# Chapter 1 Renormalization of circle diffeomorphism sequences and Markov sequences

Livre

**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 n* -periodic points of renormalization and smooth Markov

### 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-

1+

to-one correspondence between *C* conjugacy classes of Anosov diffeomorphisms

1+ 1+

and pairs of *C* circle diffeomorphism sequences that are *C n* -periodic points

of renormalization (see also [1, 8, 9, 19]). The main point in this paper is to show

1+

the existence of a one-to-one correspondence between *C* circle diffeomorphism

1+

sequences that are *C 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. AlmeidaLIAAD-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. PintoLIAAD-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.## 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 **= (*a* ) be a sequence of positive integers and let γ + (**a**) = 1/(*a* + 1/(*a*

1

*i* *i* =0

1/···)). For every *i *∈ N , let γ = γ (**a**) = 1/(*a* + 1/(*a* +1 + 1/ · · · )) and let S be a

*i i i i i* *counterclockwise oriented circle* homeomorphic to the circle S = R/(1 + γ )Z. *i* *i*

An *arc *in S is the image of a non trivial interval *I *in R by a homeomorphism

*i*

α : *I *→ S . If *I *is closed (resp. open) we say that α (*I*) is a *closed *(resp. *open*) *arc *in

*i*

S

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

*i*

1+

on S starting at the point *a *∈ S and ending at the point *b *∈ S . A *C* atlas A in

*i i i i*

S

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

*i* *i*

1+ α some chart in A , and (ii) the overlap maps are *C* compatible, for some α > 0.

*i*

Let A denote the affine atlas whose charts are isometries with respect to the

*i*

usual norm in S . Let the *rigid rotation g* : S be the affine homeomorphism,

*i i i*

*i*

*i i* *i*

A homeomorphism *h *: S is *quasisymmetric *if there exists a constant *C *> 1 → S

*i* *i*

such that for each two arcs *I* and *I* of S with a common endpoint and such that

1

2

*i*

, we have |*h*(*I* |*I* 1 | = |*I* 2 | 1 )| /|*h*(*I* 2 )| < *C*, where the lengths are measured in the

*i i i i* charts of A and A . *i i*

1+ ∞

A *C* is a sequence of triples *circle diffeomorphism sequence* (*g* , S , A )

*i i i* *i* =0

1+ α

: S is a *C* diffeomorphism, (*g* , S , A ) with the following properties: (i) *g* → S

*i i i i i i*

1+ α

with respect to the *C* atlas A , for some is quasi-symmetric

α > 0; and (ii) *g*

*i i* conjugate to the rigid rotation *g* with respect to the atlas A . *i* *i*

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

*i i i i* denote the rigid rotation (*g* .

, S , A ) by *g*

*i i* *i i*

**1.2.1 Horocycles**

Let us mark a point in S that we will denote by 0 ∈ S . Let *S* = [0 , *g* (0 )] be the

*i i i i i i* *i*

oriented closed arc in S , with endpoints 0 and *g* (0 ). For every *k *∈ {0, . . . , *a* },

*i i i i i*

- 1

*k k k* *k*

let *S* = *g* (0 ), *g* (0 ) be the oriented closed arc in S with endpoints *g* (0 )

*i i i i* *i i i* *i*

- 1 −1
*a i*+1*a i*+1

*k k k k*

and *g* (0 *i* ) and such that *S* ∩ *S* = {*g* (0)}. Let *S* = [*g* (0 *i* ), 0 *i* ] be the

*i i i i i i*

- 1

*a i* oriented closed arc in S , with endpoints *g* (0 ) and 0 . *i i i* *i*

We introduce an *equivalence relation *∼ in S by identifying the *a* + 1 points

*i i*

- 1

*a i*

*g* (0), . . . , *g* (0) and form the topological space *H* (S , *g* ) = S / ∼. We take the

*i* *i i i i* *i*

orientation in *H* as the reverse orientation of the one induced by S . We call this

*i* *i*

oriented topological space the *horocycle *and we denote it by *H* = *H* (S , *g* ). We

*i i i i*

consider the quotient topology in *H* . Let π : S → *H* be the natural projection. The

*i g i i i*

## 1 Renormalization of circle diffeomorphism sequences and Markov sequences

3 point

- 1

*a i*

= (*g* (0 )) = · · · = (*g* (0 )) ∈ *H*

ξ *i* π *g i i* π *g i i*

*i i*

*i* *k*

is called the *junction *of the horocycle *H* . For every *k *∈ {0,...,*a* }, let *S* =

*i i*

,*H*

*i*

- 1

*k* *k a*

S *S* be the projection by of the closed arc *S* . Let *R*

(S *i* , *g i* ) ⊂ *H i* π *g i i* = *S* ∪ *S*

,*H i* ,*H* ,*H*

*i* *i i i*

be the *renormalized circle *in *H* . The horocycle *H* is the union of the renormalized

*i i* *k*

*i i i*

S circle *R* with the circles *S* for every *k *∈ {1,...,*a* }.

,*H*

*i*

A *parametrization *in *H* is the image of a non trivial interval *I *in R by a homeo-

*i*

morphism : *I *→ *H* . If *I *is closed (resp. open) we say that (*I*) is a *closed *(resp.

α *i* α

*open* ) *arc *in *H* . A *chart *in *H* is the inverse of a parametrization. A *topological atlas*

*i i*

*i i*

B on the horocycle *H* is a set of charts {( *j*,*J*)}, on *H* , with the property that every

small arc is contained in the domain of a chart in B, i.e. for any open arc *K *in *H*

*i*

and any *x *∈ *K *there exists a chart {( *j*,*J*)} ∈ B such that *J *∩ *K *is a non trivial open

arc in *H* and *x *∈ *J *∩ *K*. A *C atlas *B *in H* is a topological atlas B such that the

*i i*

1+ 1+ α α

overlap maps are *C* and have *C* uniformly bounded norms, for some α > 0.

1+ 1+ Let A be a *C* atlas on S in which *g* : S → S is a *C* circle diffeomorphism.

*i i i i i*

1+

*H*

We are going to construct a *C* atlas A on *H* that we call the *extended push-*

*i* *i* *H*

A

*forward *A = ( π ) of the atlas A on S . If *x *∈ *H* \{ ξ } then there exists a

*g i i i i i i* *i* ∗

−1

sufficiently small open arc *J *⊂ *H* containing *x *and such that π (*J*) is contained in

*i* *g i*

−1

the domain of some chart (*I*, ˆ ) of A . In this case, we define (*J*, ˆ ◦ ) as a chart

ι *i* ι π

*g*

*i* −1

*H*

in A . If *x *= ξ and *J *is a small arc containing ξ , then either (i) π (*J*) is an arc

*i i* *i* *g i*

−1

in S or (ii) (*J*) is a disconnected set that consists of a union of two connected

*i* π *g*

*i* components.

−1 In case (i), (*J*) is connected and it is contained in the domain of some chart

π

*g*

*i*

−1

(*I*, ˆ ι ) ∈ A ι ◦ π

*H* . Therefore we define *J*, ˆ as a chart in A .

*i* *g i i*

−1 In case (ii), (*J*) is a disconnected set that is the union of two connected arcs

π

*g*

*i*

*L R L L l R r R*

*I* and *I* of the form *I* = (*c* , *g* (0)] and *I* = [*g* (0), *c* ), respectively, for all *l*, *r *∈

*l r l l i r i r* *L R L L R*

{1, . . . , *a* + 1}. Let *J* and *J* be the arcs in *H* defined by *J* = π (*I* ) and π (*I* )

*i i g i g i* *l r l l r* *L R L R*

respectively. Then *J *= *J* ∪ *J* is an arc in *H* with the property that *J* ∩ *J* = { ξ },

*i i* *l r l r*

for every *l*,*r *∈ {1,...,*a*

- 1}. We call such arc
*J*a (*l*,*r*)*-arc*and we denote it by

*i*

*J* . Let *j* : *J* ,*r* ,*r* ,*r* → R be defined by,

*l l l*

−1 *R* ˆ ◦ (*x*) if *x *∈ *J* ι π

*g r*

*i*

*j* (*x*) = .

*l* ,*r* −*l* −1 *r L*

ˆ ι ◦ *g* ◦ π (*x*) if *x *∈ *J*

*i* *g i* *l*

Let (*I*, ˆ ) ∈ A be a chart such that (*I*) ⊃ *J* . Then we define (*J* , *j* ) as a chart

*i*π

*g*,

*r*,

*r*,

*r*

*i l l l*

*H*

in A (see Figure 1.1). We call the atlas determined by these charts the *extended*

*i* *H*

*pushforward atlas of *A and, by abuse of notation, we will denote it by A =

*i* *i*

π *g i* *i*

A ( ) .

∗ Let the marked point 0 in S be the natural projection of 0 ∈ R onto 0 ∈ S =

*i i i i* *k k k* +1

R /(1 + )Z. Let *S* = [0 , *g* (0 )] and *S* = [*g* (0 ), *g* (0 )]. Furthermore, let γ *i i i i i*

*i i* *i i i* *k k a* +1 H

i A

S

*H* = *H* (S , *g* ) , *S* = *S* (S , *g* ) , *R* = S ∪ S and A = π .

*i i* g i *i i i* ,*H i* ,*H i i i* ,*H i* ,*H* *i i* i

∗

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

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

**H****S U** **i**

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

**S**

**)**

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

**i**

**i** **S****i**

**1****R****c**

**S****S**

**r i,H i,H**

**a**_{i}

**R****a**

**S****...** **i,H**

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

**(** **c**** )** **S**

**~** **i p****g** **r****=** **i**

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

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

**L L ****(0** **i****)**

**J****=** **p****g** **)** **i****pg( I**

**R R l l****=**

**L J****( I )**

**pg**

**L r r****i**

**I c****l****l a**_{i+1}

**S****i,H****xi**

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

**R** **j ****a **

**: J**

**R**

**l,r l,r** **^****i ****: I a **

**R**

Fig. 1.1: The horocycle *H* and the chart *j* : *J* ,*r* ,*r* → R in case (ii). The junction ξ

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

of the horocycle is equal to ξ = π (*g* (0 )) = · · · π (*g* (0 )) = π (*g* (0 )).

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

### 1.3 Renormalization

1+ S A

The *renormalization of a C circle diffeomorphism g* is the triple (*R g* , *R* , *R* )

*i i i i i i i*

*i i* *i*

S

where (i) *R* is the renormalized circle with the orientation of the horocycle *H* , i.e.

the reversed orientation of the orientation induced by S ; (ii) the *renormalized atlas*

*i* *H H*

A S

*R* S is the set of all charts in A with domains contained in *R* ; and

*i i R i i* *i i* *i i*

*i i i i i i* _{}

S S (iii) *R g* : *R* is the continuous map given by → *R*

−1 _{} *a i* +1 ◦ *g* ◦ | (*x*) *i f x *∈ *S* _{ i i ,H} π *g* π *g*

*i S i*

*i* ,*H* −1

*R g* (*x*) = .

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

π ◦ *g* ◦ π | (*x*) *i f x *∈ *S*

*g i i g i ai*+1 *i* ,*H* *S*

*i* ,*H* 1+

*i i i i i i i i i*

S A We denote the *C* renormalization (*R g* , *R* , *R* ) of *g* by *R g* .

By construction, the renormalization *R g* of the rigid rotation *g* is affine conju-

*i* *i i*

A S

gate to the rigid rotation *g* . Hence, from now on, we identify (*R g* , *R* , *R* )

*i i i* *i i*

- 1

*i i*

with (*g* , S , A ).

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

1+ 1+

α Recall that a *C circle diffeomorphism g* : S → S is a *C* diffeomorphism

*i i*

1+ α

with respect to a *C* atlas A on S , for some α > 0, that is quasi-symmetric

*i* conjugate to a rigid rotation *g *: S → S with respect to an affine atlas A on S . *i i* *i*

1+ The renormalization *R g* is a *C* circle diffeomorphism quasi-symmetric conju-

*i i*

1+ gate to the rigid rotation *g* . Hence, *R g* is quasi-symmetric conjugate to the *C*

*i i*

## 1 Renormalization of circle diffeomorphism sequences and Markov sequences

5

circle diffeomorphism *g* . The marked point 0 ∈ S determines the marked point

*i* +1 *i i*

S S = (0 ) in the circle *R* . Thus, there is a unique topological conjugacy *h*

*R* π *g i i i* *i*

*i i i* S between *R g* and *g* such that *h* (0 ) = 0 .

*i i i* +1 *i R i* +1

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

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

_{i+1}

**g**

_{i+1 i+1 i+1}

**(0 ) g**

_{i+1 (0 )}

_{a +1}

**ai****+2**

^{i+1}

**a ai+1****g (0 )**

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

**g (0 )**

_{i}

**S i S**

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

_{i+1}

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

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

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

**S****i****.**

**ai****.**

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

**.** **.**

**S . S S**

**~**

**R****.**

**i i+1 i i****.**

**.** **..**

**1****p p**

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

**i+****g**

**~****i****g** **i**

**H****i+1**

**H**_{ai+2}

### i R

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

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

**i i R i+1 i+1**

_{a}

_{i+1}

**a**_{i}

**S**

**S i+1,H**

**i,H****p (0 )**

**g**

_{i}

_{i}

**...**

**...** _{1} **p (0 )** **g**

**S i+1**

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

**xi xi+1**

*i i* +1 *i i i i* → *R i i*

S S

Fig. 1.2: The horocycles *H* and *H* , and the renormalized map *R g* : *R* .

- 1
*a i*+1

*a i* +1

Here ξ = *g* (0 ) = . . . = *g* (0 ), ξ +1 = *g* +1 (0 +1 ) = . . . = *g* (0 +1 )) and the

*i i i i i i i i* *i* +1 *i* map *R g* is identified with *g* +1 . *i i i*

1+ 1+ A *C* circle diffeomorphism *g* determines a unique *C renormalization circle*

∞ *diffeomorphism sequence ***R** (*g* ) = (*g* , S , A ) given by

*i i i*

=0

*i*

*i i i i i i*

S A

(*g* , S , A ) = (*R* ◦ . . . ◦ *R g* , *R* ◦ . . . ◦ *R* , *R* ◦ . . . ◦ *R* ).

1+

We note that the *C* renormalization circle diffeomorphism sequence **R**(*g* ) is a

1+ *C* circle diffeomorphism sequence.

∞ We say that **a **= (*a* ) a sequence is *n-periodic *if *n *is the least integer such that

*i* *i* =0

*a* +*n* = *a* , for every *i *∈ N . We observe that given a *n*-periodic sequence of positive

*i i*

∞

integers **a **= (*a* ) , γ = γ (**a**) = 1/(*a* + 1/(*a* +1 + 1/ · · · )) is equal to γ +*n* for every

*i i i i i i* *i* =0

*i* ∈ N . Hence, there exists a topological conjugacy φ : S → S +*n* such that

*i i i*

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

1+

because *g* and *g* = *R* ◦ . . . ◦ *R g* are *C* circle diffeomorphisms with the same

*i i* +*n i* +*n i i* rotation number = .

γ *i* γ *i* +*n*

1+

We say that a sequence **R**(*g* ) is a *C n-periodic point of renormalization* , if

*i*

1+ is *C* for every *i *∈ N.

### 1.4 Markov maps

Let **R**(*g* ) be the renormalization circle diffeomorphism sequence associated to the

*C* circle diffeomorphism *g* . The *Markov map M* : *H* → *H* +1 is given by

*i i i*

S (*x*) *i f x* ∈ *R* π *g i i* *i* +1

*M* (*x*) = .

*i* −*k* −1 *k*

π ◦ π ◦ *g* ◦ π (*x*) *i f x *∈ *S* , for *k *= 1, . . . , *a*

*g i g i i*

- 1
*g i i*,*H*

*i i*

1+ ∞

The *Markov sequence *(*M* ) (*g* ) associated to a *C* circle diffeomorphism *g*

*i* *i* =0

is the sequence of Markov maps *M* : *H* , for *i *∈ N . Two Markov sequences

*H*+1

*i i i*

∞ ∞

(*M* ) (*g* ) and (*N* ) (*g* ) are *quasi-symmetric conjugate *if there is a sequence

*i i* *i* =0 *i* =0

∞ of quasi-symmetric maps *h* such that *M* , for each *i *∈ N .

(*h* ) +1 ◦ *h* = *h* +1 ◦ *M*

*i i i i i i* *i* =0

∞ ∞

The*rigid Markov sequence*(

*M*) = (

*M*) (

*g*) is the Markov sequence as-

*i* *i i* =0 *i* =0

sociated to the rigid rotation *g* . The rigid Markov maps *M* : *H* are affine

*H*

*i i i* +1 *H H* with respect to the atlases A and A (see Figure 1.3).

- 1

*i i*

∞ The Markov sequence (*M* ) (*g* ) has the following properties: (i) the Markov

*i* *i* =0

1+ α

maps *M* are local *C* diffeomorphisms, for some α > 0, and (ii) the Markov

*i*

∞

sequence (*M* ) (*g* ) is quasi-symmetric conjugate to the rigid Markov sequence

*i* *i* =0

∞ (*M* ) (*g* ) because *g* is quasi-symmetric conjugate to *g* .

*i* *i i* =0 *i*

−1

*n*

The *n-extended Markov sequence *(**M** ) (*g* ) is the sequence of the *n-extended*

*i*

=0

*i*

*Markov maps ***M** (*g* ) : *H* → *H* defined by

*i i i*

−1

**M** (*g* ) = ◦ *M* ◦ · · · ◦ *M* .

*i* φ *i* +*n i* *i*

1+ We observe that a sequence **R**(*g* ) is a *C n-periodic point of renormalization*

1+

if, and only if, the *n*-extended Markov maps **M** : *H* → *H* are *C* for every *i *∈ N.

*i i i* *n* −1 *n* −1

The *rigid n-extended Markov sequence *(**M** ) = (**M** ) (*g* ) is the *n*-extended

*i* *i* =0 =0 *i i*

Markov sequence associated to the rigid rotation *g* . The rigid *n*-extended Markov

*H H*

maps **M** , . . . , **M** are affine with respect to the atlases A , . . . , A , respec-

*n* −1 *n* −1 tively, because the conjugacy maps : S → S are affine.

φ

*i i* +*n* *i*

1+ *n* −1 α

If φ : S → S +*n* is *C* then the *n*-extended Markov sequence (**M** ) (*g* ) has

*i i i* *i*

=0

*i*

1+ α

the following properties: (i) the *n*-extended Markov maps **M** are local *C* diffeo-

*i*

morphisms, for some α > 0, because the Markov maps *M* , . . . *M* −1 of the sequence

*n* *n* −1 1+

α diffeomorphisms, and (ii) the *n*-extended Markov maps (*M* ) (*g* ) are local *C*

*i* *i* =0

are quasi-symmetric conjugate to the rigid *n*-extended Markov maps **M** because **M**

*i* *i* the Markov maps *M* are quasi-symmetric conjugate to *M* .

, . . . *M* −1 , . . . , *M*

*n* −1 *n*

## 1 Renormalization of circle diffeomorphism sequences and Markov sequences

7 **~** **g****i+1****g**_{i+1}

**2****~** **g****. . .** **i+1****...** **a +1**_{i+1}

**~** **g****. . .** **i+1**_{2} **g**_{i+1}

**~** **. . .** **i+1****g**_{i+1}

**2**

**2**

**2****g g-g g g g**

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

**2****~** **~ ~ ~ ~** **g g g g****. . .** **i i i i****i**

Fig. 1.3: A representation of the rigid Markov map *M* : *H* → *H* , with respect

- 1

*i i i* *H H*

to the atlases A and A , respectively. Here we represent by ˜0 and ˜0 +1 the

- 1
*i i*

*i i* *l l*

points π (0 ) and π (0 ), respectively, and by ˜*g* the points π ◦ *g* (0 ), for

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

*i i k k*

- 1
*k*

*k* ∈ {*i*, *i *+ 1} and *l *∈ {1, . . . , *a* + 1}.

*k*

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

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**

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