Invariant measures for piecewise continuous maps C.R. Acad. Sci. Paris, Ser.I

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Dynamical systems

Invariant measures for piecewise continuous maps

Mesures invariantes pour les applications continues par morceaux

Benito Pires

¹

DepartamentodeComputaçãoeMatemática,FaculdadedeFilosofia,CiênciaseLetras,UniversidadedeSãoPaulo, 14040-901, Ribeirão Preto – SP,Brazil

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received15March2016 Acceptedafterrevision3May2016 Availableonline24May2016 PresentedbyClaireVoisin

We say that f : [⁰,1]→ [⁰,1] ^{is a piecewisecontinuous intervalmap if there exists a} partition 0=^x0<x1<· · ·<xd<xd+¹=^{1 of} [⁰,1] ^{such that f}|(x_i−1,x_i) is continuous and thelaterallimits w⁺₀ =^limx→⁰⁺f(x),w_d⁻₊₁=^limx→¹⁻f(x),w⁻_i =^limx→^x−_i f(x) and w⁺_i =^limx→^x+_i f(x) exist for eachi.We provethat every piecewisecontinuous interval map withoutconnectionsadmitsan invariant Borelprobability measure. Wealso prove thateveryinjectivepiecewisecontinuousintervalmapwithnoconnectionsandnoperi- odicorbitsistopologicallysemiconjugatetoanintervalexchangetransformation.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

r é s um é

Onditque f: [⁰,1]→ [⁰,1]^estuneapplicationd’intervallecontinueparmorceauxs’ilexiste une partition0=^x0<x1<· · ·<xd<xd+¹=^{1 de} [⁰,1] ^{telleque f}|^(xi−1,x_i) est conti- nue et telle que les limites latérales w⁺₀ =^limx→⁰⁺f(x), w_d⁻₊₁=^limx→¹⁻f(x), w⁻_i = lim_x→x−

i f(x)etw⁺_i =^limx→^x+_i f(x)existentpourchaquei.Onprouvequetouteapplica- tiond’intervallecontinueparmorceauxsansconnexionadmetunemesuredeprobabilité invariante. Onprouveégalementque touteapplicationinjectived’intervallecontinuepar morceauxsansconnexionetsansorbitepériodiqueesttopologiquementsemiconjuguéeà unéchanged’intervalles.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introduction

Muchinformationaboutthelong-termbehaviouroftheiteratesofamapisrevealedbyitsinvariantmeasures.Regard- ing piecewisecontinuous intervalmaps,thepresence ofa non-atomicinvariantBorelprobability measurecan beusedto constructtopologicalconjugaciesorsemiconjugacieswithintervalexchangetransformations(IETs).

E-mailaddress:[email protected].

1 PartiallysupportedbySãoPauloResearchFoundation(FAPESP)grant#2015/20731-5.

http://dx.doi.org/10.1016/j.crma.2016.05.002

1631-073X/©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

Transfer operatorshaveproved tobe animportanttool toobtain absolutelycontinuous invariant probabilitymeasures forpiecewisesmoothpiecewisemonotoneintervalmaps(see[1,3–5,9]).Ingeneral,thesetypesofresultsassumethateach branchofthepiecewisecontinuousmapisC^r-smooth(r

≥

^{1),monotoneandhasderivativegreaterthan1.}

TheaimofthisarticleistoprovetheexistenceofinvariantBorelprobabilitymeasuresforpiecewisecontinuousinterval maps not embraced bythe transferoperator approach.In thisway,our resultincludesgap maps,piecewise contractions and generalisedinterval exchange transformations(GIETs). No monotonicityandno smoothnessassumptions,beyond the uniform continuityofeach branchofthemap, areassumed. Ourresultisthe naturalversion oftheKryloff–Bogoliouboff Theorem(see[8])forpiecewisecontinuousintervalmaps.

We arealsointerested inconstructingtopologicalsemiconjugacybetweeninjectivepiecewisecontinuous intervalmaps and interval exchange transformations, possibly with flips. In thisregard, it is worth mentioning the result by J. Milnor and W. Thurston(see[12]), whichstatesthat anycontinuous piecewise monotone interval map ofpositive entropyhtop is topologicallysemiconjugate toamap withconstantslope equalto

±

^{ehtop.Thisresultwas}generalisedby L. Alsedàand M. Misiurewicz in[2]to piecewise continuous piecewise monotone interval mapsofpositive entropy. Concerningcount- ably piecewise continuous piecewise monotone interval maps,a necessaryandsufficient condition fortheexistence of a non-decreasingsemiconjugacytoamapofconstantslopewasprovidedbyM. MisiurewiczandS. Rothin[13].Theauthor andA. Nogueiraprovedin[14]thateveryinjectivepiecewisecontractionistopologicallyconjugatetoamapwithconstant slopeequalto

±

¹₂^.

The proof ofthe Kryloff–Bogoliouboff Theorem failsfor discontinuous maps.In thisarticle, we presenta variation of thisproofthatovercomessuchlimitation.Thehypothesisofnoconnectionscannotberemovedsincethereareexamplesof piecewisecontinuousmapsthathaveconnectionsandadmitnoBorelinvariantmeasure.Theproofpresentedheredoesnot holdforcountablypiecewisecontinuousmapssinceforsuchmapsthelaterallimitsmightnotexistatallpointsof

[

⁰

,

1

]

^.

2. Statementoftheresults

Throughoutthisarticle,assumethat f

: [

⁰

,

1

] → [

⁰

,

1

]

^{isapiecewisecontinuousintervalmap.Hence,thereexistsapar-} tition0

=

^x0

<

x1

< · · · <

xd

<

xd+¹

=

^{1 of}

[

⁰

,

1

]

^{suchthat f}

|

(x_i−1,xi)iscontinuousandthelaterallimitsw⁺₀

=

^limx→0⁺ f

(

x

)

, w⁻_d+1

=

^limx→¹⁻ f

(

x

)

,w⁻_i

=

^lim_x→x⁻

i f

(

x

)

andw⁺_i

=

^lim_x→x⁺

i f

(

x

)

existforeach i.Let D

= {

x0

, . . . ,

xd+¹

},

W

= {

w⁺₀

,

w⁻₁

,

w⁺₁

, . . . ,

w⁻_d

,

w_d⁺

,

w⁻_d+1

}.

Wesaythat f hasnoconnectionsif

w∈^W

∞ k=0

{

^fk

(

w

)} ∩

^D

= ∅.

⁽¹⁾

Wesaythatx

∈ [

⁰

,

1

]

^{isaperiodicpointof f ifthereexistsanintegerk}

≥

^{1 suchthat fk}

(

x

) =

^x.

OurfirstresultturnsouttobeaversionoftheKryloff–BogoliouboffTheorem[8]forpiecewisecontinuousintervalmaps.

Theorem2.1.Letf

: [

⁰

,

1

] → [

⁰

,

1

]

^{beapiecewisecontinuousmapwithno}connections,thenf admitsaninvariantBorelprobability measure

μ

.Moreover,if f hasnoperiodicpoints,thenthemeasure

μ

isnon-atomic.

The hypothesis ofnoconnections inthestatement ofTheorem 2.1,althoughmore readilycheckable,maysounda bit restrictivebecause,forinstance,itprohibitsthataleft-continuousmap f takesonediscontinuityintoanother.Indeed,what needs tobeavoidedfortheexistenceoftheinvariantmeasureisthepresenceofclosedconnections,amoretechnicalnotion giveninSection3.

Intheworldofgeneralisedintervalexchangetransformations,thehypothesisofnoconnectionscorrespondstothenotion ofhavingan

∞

^{-completepath.Asremarkedin[11,p. 1586],everyGIETwithsuchpropertyis}topologicallysemiconjugate toanIET.Thenextresultextendsthisclaimtopiecewisecontinuousmaps.Itcanalsobeconsideredageneralisationofthe item (a)oftheStructureTheorembyGutierrez[6,p. 18].

Corollary2.2.Let f

: [

⁰

,

1

] → [

⁰

,

1

]

^{beaninjectivepiecewisecontinuousmapwithno}connectionsandnoperiodicpoints,thenf is topologicallysemiconjugatetoanintervalexchangetransformation,possiblywithflips.

Now we presenta class of piecewise continuous interval mapsfor whichhaving no connections is a generic (in the measure-theoreticalsense)property.Werecallthatanirrationalitycriterion fortheabsenceofconnectionsinIETswithout flipswasprovidedbyM. Keanein[7].

Theorem2.3.Let

φ

1

, . . . , φ

d+¹

: [

⁰

,

1

] → (

0

,

1

)

becontinuousmapsandlet

⊂

R^{dbetheopenset}

= { (

x1

, . . . ,

xd

) ∈

R^d

|

⁰

<

x₁

< · · · <

x_d

<

1

}

^{,thenforLebesguealmostevery}

(

x₁

, . . . ,

x_d

) ∈

,thepiecewisecontinuousmap f

: [

⁰

,

1

] → (

0

,

1

)

definedby f

(

x

) = φ

i

(

x

)

ifx

∈

^Ii,whereI1

= [

⁰

,

x1

),

I2

= [

^x1

,

x2

), . . . ,

I_d

= [

^xd−1

,

x_d

)

,I_d+1

= [

^xd

,

1

]

^,hasnoconnectionsandhenceadmitsan invariantBorelprobabilitymeasure.

3. ProofofTheorem2.1

Henceforth,assumethatthemap f hasnoconnectionsandnoperiodicorbits.

Lemma3.1.Givenx

∈ [

⁰

,

1

]

^{andanintegerr}

≥

^{1,thereexistsanopen}subinterval J_xof

[

⁰

,

1

]

^{containingx suchthat}

{

^f

(

y

), . . . ,

f^r

(

y

) } ∩

^Jx

= ∅

^{for every y}

∈

^Jx

.

(2)

Proof. Firstletusprovetheresultforx

=

^xi,where1

≤

ⁱ

≤

^d.Let

γ =

r k=1

{

^fk−1

(

w⁻_i

),

f^k

(

x_i

),

f^k−1

(

w⁺_i

)}.

Bythe uniformcontinuity of f

|

(x_j−1,xj), 1

≤

^j

≤

^d

+

^{1,together with thehypothesis of no}connections, we have that for every

>

0,thereexist0

< δ <

andaninterval Jxi

= (

xi

− δ,

xi

+ δ) ⊂ [

⁰

,

1

]

^suchthat

d

f^k

(

y

), γ <

for every y

∈

Jx_i and 1

≤

k

≤

r

,

(3)

whered

f^k

(

y

), γ

₌

minz∈γ

|

^fk

(

y

) −

^z

|

^.

Let

=

¹₂^d

(

xi

, γ )

,then

>

0,otherwise f wouldhaveaconnection ora periodicorbit. Thistogetherwith(3)implies that

|

^fk

(

y

) −

^xi

| > > δ

forall y

∈

^Jx_i and1

≤

^k

≤

^{r.Hence,(2)holdsforeveryx}

=

^xi

∈

^D.

The casesin which x

=

^x0

=

^{0 or x}

=

^xd+¹

=

^{1 followslikewise,by} considering intervals ofthe form Jx0

= [

⁰

, δ)

and J_xd+1

= (

1

− δ,

1

]

^,respectively.

Itremainstoconsiderthecaseinwhichx

∈ { /

^x0

, . . . ,

xd+¹

}

^{.Duetothehypothesisofno}connections,thereareonlytwo possibilities:either

{

^fk

(

x

) :

^k

≥

⁰

} ∩ {

^x0

, . . . ,

x_d+1

} = ∅

^{orthereexistk}

≥

^{1 (taketheleastvalue)and0}

≤

ⁱ

≤

^d

+

^{1 suchthat} f^k

(

x

) =

^xi.As forthefirst possibility,take

γ = {

^f

(

x

), . . . ,

f^r

(

x

) }

^{,then f iscontinuous on}

{

^x

} ∪ γ

.Moreover, since f has noperiodicpoints,wehavethat x

∈ / γ

.Therefore,forevery

>

0,thereexist0

< δ <

andaninterval Jx

= (

x

− δ,

x

+ δ)

suchthat (3)holdsfor Jx intheplaceof Jx_i.Toconcludetheproof, proceedasbefore.Concerningthesecondpossibility, let Jxi

= (

xi

− δ,

xi

+ δ)

beasinthebeginningoftheproof,then,asalreadyproved,

{

f

(

y

), . . . ,

f^r

(

y

)} ∩

Jx_i

= ∅

for every y

∈

Jx_i

.

(4)

Moreover, since k is the least value, f is locally continuous around

{

^x

,

f

(

x

), . . . ,

f^k−1

(

x

)}

^{, thus there exists an interval} Jx

= (

x

− η ,

x

+ η )

suchthat Jx

,

f

(

Jx

), . . . ,

f^k

(

Jx

)

arepairwisedisjointintervalsand f^k

(

Jx

) ⊂

^Jx_i.Now(4)impliesthat(2) holdsforevery y

∈

^Jx,whichconcludestheproof. 2

Letq

∈ [

⁰

,

1

]

^{begiven.Since f hasnoperiodicorbits,thereexists}

≥

^{0 suchthat}

f^k

(

q

) :

^k

≥

∩

^D

= ∅

^{.Hereafter,set} p

=

^f

(

q

)

,then

{

p

,

f

(

p

),

f²

(

p

), . . .} ∩

D

= ∅.

(5)

Denoteby

( μ

n

)

_n^∞₌₁thesequenceofBorelprobabilitymeasureson

[

⁰

,

1

]

^definedby

μ

n

=

¹ n

n−¹ k=⁰

δ

_fk(p)

,

where

δ

_fk(p)istheDiracprobabilitymeasureon

[

⁰

,

1

]

concentratedat f^k

(

p

)

.

Bythe Banach–AlaogluTheorem, thespaceof Borelprobability measureson a compactmetric spaceiscompact with respect tothe weak^∗ topology. Hence,there exist aBorelprobability measure on

[

⁰

,

1

]

^{,denoted henceforthby}

μ

,anda subsequenceof

{ μ

n

}

^{,denotedhenceforthby}

{ μ

n_j

}

^∞_j=1^{,thatconvergesto}

μ

intheweak^∗ topology.

Thenextresultisgoingtobeusedtwice,inLemma 3.3aswellasinLemma 3.5.

Lemma3.2.Letx

∈ [

⁰

,

1

]

^.Given

>

0,thereexistanopensubinterval Jxof

[

⁰

,

1

]

^{containingx,andanintegerj}0

≥

^1suchthat

μ

n_j

(

Jx

) <

for every j

≥

j0

.

(6)

Proof. Letr

≥

^{1 beanintegersogreatthat 2}

r

<

.Since

{

ⁿj

}

^∞_j=1 ^isasubsequenceof

{

¹

,

2

, . . .}

^{,thereexists j}0

≥

^{1 suchthat} n_j

>

rforevery j

≥

^j0.Let J_xbeasinthestatementofLemma 3.1.Let j

≥

^j0 and

=

^#

{

⁰

≤

^k

≤

ⁿj

−

¹

|

^fk

(

p

) ∈

^Jx

}

^,where

# denotescardinality.By(2),wehavethat

( −

¹

)

r

≤

ⁿj,thus

μ

_nj

(

J_x

) =

¹ n_j

n_j−¹ k=⁰

δ

_fk(p)

(

J_x

) =

^#

{

⁰

≤

^k

≤

ⁿj

−

¹

|

^fk

(

p

) ∈

^Jx

}

n_j

≤

²

r

<

for every j

≥

^j0

. 2

Lemma3.3.Themeasure

μ

isnon-atomic.

Proof. Let x

∈ (

0

,

1

)

. Given

>

0,let Jx be an opensubintervalof

[

⁰

,

1

]

^{containing x asinthestatement ofLemma 3.2.}

Since the set S

= {

z

∈ [

0

,

1

] : μ ({

z

}) >

0

}

is atmost countable, thereexists a subinterval J_x

⊂

Jx containing xsuch that

μ

∂

J_x

=

^0,where

∂

J_xdenotestheendpointsoftheinterval J_x.By[15,Theorem6.1,p.40]andby(6),

μ ( {

^x

} ) ≤ μ (

J_x

) =

^lim

j→∞

μ

n_j

(

J_x

) ≤

^{lim sup}

j→∞

μ

n_j

(

Jx

) ≤ .

Thefactthat isarbitraryyields

μ ( {

^x

} ) =

^0.

Now let A1

⊂

^A2

⊂ · · ·

^{be asequence ofsubsetsof}

[

⁰

,

1

]

^suchthat

k≥¹Ak

= (

0

,

1

)

and

∂

Ak

∩

^S

= ∅

^foreveryk

≥

^1.

By(5),wehavethat

μ

nj

(

Ak

) =

^{1 forevery j}

,

k

≥

^{1.By[15,Theorem6.1,p.40]oncemore,wehavethat}

μ (

A_k

) =

^lim

j→∞

μ

n_j

(

A_k

) =

1 for every k

≥

¹

.

Inthisway,

μ ((

0

,

1

)) =

lim

k→∞

μ (

A_k

) =

1

,

thus

μ ({

0

}) = μ ({

1

}) =

0

. 2

The convergenceof

{ μ

nj

}

^∞_j=1 ^to

μ

intheweak^∗ topology impliesthat limj→∞

φ

d

μ

nj

=

φ

d

μ

foreverycontinuous function

φ : [

⁰

,

1

] →

R^{.Thenextlemmaextendsthisclaimforthepiecewisecontinuousmap}

φ = ϕ ◦

^f.

Remark3.4.AspointedoutbyC.Liveraniin[10,p. 4],thepointwheretheproofoftheKryloff–BogoliouboffTheoremfails isLemma 3.5,whichisautomaticforcontinuousfunctions.

Lemma3.5.Foreverycontinuousfunction

ϕ : [

⁰

,

1

] →

R^,

jlim→∞

ϕ ◦

fd

μ

n_j

=

ϕ ◦

fd

μ .

Proof. Let

>

0 be arbitrarilysmall.ByLemma 3.3,we havethat

μ ({

^xi

}) =

^{0 forevery 1}

≤

ⁱ

≤

^{d.Hence,there existsan} open interval J_xi containing xi such that

μ (

J_xi

) <

forevery 1

≤

ⁱ

≤

^{d.By Lemma 3.2,there existan open interval J}xi

containing xi,andaninteger j0

≥

^{1 suchthat}

μ

n_j

J_x

i

<

for every j

≥

^j0 and 1

≤

ⁱ

≤

^d

.

Set J_xi

=

^Jx_i

∩

^Jx_i.Thefunction

ϕ ◦

^{f isboundedbysome constantM andcontinuousoneachinterval}

(

x_i−1

,

x_i

)

forevery 1

≤

ⁱ

≤

^d

+

^{1. In this way, there exists a continuous function h}

: [

⁰

,

1

] → [−

^M

,

M

]

^{such that h}

(

x

) = ϕ ◦

^f

(

x

)

forevery x

∈ [

⁰

,

1

] \

d

i=1 Jxi.Puttingitalltogetheryields

ϕ ◦

^fd

μ

nj

−

hd

μ

nj

≤

| ϕ ◦

^f

−

^h

|

^d

μ

nj

≤

^2Md

for every j

≥

^j0

,

(7)

and

ϕ _◦

fd

μ ₋

hd

μ _≤

2Md

.

(8)

Finally,sincehiscontinuouson

[

⁰

,

1

]

^and

μ

nj convergesto

μ

intheweak^*topology,thereexists j1

≥

^j0suchthat

hd

μ

n_j

−

hd

μ _≤

for every j

≥

^j1

.

(9)

Itfollowsfromtheequations(7),(8)and(9)that

ϕ ◦

fd

μ

n_j

−

ϕ ◦

fd

μ ≤ (

4Md

+

1

)

for every j

≥

j1

,

whichconcludestheproof. 2

Lemma3.6([16,Theorem6.2,p.147]).Letm1andm2betwoBorelprobabilitymeasureson

[

⁰

,

1

]

^.If

ϕ

dm1

=

ϕ

dm2forevery continuousfunction

ϕ : [

⁰

,

1

] →

R^,thenm1

=

^m2.

Givena Borelprobabilitymeasure m on

[

⁰

,

1

]

^{andaninteger k}

≥

^1,letm

◦

^{f−k denotethe Borelmeasuredefinedby}

(

m

◦

^f−k

)(

B

) =

^m

f^−k

(

B

)

foranyBorelsetB.Inparticular,form

= δ

p wehavethat

δ

p

◦

^f−k

= δ

_fk(p).

Lemma3.7([16,Lemma6.6,p.150]).Let

ψ : [

⁰

,

1

] →

R^beaBorel-measurablefunction,k

≥

^{1aninteger,andm aBorel}probability measureon

[

⁰

,

1

]

^,then

ψ ◦

^fkdm

=

ψ

d

(

m

◦

^f−k

).

Lemma3.8.Themeasure

μ

isinvariantbyf .

Proof. ByLemma 3.6andLemma 3.7(taking

ψ = ϕ

,k

=

^{1 andm}

= μ

),itsufficestoshowthat

ϕ _◦

fd

μ ₌

ϕ

d

μ

(10)

foreverycontinuousfunction

ϕ : [

⁰

,

1

] →

R^{.ByLemma 3.5,forevery}continuous function

ϕ : [

⁰

,

1

] →

R^,

ϕ _◦

fd

μ ₋

ϕ

d

μ ₌

lim

j→∞

ϕ _◦

fd

μ

_nj

₋

ϕ

d

μ

_nj

.

(11)

ByLemma 3.7oncemore(nowtaking

ψ = ϕ ◦

^{f andm}

= δ

p),wereach

ϕ ◦

^fd

μ

nj

=

¹ n_j

nj−1 k=⁰

ϕ ◦

^fd

(δ

p

◦

^f−k

) =

¹ n_j

nj−1 k=⁰

ϕ ◦

^fk+1d

δ

p

.

(12)

Likewise,

ϕ

d

μ

n_j

=

¹ n_j

n_j−¹ k=⁰

ϕ

d

(δ

p

◦

^f−k

) =

¹ n_j

n_j−¹ k=⁰

ϕ _◦

f^kd

δ

p

.

(13)

Itfollowsfrom(11),(12)and(13)that

ϕ ◦

^fd

μ −

ϕ

d

μ =

^lim

j→∞

^n1j

^nj−1

k=0

ϕ ◦

^fk+1

− ϕ ◦

^fk

d

δ

p

=

^lim

j→∞

_n¹_j

ϕ _◦

f^nj

− ϕ

d

δ

p

≤

^lim

j→∞

2

^f

n_j

=

⁰

.

Hence,(10)holds,whichconcludestheproof. 2

Remark3.9.TheproofofTheorem 2.1followsfromLemmas 3.3 and3.8.

4. Proofoftheotherresults

Corollary2.2.Letf

: [

⁰

,

1

] → [

⁰

,

1

]

^{beaninjectivepiecewisecontinuousmapwithno}connectionsandnoperiodicorbits,then f is topologicallysemiconjugatetoanintervalexchangetransformation,possiblywithflips.

Proof. ByTheorem 2.1, f admitsanon-atomicBorelprobabilitymeasure

μ

invariantby f.Leth

: [

⁰

,

1

] → [

⁰

,

1

]

^bedefined byh

(

x

) = μ ( [

⁰

,

x

] )

,thenhisacontinuousnon-decreasingsurjectivemap.Let1

≤

ⁱ

≤

^d

+

^{1 andx}

,

y

∈ (

xi−¹

,

xi

)

besuchthat h

(

x

) =

^h

(

y

)

.Weclaimthath

(

f

(

x

)) =

^h

(

f

(

y

))

.Assumethat x

≤

^{y and f}

(

x

) ≤

^f

(

y

)

,then,theinjectivity of f togetherwith thecontinuityof f

|

(x_i−1,xi)yields

[

^x

,

y

] =

^f−1

([

^f

(

x

),

f

(

y

)])

^.Hence,since

μ

isnon-atomic,

|

^h

(

f

(

y

)) −

^h

(

f

(

x

)) | = μ _([

f

(

x

),

f

(

y

) ]) = μ

f⁻¹

([

^f

(

x

),

f

(

y

) ])

= μ _([

x

,

y

]) = |

^h

(

y

) −

^h

(

x

) | .

(14) Asfortheothercases,toproceed likewisetoshowthat(14)stillholds.Hence,theclaimistrue.

LetT

: [

⁰

,

1

] → [

⁰

,

1

]

^{bedefinedby T}

(

h

(

x

)) =

^h

(

f

(

x

))

.Bytheclaim, T iswell defined. Lett0

,

t1

, . . . ,

td+¹ bedefinedby t₀

=

^0,td+¹

=

^{1 andt}i

=

^h

(

x_i

)

forevery1

≤

ⁱ

≤

^{d.By(14),wehavethatforeveryt}

,

s

∈ (

t_i−1

,

t_i

)

,thereexistx

,

y

∈ (

x_i−1

,

x_i

)

suchthatt

=

^h

(

x

)

,s

=

^h

(

y

)

and

|

^T

(

t

) −

^T

(

s

) | = |

^h

(

f

(

x

)) −

^h

(

f

(

y

)) | = |

^h

(

x

) −

^h

(

y

) | = |

^t

−

^s

|

^{for every t}

,

s

∈ (

t_i−1

,

t_i

).

Thisprovesthat T

|

(ti−1,ti)isanisometry;therefore,T isanintervalexchangetransformation,possiblywithflips.Bydefini- tion, T

◦

^h

=

^h

◦

^{f,thus f is}topologicallysemiconjugatetoT. 2

Theorem2.3.Let

φ

1

, . . . , φ

d+¹

: [

⁰

,

1

] → (

0

,

1

)

becontinuousmapsandlet

⊂

R^{dbetheopenset}

= { (

x1

, . . . ,

xd

) ∈

R^d

|

⁰

<

x1

< · · · <

x_d

<

1

}

^{,thenforLebesguealmostevery}

(

x1

, . . . ,

x_d

) ∈

,thepiecewisecontinuousmap f

: [

⁰

,

1

] → (

0

,

1

)

definedby f

(

x

) = φ

i

(

x

)

ifx

∈

^Ii,whereI1

= [

⁰

,

x1

),

I2

= [

^x1

,

x2

), . . . ,

Id

= [

^xd−¹

,

xd

)

,Id+¹

= [

^xd

,

1

]

^,hasnoconnectionsandhenceadmitsan invariantBorelprobabilitymeasure.

Proof. Denoteby Idtheidentitymapon

[

⁰

,

1

]

^.SetC0

= {

^Id

}

^.Let Ck

= {φ

i

◦

^h

|

¹

≤

ⁱ

≤

^d

+

¹

,

h

∈

Ck−¹

},

^k

≥

¹

.

Foreach 0

≤

ⁱ

≤

^d

+

^1,1

≤

^j

≤

^d,wi

∈ {

^w−_i

,

w⁺_i

}

^andh

∈

k≥⁰Ck,theset

{ (

x1

, . . . ,

x_d

) ∈ |

^xj

=

^h

(

wi

) }

^{isthegraphofa} continuous function definedon

[

⁰

,

1

]

^{,thusit isaLebesguenullset.Thistogether withthefactthat x}0

=

^{0 andx}d+¹

=

¹ donotbelongtotherangeofanyh

∈

k≥¹Ckimplies thatthesetofparameters

(

x1

, . . . ,

x_d

) ∈

forwhichthemap f has connections isaLebesguenullset,denotedby N.Let

(

x1

, . . . ,

xd

) ∈ \

^{N,theneither f hasaperiodicpointor f hasno} periodicpointsandnoconnections.Inthefirstcase, f hasan invariantBorelprobabilitymeasuresupportedonitsperiodic orbits, whileinthesecondcase,byTheorem 2.1, f admitsaninvariantnon-atomicBorelprobabilitymeasure. 2

3. Finalremarks

The claimofTheorem 2.1 keepstrueifin itsstatement theterm“noconnections” isreplaced bythe term“no closed connections” definedbelow. Let f

: [

⁰

,

1

] → [

⁰

,

1

]

^{be asin(1) andlet}

¯

_f

:

P

( [

⁰

,

1

] ) →

P

( [

⁰

,

1

] )

bethe map definedon eachset A

⊂ [

⁰

,

1

]

^by

¯

f

(

A

) =

x∈^A

lim

→⁰⁺f

(

x

− ),

lim

→⁰⁺f

(

x

+ )

,

wherelim→0⁺ f

( − ) :=

^f

(

0

)

andlim→0⁺ f

(

1

+ ) :=

^f

(

1

)

.Wesaythat themap f hasaclosedconnectionifthereexist 0

≤

ⁱ

≤

^d

+

^{1 andk}

≥

^{1 suchthat x}i

∈

k≥¹

¯

f^k

({

^xi

})

^.

Theexistenceofconnectionsneitherimpliesnorisimpliedbytheexistenceofperiodicpoints.Infact,let f1

,

f2

: [

⁰

,

1

] → [

0

,

1

]

bethepiecewisecontinuousmapsdefinedby

f₁

(

x

) =

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

x 2

+

¹

8 if 0

≤

^x

<

¹ 2 x

2

+

³ 8 if 1

2

≤

^x

≤

¹

,

f₂

(

x

) =

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

x 2

+

¹

4 if 0

≤

^x

<

¹ 2 x

2 if 1

2

≤

^x

≤

¹

.

The map f1 hastwo periodicpoints andnoconnections. Themap f2 hasaclosedconnection butnoperiodicpoints.

Moreover,itdoesnotadmitanyinvariantBorelprobabilitymeasure.

Acknowledgements

IamverygratefultoKleyberCunhaandCarlangeloLiveraniforthepromptreplytomyquestions.Iamequallythankful fortheprecisecommentsprovidedbytheanonymousreferee.

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