Chapter 1 Renormalization of circle diffeomorphism sequences and Markov sequences

Chapter 1

Renormalization of circle diffeomorphism

sequences and Markov sequences

João P. Almeida, Alberto A. Pinto and David A. Rand

Abstract We show a one-to-one correspondence between circle diffeomorphism

sequences that areC1+ n-periodic points of renormalization and smooth Markov sequences.

1.1 Introduction

Following [2–9, 20–23], we present the concept of renormalization applied to cir-cle diffeomorphism sequences. These concepts are essential to extend the results presented in [8, 9] to all Anosov diffeomorphisms on surfaces, i.e. to prove a one-to-one correspondence betweenC1+conjugacy classes of Anosov diffeomorphisms and pairs ofC1+circle diffeomorphism sequences that areC1+n-periodic points of renormalization (see also [1, 8, 9, 19]). The main point in this paper is to show the existence of a one-to-one correspondence betweenC1+circle diffeomorphism sequences that areC1+n-periodic points of renormalization and smooth Markov se-quences. This correspondence is a key step in passing from circle diffeomorphisms to Anosov diffeomorphisms because the Markov sequences encode the smooth

in-João P. Almeida

LIAAD-INESC TEC and Department of Mathematics, School of Technology and Management, Polytechnic Institute of Bragança. Campus de Santa Apolónia, Ap. 1134, 5301-857 Bragança, Portugal. e-mail: jpa@ipb.pt

Alberto A. Pinto

LIAAD-INESC TEC and Department of Mathematics, Faculty of Sciences, University of Porto. Rua do Campo Alegre, 4169-007 Porto, Portugal. e-mail: aapinto@fc.up.pt

David A. Rand

Warwick Systems Biology & Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom.

e-mail: dar@maths.warwick.ac.uk

2 João P. Almeida, Alberto A. Pinto and David A. Rand

formation of the expanding and contracting laminations of the Anosov diffeomor-phisms [10–18]).

1.2 Circle difeomorphisms

Leta= (ai)∞i=0be a sequence of positive integers and letγ(a) =1/(a0+1/(a1+ 1/· · ·)). For everyi∈N0, letγ_i=γ_{i(a) =}1_/(ai+1/(ai+1+1/· · ·))and letSibe a counterclockwise oriented circlehomeomorphic to the circleS_i=R_/(1₊γ_i)Z.

AnarcinS_iis the image of a non trivial intervalIinRby a homeomorphism

α:I→Si. IfIis closed (resp. open) we say thatα(I)is aclosed(resp.open)arcin

S_{i. We denote by(a,b)(resp.[a,b]) the positively oriented open (resp. closed) arc}

onS_{istarting at the pointa∈}S_{iand ending at the pointb∈}S_{i. AC}1+_atlasA_iin

S_{iis a set of charts such that (i) every small arc of}S_{iis contained in the domain of}

some chart inA_{i, and (ii) the overlap maps areC}1+α_{compatible, for some}α_>0.

LetA

i denote the affine atlas whose charts are isometries with respect to the usual norm inS

i. Let therigid rotation g_i :Si→Sibe the affine homeomorphism, with respect to the atlasA

i, with rotation numberγi/(1+γi). A homeomorphismh:S

i→Siisquasisymmetricif there exists a constantC>1 such that for each two arcsI1andI2of S

iwith a common endpoint and such that |I1|i=|I2|i, we have|h(I1)|i/|h(I2)|i<C, where the lengths are measured in the charts ofA_iandA_i.

AC1+ circle diffeomorphism sequence (gi,Si,Ai)∞i=0 is a sequence of triples

(gi,Si,Ai)with the following properties: (i)gi:Si→Siis aC1+αdiffeomorphism, with respect to theC1+α atlasAi, for someα_>0; and (ii)giis quasi-symmetric conjugate to the rigid rotationg

iwith respect to the atlasAi.

We denote theC1+ circle diffeomorphism (gi,Si,Ai)by gi. In particular, we denote the rigid rotation(g

i,Si,Ai)bygi.

1.2.1 Horocycles

Let us mark a point inS_{ithat we will denote by 0i∈}S_{i. LetS}0

i = [0i,gi(0i)]be the oriented closed arc in Si, with endpoints 0i and_gi(0i₎. For every_{k∈ {}0_{, . . . ,ai}},

let Sk i =

gk

i(0i),gki+1(0i)

_{be the oriented closed arc in S}

i with endpointsgki(0i) andgk_i+1(0i)and such thatSk_i∩S_ik−1={gk_i(0)}. LetSai+1

i = [g ai+1

i (0i),0i]be the oriented closed arc inSi, with endpoints_gai+1

i (0i)and 0i.

point

ξi=πgi(gi(0i)) =· · ·=πgi(g ai+1

i (0i))∈Hi

is called the junction of the horocycle Hi. For every k∈ {0, . . . ,ai}, let Ski,H = Sk_i,H(S_{i,gi)⊂Hibe the projection by}π_{gi of the closed arcS}k

i. LetRiSi=S0i,H∪S a+1 i,H be therenormalized circleinHi. The horocycleHiis the union of the renormalized circleRiSiwith the circlesSik,Hfor everyk∈ {1, . . . ,ai}.

AparametrizationinHiis the image of a non trivial intervalIinRby a

homeo-morphismα:I→Hi. IfIis closed (resp. open) we say thatα(I)is aclosed(resp. open)arcinHi. AchartinHiis the inverse of a parametrization. Atopological atlas

Bon the horocycleHiis a set of charts{(j,J)}, onHi, with the property that every small arc is contained in the domain of a chart inB, i.e. for any open arcKinHi and anyx∈Kthere exists a chart{(j,J)} ∈Bsuch thatJ∩Kis a non trivial open arc inHiandx∈J∩K. AC1+atlasBin Hiis a topological atlasBsuch that the overlap maps areC1+α and haveC1+αuniformly bounded norms, for someα>0.

LetA_ibe aC1+atlas onS_iin whichgi:Si→Siis aC1+circle diffeomorphism. We are going to construct aC1+ atlasAH

i onHi that we call theextended push-forwardAH

i = (πgi)_∗Ai of the atlasAi on Si. Ifx∈Hi\{ξi} then there exists a sufficiently small open arcJ⊂Hicontainingxand such thatπg−i1(J)is contained in the domain of some chart(I,ιˆ)ofAi. In this case, we define₍J,ιˆ◦π−1

gi )as a chart inAH

i . Ifx=ξiandJis a small arc containingξi, then either (i)πgi−1(J)is an arc inS_{i or (ii)}π−1

gi (J)is a disconnected set that consists of a union of two connected components.

In case (i),π−1

gi (J)is connected and it is contained in the domain of some chart

(I,ιˆ)∈A_{i. Therefore we define J,}ιˆ_◦π−1

gi

as a chart inAH i . In case (ii),π−1

gi (J)is a disconnected set that is the union of two connected arcs IL

l andIrRof the formIlL= (cLl,gli(0)]andIrR= [gri(0),cRr), respectively, for alll,r∈ {1, . . . ,ai+1}. LetJ_lLandJrRbe the arcs inHidefined byJ_lL=πgi(I

L

l)andπgi(I R r) respectively. ThenJ=JL

l ∪JrRis an arc inHiwith the property thatJ_lL∩JrR={ξi}, for everyl,r∈ {1, . . . ,ai+1}. We call such arcJa(l,r)-arcand we denote it by Jl,r. Let jl,r:Jl,r→Rbe defined by,

jl,r(x) =

_ˆι

◦π−1

gi (x) ifx∈J R r ˆ

ι◦gir−l◦πg−i1(x)ifx∈J L l

.

Let(I,ιˆ)∈A_{ibe a chart such that}π_{gi(I)⊃Jl,r. Then we define(Jl,r,jl,r)as a chart}

inAH

i (see Figure 1.1). We call the atlas determined by these charts theextended pushforward atlas of A_i and, by abuse of notation, we will denote it by AH

i =

(πgi)∗Ai.

Let the marked point 0_iinS

ibe the natural projection of 0∈Ronto 0i∈Si=

R_/(1₊γ_i)Z. LetS0_i = [0i,g

i(0i)]andS k

i = [gk_i(0i),gk_i+1(0i)]. Furthermore, let

H_i=Hi(Si,g_i), Ski,H=Ski,H(Si,g_i), RiSi=Si0,H∪Sai,+H1andA

Hi₌π_g i

∗

4 João P. Almeida, Alberto A. Pinto and David A. Rand

x_i

..

.

0i

ga_ii₍₀

i)

gi(0i)

S

g_iai+1_(0i)

p_g i

H

S_ia_i+1

j_l,r : J_l,ra R

i : I a R

0 R I_rR

J_rR=p_g

i(Ir R

) Jl

L =p_g(I_lL )

p_g i(cl

L ₎

I l L

p_g

i(0i) p_g

i(cr

R ₎

crR

cLl

^

S_ia_i

S_i,Ha_i+1

Si,H a_i

Si,H 0

S_i,H1

...

U

RiSi=S_i,H

0

S_i,Ha_i+1

Fig. 1.1: The horocycleHi and the chart jl,r:Jl,r→Rin case (ii). The junctionξi of the horocycle is equal toξi=πgi(gi(0i)) =· · ·πgi(g

ai

i (0i)) =πgi(g ai+1 i (0i)).

1.3 Renormalization

Therenormalization of a C1+circle diffeomorphism giis the triple(Rigi,RiSi,RiAi) where (i)RiSiis the renormalized circle with the orientation of the horocycleHi, i.e. the reversed orientation of the orientation induced bySi; (ii) therenormalized atlas RiAi=AiH|RiSi is the set of all charts inA

H

i with domains contained inRiSi; and (iii)Rigi:RiSi→RiSiis the continuous map given by

Rigi(x) =



 

 

πgi◦gaii +1◦

πgi|_S0

i,H

−1

(x)i f x∈S0_i,H

πgi◦gi◦

πgi|_Sai+1

i,H

−1

(x) i f x∈Sai_i,H+1

.

We denote theC1+renormalization(Rigi,RiSi,RiAi)ofgibyRigi.

By construction, the renormalizationRig_iof the rigid rotationg_iis affine conju-gate to the rigid rotationg

i+1. Hence, from now on, we identify(Rig_i,RiSi,RiAi) with(g

i+1,Si+1,Ai+1).

Recall that aC1+circle diffeomorphism g:S_i→S_{i is aC}1+α _{diffeomorphism}

with respect to aC1+α atlas A _on S_{i, for some} α _{>0, that is quasi-symmetric}

conjugate to a rigid rotationg:S_i→S_iwith respect to an affine atlasA onS

i. The renormalizationRigiis aC1+circle diffeomorphism quasi-symmetric conju-gate to the rigid rotationg

circle diffeomorphismgi+1. The marked point 0i∈Sidetermines the marked point 0RiS_i =πgi(0i)in the circleRiSi. Thus, there is a unique topological conjugacyhi betweenRigiandgi+1such thathi(0RiS_i) =0i₊1.

.

.

.

.

..

S

gaii+2(0i)

g_ia_{i+1 g}

i

R

g

~

g

i+1

0i+1

S

R

S

.

.

.

.

.

.

gi+1

pgi(0i) pgi(giai+2(0i)) RiSi

pgi

g_i(0i)

g_i+1(0i+1)

pgi+1

H

Ri+1Si+1

S1

i+1,H pgi+1(0i+1)

H

pgi+1(gi+1

ai+1+1₍₀

i+1))

x_i x_i+1

ga_i+1+1

i+1

g_iai+1₍₀

i)

g_iai₍₀

i) Si a_i S0 i Si a_i+1 Si,H a_i Si,H 1

...

...

i+1 (0i+1)

ai+1+2

g_i+1 (0i+1)

ai+1+1

Si+1

ai+1+1

Si+1

ai+1

g_i+1ai+1(0_i+1)

Fig. 1.2: The horocyclesHiandHi+1, and the renormalized mapRigi:RiSi→RiSi. Hereξi=gi(0i) =. . .=gai_i+1(0i),ξi+1=gi+1(0i+1) =. . .=gai+1+1

i+1 (0i+1))and the mapRigiis identified withgi+1.

AC1+circle diffeomorphismg0determines a uniqueC1+renormalization circle diffeomorphism sequence R(g0) = (gi,Si,Ai)∞i=0given by

(gi,Si,Ai) = (Ri◦. . .◦R0g0,Ri◦. . .◦R0S0,Ri◦. . .◦R0A0).

We note that theC1+renormalization circle diffeomorphism sequenceR(g0)is a C1+circle diffeomorphism sequence.

We say thata= (ai)∞_i=0a sequence isn-periodicifnis the least integer such that ai+n=ai, for everyi∈N0. We observe that given an-periodic sequence of positive integersa= (ai)∞i=0,γi=γi(a) =1/(ai+1/(ai+1+1/· · ·))is equal toγi+nfor every

i∈N0. Hence, there exists a topological conjugacyφ_i:S_i→S_i+nsuch that

6 João P. Almeida, Alberto A. Pinto and David A. Rand

becausegiandgi+n=Ri+n◦. . .◦RigiareC1+circle diffeomorphisms with the same rotation numberγi=γi+n.

We say that a sequenceR(g0)is aC1+n-periodic point of renormalization, ifφi isC1+for everyi∈N.

1.4 Markov maps

LetR(g0)be the renormalization circle diffeomorphism sequence associated to the C1+circle diffeomorphismg0. TheMarkov map Mi:Hi→Hi+1is given by

Mi(x) =

π

gi+1(x) i f x∈RiSi

πgi+1◦πgi◦g

−k

i ◦πgi−1(x)i f x∈Ski,Hi, for k=1, . . . ,ai

.

TheMarkov sequence(Mi)∞i=0(g0)associated to aC1+circle diffeomorphismg0 is the sequence of Markov mapsMi:Hi→Hi+1, fori∈N0. Two Markov sequences (Mi)∞i=0(g0)and(Ni)∞i=0(g0)arequasi-symmetric conjugateif there is a sequence

(hi)∞i=0of quasi-symmetric mapshisuch thatMi+1◦hi=hi+1◦Mi, for eachi∈N_0.

Therigid Markov sequence(M_i)∞_i=0= (Mi)∞i=0(g₀)is the Markov sequence as-sociated to the rigid rotationg₀. The rigid Markov mapsM_i:H_i→H_i+1are affine with respect to the atlasesAH_i andAH_i+1(see Figure 1.3).

The Markov sequence(Mi)∞i=0(g0)has the following properties: (i) the Markov mapsMi are localC1+α diffeomorphisms, for someα >0, and (ii) the Markov sequence(Mi)∞i=0(g0)is quasi-symmetric conjugate to the rigid Markov sequence

(M_i)∞_i=0(g₀)becausegiis quasi-symmetric conjugate tog_i.

Then-extended Markov sequence(Mi)n_i=−₀1(g0)is the sequence of then-extended Markov maps Mi(g0):Hi→Hidefined by

Mi(g0) =φi−1◦Mi+n◦ · · · ◦Mi.

We observe that a sequenceR(g0)is aC1+n-periodic point of renormalization if, and only if, then-extended Markov mapsMi:Hi→HiareC1+for everyi∈N.

Therigid n-extended Markov sequence(Mi)ni=−01= (Mi)ni=−01(g0)is then-extended Markov sequence associated to the rigid rotationg₀. The rigidn-extended Markov mapsM0, . . . ,Mn−1 are affine with respect to the atlases AH0, . . . ,AHn−1, respec-tively, because the conjugacy mapsφ_i:S

i→Si+nare affine.

Ifφi:Si→Si+nisC1+α then then-extended Markov sequence(Mi)n_i=−₀1(g0)has the following properties: (i) then-extended Markov mapsMiare localC1+α diffeo-morphisms, for someα>0, because the Markov mapsM0, . . .Mn−1of the sequence

(Mi)n_i=−01(g0)are localC1+α diffeomorphisms, and (ii) then-extended Markov maps

0

g

i+1

g

~

~

~

g

~

g

~

~

0

g

~

g2 i g-g

2 i i g

2

i gi gi

g

~

g

~

g

~

. . .

. . .

...

. . .

. . .

g

i+1

g2

i+1

g

i+1

Fig. 1.3: A representation of the rigid Markov mapM_i:H_i→H_i+1, with respect to the atlasesAH

i andAHi+1, respectively. Here we represent by ˜0i and ˜0i+1the pointsπg

i(0i)andπgi+1(0i+1), respectively, and by ˜g

l

k the points πgk◦g l

k(0k), for k∈ {i,i+1}andl∈ {1, . . . ,ak+1}.

Acknowledgements We thank the financial support of LIAAD–INESC TEC through program

PEst, USP-UP project, Faculty of Sciences, University of Porto, Calouste Gulbenkian Founda-tion, FEDER and COMPETE Programmes, PTDC/MAT/121107/2010 and Fundação para a Ciên-cia e a Tecnologia (FCT). J. P. Almeida acknowledges the FCT support given through grant SFRH/PROTEC/49754/2009.

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