θg≥. . .≥ −θ3≥ −θ2>−θ1,

where2g= dimHπ ^and0^ourswiththemultipliityκ−1 = dim ker Ωπ ^(seee.g.

[35℄and[36℄). Moreover,ρ1:= expθ1^isthePerron-FrobeniuseigenvalueofA^{. We}

willusesometimesthesymbolθi(T)^insteadofθi ^{toemphasizethatitisassoiated}

toT^.

Denition. AnIETofperioditypeT(π,λ)^hasnon-degeneratedspetrumifθg>0^.

2.3. Growth ofBV oyles.

The reurreneof aoyle ϕ ^{with valuesin} R^{ℓ is related to the growth of} ϕ⁽ⁿ⁾

whenn^tendsto∞^.

ForanirrationalrotationT :x→x+αmod 1^{(thisanbeviewedasanexhange}

of 2intervals),when ϕ ^{hasa bounded variation, thegrowth of} ϕ^{(n) is ontrolled}

bytheDenjoy-Koksmainequality: ifϕ^{isazeromeanfuntion on}X =R/Z^with

boundedvariationVarϕ^{, and}(qn)^thedenominators(oftheonvergents)givenby theontinuedfrationexpansionof α^{,thenthefollowinginequalityholds:}

|

qXn−1 j=0

ϕ(x+jα)| ≤Varϕ,∀x∈X.

(2.4)

Thisinequalityimpliesobviouslyreurreneoftheoyleϕ^(·)andifα^hasbounded

partialquotients(wesayforbrevitybpq)

Pn−1

j=0 ϕ(x+jα) =O(logn)^uniformlyin x∈X^.

Itismuhmorediulttogetapreiseupperboundforthegrowthofaoyle

overanIET. Thefollowingtheorem (provedin Appendix A) givesfor an IET of

periodi type aontrol on the growth of aBV oylein termsof the Lyapunov

exponentsofthematrixA^.

Theorem2.2. Supposethat T(π,λ):I→I ^{isan intervalexhange}transformation ofperiodi type, 0≤θ2< θ1^{arethe twolargestLyapunovexponents,and}M ^isthe

maximal size of Jordan bloks in the Jordan deomposition of its periodi matrix

A^{. Then thereexists}C >0 ^suhthat

kϕ⁽ⁿ⁾k^sup≤C·log^M+1n·n^θ2/θ1·Varϕ

for every funtion ϕ : I → R ^{of bounded variation with zero mean and for eah}

naturaln^.

For ourpurpose,this inequalityisusefulwhenθ2(T)/θ1(T)^{issmall. InAppen-}

dixBwewillgiveexampleswitharbitrarysmallvaluesof thisratio.

2.4. Reurrene,essentialvalues, and ergodiityof oyles.

Inthis subsetionwereall somegeneralfats aboutoyles. Forrelevantbak-

groundmaterialonerningskewprodutsandinnitemeasure-preservingdynam-

ialsystems,wereferthereaderto[28℄and [1℄.

DenotebyG^theonepointompatiationofthegroupG^{. An element}g∈G

issaidto beanessential value ofϕ^{,ifforeveryopen}neighbourhood Vg ^ofg ⁱⁿG

andanyset B∈ B^,µ(B)>0^,thereexistsn∈Z^suhthat µ(B∩T⁻ⁿB∩ {x∈X :ϕ⁽ⁿ⁾(x)∈Vg})>0.

(2.5)

Thesetofessentialvaluesofϕ^{willbedenotedby}E(ϕ)^{. Thesetofniteessential}

valuesE(ϕ) :=G∩E(ϕ)^{isalosedsubgroupof}G^{. Wereallbelowsomeproperties}

ofE(ϕ)^(see[28℄).

Two oyles ϕ, ψ : X → G ^{are alled} ohomologous for T ^{if there exists a}

measurable funtion g: X →G ^{suh that} ϕ=ψ+g−g◦T^{. The}orresponding skew produts Tϕ ^and Tψ ^{are then} measure-theoretially isomorphi. A oyle

ϕ:X →G^{isaoboundaryifitis}ohomologoustothezerooyle.

Ifϕ^and ψ^areohomologousthenE(ϕ) =E(ψ)^{. Moreover,}ϕ ^isaoboundary

ifandonlyifE(ϕ) ={0}^.

Aoyleϕ:X→G^{isreurrent(asdened inthe}introdution)ifandonlyif, foreahopenneighborhoodV0 ^of0^{, (2.5)holdsforsome}n6= 0^{. Thisisequivalent}

to the onservativity of the skew produt Tϕ ^{(f. [28℄). Let} ϕ : X → R^{ℓ be an}

integrablefuntion. If itis reurrent, then

R

Xϕ dµ = 0^{; moreover,for} ℓ = 1^this

onditionissuientforreurrenewhenT ^isergodi.

The group E(ϕ) ^{oinides with the group of periods of} Tϕ^{-invariant funtions}

i.e.theset ofallg0∈G^{suhthat, if}f :X×G→R^isaTϕ^{-invariantmeasurable}

funtion,thenf(x, g+g0) =f(x, g)µ×mG^{-a.e. Inpartiular,}Tϕ^{isergodiifand}

onlyifE(ϕ) =G^.

Proposition 2.3 (see Corollary1.2 in [5℄). If ϕ:X →R^{ℓ is asquareintegrable}

oyle for an automorphism T : (X,B, µ) → (X,B, µ) ^{suh that} kϕ⁽ⁿ⁾kL²(µ) =

o(n^1/ℓ)^{,thenitisreurrent.}

InviewofTheorem2.2,asaonsequenewehavethefollowing.

Corollary2.4. IfT :I→I^{isanIETofperioditypesuhthat}θ2(T)/θ1(T)<1/ℓ

for an integer ℓ ≥1^{, then every oyle} ϕ :I → R^{ℓ over} T ^{of boundedvariation}

with zero mean is reurrent. If, for j = 1, . . . , ℓ^, Tj : I^(j) → I^{(j) are interval}

exhangetransformationsofperioditypesuhthatθ2(Tj)/θ1(Tj)<1/ℓ^,thenevery

"produt"oyleϕ= (ϕ1, . . . , ϕℓ) :I⁽¹⁾×. . .×I^(ℓ)→R^{ℓ ofboundedvariationwith}

zeromeanoverT1×...×Tℓ ^isreurrent.

Weontinuethesepreliminariesbysomeusefulobservationsforprovingtheer-

godiity of oyles. Let (X, d) ^{bea ompat metrispae. Let} B ^{standfor the} σ^{algebraofallBorelsetsandlet}µ^beaprobabilityBorelmeasureonX^{. By}χB^we

willdenotetheindiatorfuntionofasetB^{. Supposethat}T : (X,B, µ)→(X,B, µ)

is anergodi measurepreserving automorphismand there exist an inreasingse-

queneofnaturalnumbers(qn)^{andasequeneofBorelsets}(Cn)^suhthat µ(Cn)→α >0, µ(Cn△T⁻¹Cn)→0 ^and sup

x∈Cn

d(x, T^qnx)→0.

Assume that G ⊂ R^{ℓ for some} ℓ ≥ 1^{. Let} ϕ : X → G ^{be a Borel integrable}

oyleforT ^{withzeromean. Supposethatthesequene}(R

Cn|ϕ^(qn)(x)|dµ(x))n≥1

isbounded. Asthedistributions

(µ(Cn)⁻¹(ϕ^(qn)|^Cn)∗(µ|^Cn), n∈N)

areuniformlytight,bypassingtoafurthersubsequeneifneessaryweanassume

thattheyonvergeweaklyto aprobabilityBorelmeasureP ^onG^.

Lemma 2.5. The topologial support of the measure P ^{is inluded in the group} E(ϕ) ^{ofessential values ofthe oyle}ϕ^.

Proof. Suppose that g ∈ supp(P)^{. Let} Vg ^{be an open} neighborhood of g^{. Let} ψ : G → [0,1] ^{be a ontinuous funtion suh that} ψ(g) = 1 ^and ψ(h) = 0 ^for h∈G\Vg^{. Thus}R

Gψ(g)dP(g)>0^{. ByLemma 5in [13℄, for every}B ∈ B ^with µ(B)>0^wehave

µ(B∩T^−qnB∩(ϕ^(qn)∈Vg))≥ Z

Cn

ψ

ϕ^(qn)(x)

χB(x)χB(T^qnx)dµ(x)

→α Z

X

Z

G

ψ(g)χB(x)dP(g)dµ(x) =αµ(B) Z

G

ψ(g)dP(g)>0,

andheneg∈E(ϕ)^.

Corollary 2.6 (see also [6℄). If ϕ^(qn)(x) = gn ^{for all} x ∈Cn ^and gn →g^{, then} g∈E(ϕ)^.

Proposition 2.7 (see Proposition 3.8 in [28℄). Let T : (X,B, µ) → (X,B, µ) ^be

an ergodi automorphism and let ϕ : X → G ^{be a measurable oyle for} T^{. If} K⊂G^{isaompatsetsuhthat}K∩E(ϕ) =∅^{,thenthereexists}B ∈ B^suhthat µ(B)>0^and

µ(B∩T⁻ⁿB∩(ϕ⁽ⁿ⁾∈K)) = 0 ^forevery n∈Z.