**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 are*C*1+ *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 between*C*1+conjugacy classes of Anosov diffeomorphisms
and pairs of*C*1+circle diffeomorphism sequences that are*C*1+*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 between*C*1+circle diffeomorphism
sequences that are*C*1+*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**

Let**a**= (*ai*)∞*i*=0be a sequence of positive integers and letγ(a) =1/(*a*0+1/(*a*1+
1/· · ·)). For every*i*∈N0, letγ_{i}_{=}γ_{i}_{(a) =}1_{/}_{(}*ai*+1/(*ai*+1+1/· · ·))and letS*i*be a
*counterclockwise oriented circle*homeomorphic to the circleS_{i}_{=}R_{/}_{(}1_{+}γ_{i}_{)}Z.

An*arc*inS* _{i}*is the image of a non trivial interval

*I*inRby a homeomorphism

α:*I*→S*i. IfI*is closed (resp. open) we say thatα(*I*)is a*closed*(resp.*open)arc*in

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

onS_{i}_{starting at the point}_{a}_{∈}S_{i}_{and ending at the point}_{b}_{∈}S* _{i. A}_{C}*1+

_{atlas}A

_{i}_{in}

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

some chart inA* _{i, and (ii) the overlap maps are}_{C}*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 the*rigid rotation g _{i}* :S

*i*→S

*i*be the affine homeomorphism, with respect to the atlasA

*i*, with rotation numberγ*i*/(1+γ*i*).
A homeomorphism*h*:S

*i*→S*i*is*quasisymmetric*if there exists a constant*C*>1
such that for each two arcs*I1*and*I2*of S

*i*with 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_{i}_{and}A_{i.}

A*C*1+ *circle diffeomorphism sequence* (*gi*,S*i*,A*i*)∞*i*=0 is a sequence of triples

(*gi*,S*i*,A*i*)with the following properties: (i)*gi*:S*i*→S*i*is a*C*1+αdiffeomorphism,
with respect to the*C*1+α atlasA*i, for some*α_{>}0; and (ii)*gi*is quasi-symmetric
conjugate to the rigid rotation*g*

*i*with respect to the atlasA*i*.

We denote the*C*1+ circle diffeomorphism (*gi*,S*i*,A*i*)by *gi. In particular, we*
denote the rigid rotation(*g*

*i*,S*i*,A*i*)by*gi*.

**1.2.1 Horocycles**

**1.2.1 Horocycles**

Let us mark a point inS_{i}_{that we will denote by 0i}_{∈}S* _{i. Let}_{S}*0

*i* = [0i,*gi*(0i)]be the
oriented closed arc in S*i, 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 endpoints*gki*(0i)
and*gk _{i}*+1(0i)and such that

*Sk*∩

_{i}*S*−1={

_{i}k*gk*(0)}. Let

_{i}*Sai*+1

*i* = [*g*
*ai*+1

*i* (0i),0i]be the
oriented closed arc inS*i, with endpoints _{g}ai*+1

*i* (0i)and 0i.

point

ξ*i*=π*gi*(*gi*(0i)) =· · ·=π*gi*(*g*
*ai*+1

*i* (0i))∈*Hi*

is called the *junction* of the horocycle *Hi. For every* *k*∈ {0, . . . ,*ai*}, let *Ski*,*H* =
*Sk _{i}*

_{,}

*(S*

_{H}

_{i}_{,}

_{g}_{i}_{)}

_{⊂}

_{H}_{i}_{be the projection by}π

_{gi}_{of the closed arc}

_{S}k*i*. Let*Ri*S*i*=*S*0*i*,*H*∪*S*
*a*+1
*i*,*H*
be the*renormalized circle*in*Hi. The horocycleHi*is the union of the renormalized
circle*Ri*S*i*with the circles*Sik*,*H*for every*k*∈ {1, . . . ,*ai*}.

A*parametrization*in*Hi*is the image of a non trivial interval*I*inRby a

homeo-morphismα:*I*→*Hi. IfI*is closed (resp. open) we say thatα(*I*)is a*closed*(resp.
*open)arc*in*Hi. Achart*in*Hi*is the inverse of a parametrization. A*topological atlas*

Bon the horocycle*Hi*is a set of charts{(*j*,*J*)}, on*Hi, with the property that every*
small arc is contained in the domain of a chart inB, i.e. for any open arc*K*in*Hi*
and any*x*∈*K*there exists a chart{(*j*,*J*)} ∈Bsuch that*J*∩*K*is a non trivial open
arc in*Hi*and*x*∈*J*∩*K. AC*1+*atlas*B*in Hi*is a topological atlasBsuch that the
overlap maps are*C*1+α and have*C*1+αuniformly bounded norms, for someα>0.

LetA* _{i}*be a

*C*1+atlas onS

*in which*

_{i}*gi*:S

*i*→S

*i*is a

*C*1+circle diffeomorphism. We are going to construct a

*C*1+ atlasA

*H*

*i* on*Hi* that we call the*extended *
*push-forward*A*H*

*i* = (π*gi*)_{∗}A*i* of the atlasA*i* on S*i. Ifx*∈*Hi*\{ξ*i*} then there exists a
sufficiently small open arc*J*⊂*Hi*containing*x*and such thatπ*g*−*i*1(*J*)is contained in
the domain of some chart(*I*,ιˆ)ofA*i. In this case, we define*_{(}*J*,ιˆ◦π−1

*gi* )as a chart
inA*H*

*i* . If*x*=ξ*i*and*J*is a small arc containingξ*i, then either (i)*π*gi*−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 inA*H*
*i* .
In case (ii),π−1

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

*l* and*IrR*of the form*IlL*= (*cLl*,*gli*(0)]and*IrR*= [*gri*(0),*cRr*), respectively, for all*l*,*r*∈
{1, . . . ,*ai*+1}. Let*J _{l}L*and

*JrR*be the arcs in

*Hi*defined by

*J*=π

_{l}L*gi*(

*I*

*L*

*l*)andπ*gi*(*I*
*R*
*r*)
respectively. Then*J*=*JL*

*l* ∪*JrR*is an arc in*Hi*with the property that*J _{l}L*∩

*JrR*={ξ

*i*}, for every

*l*,

*r*∈ {1, . . . ,

*ai*+1}. We call such arc

*J*a(

*l*,

*r*)

*-arc*and we denote it by

*Jl*,

*r. Let*

*jl*,

*r*:

*Jl*,

*r*→Rbe defined by,

*jl*,*r*(*x*) =

_{ˆ}ι

◦π−1

*gi* (*x*) if*x*∈*J*
*R*
*r*
ˆ

ι◦*gir*−*l*◦π*g*−*i*1(*x*)if*x*∈*J*
*L*
*l*

.

Let(*I*,ιˆ)∈A_{i}_{be a chart such that}π_{gi}_{(}_{I}_{)}_{⊃}_{J}_{l}_{,}_{r. Then we define}_{(}_{J}_{l}_{,}_{r}_{,}_{j}_{l}_{,}_{r}_{)}_{as a chart}

inA*H*

*i* (see Figure 1.1). We call the atlas determined by these charts the*extended*
*pushforward atlas of* A* _{i}* and, by abuse of notation, we will denote it by A

*H*

*i* =

(π*gi*)∗A*i.*

Let the marked point 0* _{i}*inS

*i*be the natural projection of 0∈Ronto 0*i*∈S*i*=

R_{/}_{(}1_{+}γ_{i}_{)}Z. Let*S*0* _{i}* = [0i,

*g*

*i*(0i)]and*S*
*k*

*i* = [*gk _{i}*(0i),

*gk*+1(0i)]. Furthermore, let

_{i}*H _{i}*=

*Hi*(S

*i*,

*g*),

_{i}*Ski*,

*H*=

*Ski*,

*H*(S

*i*,

*g*),

_{i}*Ri*S

*i*=S

*i*0,

*H*∪S

*ai*,+

*H*1andA

Hi_{=}π_{g}
i

∗

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

**x**_{i}

**..**

**.**

**0****i**

**g****a**_{i}**i**_{(0}

**i****)**

**g****i****(0****i****)**

**S**

_{i}**g**_{i}**a****i+1**_{(0}_{i}_{)}

**p**_{g}**i**

**H**

**H**

_{i}

**S****0**

**i****~**

**=**

**S**_{i}**a**_{i+1}

**j**_{l,r }**: ****J**_{l,r}**a R**

**i ****: ****I ****a R**

**0** ** R**
**I**_{r}**R**

**J**_{r}**R****=p _{g}**

**i****(****I****r****R **

**)** **J****l**

**L****=p _{g(}**

**I**_{l}**L****)**

**p**_{g}**i****(****c****l**

**L **_{ )}

**I****l****L**

**p**_{g}

**i****(0****i****)**
**p**_{g}

**i****(****c****r**

**R**_{ )}

**c****r****R**

**c****L****l**

**^**

**S**_{i}**a**_{i}

**S**_{i,H}**a**_{i+1}

**S****i,H****a**_{i}

**S****i,H****0**

**S**_{i,H}**1**

**...**

**U**

**R****i****S****i****=****S**_{i,H}

**0**

**S**_{i,H}**a**_{i+1}

Fig. 1.1: The horocycle*Hi* and the chart *jl*,*r*:*Jl*,*r*→Rin case (ii). The junctionξ*i*
of the horocycle is equal toξ*i*=π*gi*(*gi*(0i)) =· · ·π*gi*(*g*

*ai*

*i* (0i)) =π*gi*(*g*
*ai*+1
*i* (0i)).

**1.3 Renormalization**

The*renormalization of a C*1+*circle diffeomorphism gi*is the triple(*Rigi*,*Ri*S*i*,*Ri*A*i*)
where (i)*Ri*S*i*is the renormalized circle with the orientation of the horocycle*Hi, i.e.*
the reversed orientation of the orientation induced byS*i; (ii) therenormalized atlas*
*Ri*A*i*=A*iH*|*Ri*S*i* is the set of all charts inA

*H*

*i* with domains contained in*Ri*S*i; and*
(iii)*Rigi*:*Ri*S*i*→*Ri*S*i*is the continuous map given by

*Rigi*(*x*) =

π*gi*◦*gaii* +1◦

π*gi*|* _{S}*0

*i*,*H*

−1

(*x*)*i f x*∈*S*0_{i}_{,}_{H}

π*gi*◦*gi*◦

π*gi*|* _{S}ai*+1

*i*,*H*

−1

(*x*) *i f x*∈*Sai _{i}*

_{,}

*+1*

_{H}.

We denote the*C*1+renormalization(*Rigi*,*Ri*S*i*,*Ri*A*i*)of*gi*by*Rigi.*

By construction, the renormalization*Rig _{i}*of the rigid rotation

*g*is affine conju-gate to the rigid rotation

_{i}*g*

*i*+1. Hence, from now on, we identify(*Rig _{i}*,

*Ri*S

*i*,

*Ri*A

*i*) with(

*g*

*i*+1,S*i*+1,A*i*+1).

Recall that a*C*1+*circle diffeomorphism g*:S_{i}_{→}S_{i}_{is a}* _{C}*1+α

_{diffeomorphism}

with respect to a*C*1+α atlas A _{on} S* _{i, for some}* α

_{>}

_{0, that is quasi-symmetric}

conjugate to a rigid rotation*g*:S_{i}_{→}S* _{i}*with respect to an affine atlasA onS

*i*.
The renormalization*Rigi*is a*C*1+circle diffeomorphism quasi-symmetric
conju-gate to the rigid rotation*g*

circle diffeomorphism*gi*+1. The marked point 0i∈S*i*determines the marked point
0*Ri*S* _{i}* =π

*gi*(0i)in the circle

*Ri*S

*i. Thus, there is a unique topological conjugacyhi*between

*Rigi*and

*gi*+1such that

*hi*(0

*Ri*S

*) =0i*

_{i}_{+}1.

**.**

**.**

**.**

**.**

**..**

**0**

**i****S**

_{i}**g****ai****i****+2(0****i****)**

**g**_{i}**a**_{i+1}_{g}

**i**

**R**

**R**

**i****g**

**i****~**

**g**

**i+****1**

**0****i+1**

**S**

_{i+1 }**~**

**R**

**R**

**i****S**

**i****.**

**.**

**.**

**.**

**.**

**.**

**S****0**

**i+1****g****i****+****1**

**pg****i****(0****i****)**
**pg****i****(g****i****ai+2****(0****i****))**
**R****i****S****i**

**pgi**

**g**_{i}**(0****i****)**

**g**_{i+1}**(0****i+1****)**

**pgi+1**

**H**

**H**

_{i+1}**R****i+1****S****i+1**

**S****1**

**i+1,H****pg****i+1****(0****i+1****)**

**H**

**H**

**i****pg****i+1****(g****i+1**

**a****i+1+1**_{(0}

**i+1****))**

**x**_{i}**x**_{i+1}

**g****a**_{i+1+1}

**i+1**

**g**_{i}**ai+1**_{(0}

**i****)**

**g**_{i}**ai**_{(0}

**i****)**
**S****i****a**_{i}**S****0****i****S****i****a**_{i+1}**S****i,H****a**_{i}**S****i,H****1**

**...**

**...**

**S**

**i+1,H**

**a****i+1****g**

*i+1 ***(0****i+1****)**

**a****i+1****+2**

**g**_{i+1}_{ }**(0****i+1****)**

**a****i+1****+1**

**S****i+1**

**a****i+1****+1**

**S****i+1**

**a****i+1**

**g**_{i+1}**a****i+1****(0**_{i+1}**)**

Fig. 1.2: The horocycles*Hi*and*Hi*+1, and the renormalized map*Rigi*:*Ri*S*i*→*Ri*S*i.*
Hereξ*i*=*gi*(0i) =. . .=*gai _{i}*+1(0i),ξ

*i*+1=

*gi*+1(0i+1) =. . .=

*gai*+1+1

*i*+1 (0i+1))and the
map*Rigi*is identified with*gi*+1.

A*C*1+circle diffeomorphism*g*0determines a unique*C*1+*renormalization circle*
* diffeomorphism sequence R*(

*g*0) = (

*gi*,S

*i*,A

*i*)∞

*i*=0given by

(*gi*,S*i*,A*i*) = (*Ri*◦. . .◦*R0g0*,*Ri*◦. . .◦*R0*S0,*Ri*◦. . .◦*R0*A0).

We note that the*C*1+renormalization circle diffeomorphism sequence**R(***g0*)is a
*C*1+circle diffeomorphism sequence.

We say that**a**= (*ai*)∞_{i}_{=}0a sequence is*n-periodic*if*n*is the least integer such that
*ai*+*n*=*ai, for everyi*∈N0. We observe that given a*n-periodic sequence of positive*
integers**a**= (*ai*)∞*i*=0,γ*i*=γ*i*(a) =1/(*ai*+1/(*ai*+1+1/· · ·))is equal toγ*i*+*n*for every

*i*∈N0. Hence, there exists a topological conjugacyφ* _{i}*:S

_{i}_{→}S

_{i}_{+}

*such that*

_{n}6 João P. Almeida, Alberto A. Pinto and David A. Rand

because*gi*and*gi*+*n*=*Ri*+*n*◦. . .◦*Rigi*are*C*1+circle diffeomorphisms with the same
rotation numberγ*i*=γ*i*+*n.*

We say that a sequence**R(***g*0)is a*C*1+*n-periodic point of renormalization, if*φ*i*
is*C*1+for every*i*∈N.

**1.4 Markov maps**

Let**R(***g*0)be the renormalization circle diffeomorphism sequence associated to the
*C*1+circle diffeomorphism*g*0. The*Markov map Mi*:*Hi*→*Hi*+1is given by

*Mi*(*x*) =

π

*gi*+1(*x*) *i f x*∈*Ri*S*i*

π*gi*+1◦π*gi*◦*g*

−*k*

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

.

The*Markov sequence*(*Mi*)∞*i*=0(*g*0)associated to a*C*1+circle diffeomorphism*g*0
is the sequence of Markov maps*Mi*:*Hi*→*Hi*+1, for*i*∈N0. Two Markov sequences
(*Mi*)∞*i*=0(*g0*)and(*Ni*)∞*i*=0(*g0*)are*quasi-symmetric conjugate*if there is a sequence

(*hi*)∞*i*=0of quasi-symmetric maps*hi*such that*Mi*+1◦*hi*=*hi*+1◦*Mi, for eachi*∈N_{0.}

The*rigid Markov sequence*(*M _{i}*)∞

_{i}_{=}0= (

*Mi*)∞

*i*=0(

*g*

_{0})is the Markov sequence as-sociated to the rigid rotation

*g*

_{0}. The rigid Markov maps

*M*:

_{i}*H*→

_{i}*H*

_{i}_{+}1are affine with respect to the atlasesA

*H*andA

_{i}*H*

_{i}_{+}

_{1}(see Figure 1.3).

The Markov sequence(*Mi*)∞*i*=0(*g*0)has the following properties: (i) the Markov
maps*Mi* are local*C*1+α diffeomorphisms, for someα >0, and (ii) the Markov
sequence(*Mi*)∞*i*=0(*g*0)is quasi-symmetric conjugate to the rigid Markov sequence

(*M _{i}*)∞

_{i}_{=}

_{0}(

*g*

_{0})because

*gi*is quasi-symmetric conjugate to

*g*.

_{i}The*n-extended Markov sequence*(M*i*)*n _{i}*

_{=}−

_{0}1(

*g0*)is the sequence of the

*n-extended*

*(*

**Markov maps M**i*g*0):

*Hi*→

*Hi*defined by

**M***i*(*g*0) =φ*i*−1◦*Mi*+*n*◦ · · · ◦*Mi*.

We observe that a sequence**R(***g*0)is a*C*1+*n-periodic point of renormalization*
if, and only if, the*n-extended Markov maps***M***i*:*Hi*→*Hi*are*C*1+for every*i*∈N.

The*rigid n-extended Markov sequence*(M*i*)*ni*=−01= (M*i*)*ni*=−01(*g*0)is the*n-extended*
Markov sequence associated to the rigid rotation*g*_{0}. The rigid*n-extended Markov*
maps**M**0, . . . ,**M***n*−1 are affine with respect to the atlases A*H*0, . . . ,A*Hn*−1,
respec-tively, because the conjugacy mapsφ* _{i}*:S

*i*→S*i*+*n*are affine.

Ifφ*i*:S*i*→S*i*+*n*is*C*1+α then the*n-extended Markov sequence*(M*i*)*n _{i}*

_{=}−

_{0}1(

*g0*)has the following properties: (i) the

*n-extended Markov maps*

**M**

*i*are local

*C*1+α diffeo-morphisms, for someα>0, because the Markov maps

*M*0, . . .

*Mn*−1of the sequence

(*Mi*)*n _{i}*

_{=}−01(

*g*0)are local

*C*1+α diffeomorphisms, and (ii) the

*n-extended Markov maps*

**0**

**0**

_{i+1}**g**

**g**

**2****i+1**

**g**

**g**

**~**

_{i+1}**~**

**~**

**g**

**g**

**~**

_{i+1}**a****i+1****+1****g**

**g**

**~**

_{i+1}**~**

**0**

**0**

_{i}**g**

**g**

**~**

_{i}**a****i****+1****g****2****i****g-g**

**2****i****i****g**

**2**

**i****g****i****g****i**

**g**

**g**

**~**

_{i}**2****g**

**g**

**~**

_{i}**g**

**g**

**~**

_{i}**. . .**

**. . .**

**...**

**. . .**

**. . .**

**g**

**i+1**

**g****2**

**i+1**

**g**

**i+1**

Fig. 1.3: A representation of the rigid Markov map*M _{i}*:

*H*→

_{i}*H*

_{i}_{+}

_{1}, with respect to the atlasesA

*H*

*i* andA*Hi*+1, respectively. Here we represent by ˜0i and ˜0i+1the
pointsπ*g*

*i*(0*i*)andπ*gi*+1(0*i*+1), respectively, and by ˜*g*

*l*

*k* the points π*gk*◦*g*
*l*

*k*(0*k*), for
*k*∈ {*i*,*i*+1}and*l*∈ {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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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. IHES**49**, 1979, 5-233.

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 with*C*1+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,**5**3 639-662 (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)