Chapter 1 Renormalization of circle diffeomorphism sequences and Markov sequences

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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

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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, letγi=γi(a) =1/(ai+1/(ai+1+1/· · ·))and letSibe a counterclockwise oriented circlehomeomorphic to the circleSi=R/(1+γi)Z.

AnarcinSiis 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

Si. We denote by(a,b)(resp.[a,b]) the positively oriented open (resp. closed) arc

onSistarting at the pointaSiand ending at the pointbSi. AC1+atlasAiin

Siis a set of charts such that (i) every small arc ofSiis contained in the domain of

some chart inAi, and (ii) the overlap maps areC1+αcompatible, for someα>0.

LetA

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

i. Let therigid rotation gi :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 ofAiandAi.

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 inSithat we will denote by 0iSi. LetS0

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

let Sk i =

gk

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

be the oriented closed arc in S

i with endpointsgki(0i) andgki+1(0i)and such thatSkiSik−1={gki(0)}. LetSai+1

i = [g ai+1

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

i (0i)and 0i.

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point

ξigi(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 = Ski,H(Si,gi)Hibe the projection byπgi of the closed arcSk

i. LetRiSi=S0i,HS 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:IHi. 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 anyxKthere exists a chart{(j,J)} ∈Bsuch thatJKis a non trivial open arc inHiandxJK. AC1+atlasBin Hiis a topological atlasBsuch that the overlap maps areC1+α and haveC1+αuniformly bounded norms, for someα>0.

LetAibe aC1+atlas onSiin 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. IfxHi\{ξi} then there exists a sufficiently small open arcJHicontainingxand such thatπgi1(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 . IfxiandJis a small arc containingξi, then either (i)πgi−1(J)is an arc inSi 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)∈Ai. Therefore we define J,ιˆπ−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}. LetJlLandJrRbe the arcs inHidefined byJlLgi(I

L

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

lJrRis an arc inHiwith the property thatJlLJrR={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) ifxJ R r ˆ

ι◦girl◦πgi1(x)ifxJ L l

.

Let(I,ιˆ)∈Aibe a chart such thatπgi(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 Ai and, by abuse of notation, we will denote it by AH

i =

gi)∗Ai.

Let the marked point 0iinS

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

R/(1+γi)Z. LetS0i = [0i,g

i(0i)]andS k

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

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

Hi=πg i

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

xi

..

.

0i

gaii(0

i)

gi(0i)

S

i

giai+1(0i)

pg i

H

i S0 i ~ =

Siai+1

jl,r : Jl,ra R

i : I a R

0 R IrR

JrR=pg

i(Ir R

) Jl

L =pg(IlL )

pg i(cl

L )

I l L

pg

i(0i) pg

i(cr

R )

crR

cLl

^

Siai

Si,Hai+1

Si,H ai

Si,H 0

Si,H1

...

U

RiSi=Si,H

0

Si,Hai+1

Fig. 1.1: The horocycleHi and the chart jl,r:Jl,r→Rin case (ii). The junctionξi of the horocycle is equal toξigi(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:RiSiRiSiis the continuous map given by

Rigi(x) =

 

 

πgigaii +1◦

πgi|S0

i,H

−1

(x)i f xS0i,H

πgigi

πgi|Sai+1

i,H

−1

(x) i f xSaii,H+1

.

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

By construction, the renormalizationRigiof the rigid rotationgiis affine conju-gate to the rigid rotationg

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

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

Recall that aC1+circle diffeomorphism g:SiSi is aC1+α diffeomorphism

with respect to aC1+α atlas A on Si, for some α >0, that is quasi-symmetric

conjugate to a rigid rotationg:SiSiwith respect to an affine atlasA onS

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

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

.

.

.

.

..

0i

S

i

gaii+2(0i)

giai+1 g

i

R

i

g

i

~

g

i+1

0i+1

S

i+1 ~

R

i

S

i

.

.

.

.

.

.

S0 i+1

gi+1

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

pgi

gi(0i)

gi+1(0i+1)

pgi+1

H

i+1

Ri+1Si+1

S1

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

H

i

pgi+1(gi+1

ai+1+1(0

i+1))

xi xi+1

gai+1+1

i+1

giai+1(0

i)

giai(0

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

...

...

S i+1,H ai+1 g

i+1 (0i+1)

ai+1+2

gi+1 (0i+1)

ai+1+1

Si+1

ai+1+1

Si+1

ai+1

gi+1ai+1(0i+1)

Fig. 1.2: The horocyclesHiandHi+1, and the renormalized mapRigi:RiSiRiSi. Hereξi=gi(0i) =. . .=gaii+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,γii(a) =1/(ai+1/(ai+1+1/· · ·))is equal toγi+nfor every

i∈N0. Hence, there exists a topological conjugacyφi:SiSi+nsuch that

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

becausegiandgi+n=Ri+n◦. . .◦RigiareC1+circle diffeomorphisms with the same rotation numberγii+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:HiHi+1is given by

Mi(x) =

π

gi+1(x) i f xRiSi

πgi+1◦πgig

k

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

.

TheMarkov sequence(Mi)∞i=0(g0)associated to aC1+circle diffeomorphismg0 is the sequence of Markov mapsMi:HiHi+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∈N0.

Therigid Markov sequence(Mi)∞i=0= (Mi)∞i=0(g0)is the Markov sequence as-sociated to the rigid rotationg0. The rigid Markov mapsMi:HiHi+1are affine with respect to the atlasesAHi andAHi+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

(Mi)∞i=0(g0)becausegiis quasi-symmetric conjugate togi.

Then-extended Markov sequence(Mi)ni=01(g0)is the sequence of then-extended Markov maps Mi(g0):HiHidefined 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:HiHiareC1+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 mapsφi:S

i→Si+nare affine.

Ifφi:Si→Si+nisC1+α then then-extended Markov sequence(Mi)ni=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)ni=−01(g0)are localC1+α diffeomorphisms, and (ii) then-extended Markov maps

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0

i+1

g

2

i+1

g

~

i+1

~

~

g

~

i+1ai+1+1

g

~

i+1

~

0

i

g

~

iai+1

g2 i g-g

2 i i g

2

i gi gi

g

~

i2

g

~

i

g

~

i

. . .

. . .

...

. . .

. . .

g

i+1

g2

i+1

g

i+1

Fig. 1.3: A representation of the rigid Markov mapMi:HiHi+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 πgkg 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.

References

1. Almeida, J. P., Fisher, A. M., Pinto, A. A. and Rand, D. A.: Anosov and circle diffeomor-phisms, Dynamics Games and Science I, Springer Proceedings in Mathematics (eds. M. Peixoto, A. Pinto and D. Rand), Springer Verlag, 11-23 (2011).

2. Arnol’d, V. I.: Small denominators I: on the mapping of a circle into itself.Investijia Akad. Nauk. Math., 25, 1, (1961), 21-96. Transl. A.M.S., 2nd series, 46, 213-284.

3. Coullet, P. and Tresser, C.: Itération d’endomorphismes et groupe de renormalisation. J. de Physique, C5 25 (1978).

4. Herman, M. R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rota-tions. Publ. IHES49, 1979, 5-233.

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

6. Masur, H.: Interval exchange transformations and measured foliations. The Annals of Math-ematics. 2nd Ser.,115(1), 169–200 (1982).

7. Penner, R. C. and Harer, J. L.:Combinatorics of Train-Tracks. Princeton University Press,

Princeton, New Jersey (1992).

8. Pinto, A. A., Almeida, J. P. and Portela, A.: Golden tilings. Transactions of the American Mathematical Society, 364 (2012), 2261-2280.

9. Pinto, A. A., Almeida, J. P. and Rand, D. A.: Anosov and renormalized circle diffeomor-phisms. pp. 1–33 (2012) (Submitted).

10. Pinto, A. A. and Rand, D. A.: Train-tracks withC1+self-renormalisable structures. Journal

of Difference Equations and Applications, 16(8), 945-962 (2010).

11. Pinto, A. A. and Rand, D. A.: Solenoid functions for hyperbolic sets on surfaces. Dynam-ics, Ergodic Theory and Geometry, Boris Hasselblat (ed.), MSRI Publications, 54 145-178 (2007).

12. Pinto, A. A. and Rand, D. A.: Rigidity of hyperbolic sets on surfaces. J. London Math. Soc., 71 2 481-502 (2004).

13. Pinto, A. A. and Rand, D. A.: Smoothness of holonomies for codimension 1 hyperbolic dy-namics. Bull. London Math. Soc., 34 341-352 (2002).

14. Pinto, A. A. and Rand, D. A.: Teichmüller spaces and HR structures for hyperbolic surface dynamics. Ergod. Th. Dynam. Sys, 22 1905-1931 (2002).

15. Pinto, A. A. and Rand, D. A.: Existence, uniqueness and ratio decomposition for Gibbs states via duality. Ergod. Th. & Dynam. Sys., 21 533-543 (2001).

16. Pinto, A. A. and Rand, D. A.: Characterising rigidity and flexibility of pseudo-Anosov and other transversally laminated dynamical systems on surfaces. Warwick Preprint 1995. 17. Pinto, A. A., Rand, D. A. and Ferreira, F.: Arc exchange systems and renormalization. Journal

of Difference Equations and Applications 16 (2010), no. 4, 347-371.

18. Pinto, A. A., Rand, D. A. and Ferreira, F.: Cantor exchange systems and renormalization. Journal of Differential Equations, 243 593-616 (2007).

19. Pinto, A. A. Rand, D. A., Ferreira, F.:Fine structures of hyperbolic diffeomorphisms. Springer

Monographs in Mathematics, Springer (2010).

20. Pinto, A. A. and Sullivan, D.: The circle and the solenoid. Dedicated to Anatole Katok On the Occasion of his 60th Birthday.DCDS A, 16 2 463-504 (2006).

21. Rand, D. A.: Existence, nonexistence and universal breakdown of dissipative golden invariant tori. I. Golden critical circle maps. Nonlinearity,53 639-662 (1992).

22. Veech, W.: Gauss measures for transformations on the space of interval exchange maps. The Annals of Mathematics, 2nd Ser. 115 2 201-242 (1982).

23. Yoccoz J. C.: Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Scient. Éc. Norm. Sup., 4 série, t,17(1984)

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