Chapter 2 Anosov and Circle

Joao˜ P. Almeida, Albert M. Fisher, Alberto A. Pinto and David A. Rand

Abstract We present an infinite dimensional space of C1+ smooth conjugacy classes of circle diffeomorphisms that are C1+ fixed points of renormalization. We exhibit a one-to-one correspondence between these C1+ fixed points of renormaliz- ation and C1+ conjugacy classes of Anosov diffeomorphisms.

2.1 Introduction

The link between Anosov diffeomorphisms and diffeomorphisms of the circle is due to D. Sullivan and E. Ghys through the observation that the holonomies of Anosov diffeomorphisms give rise to C1+ circle diffeomorphisms that are C1+ fixed points of renormalization (see also [1]). A. Pinto and D. Rand [21] proved that this observation gives an one-to-one correspondence between the corresponding smooth

Joao˜ P. Almeida LIAAD-INESC Porto LA; Departamento de Matematica,´ Escola Suoerior de Tecnologia e de Gestao,˜ Instituto Politecnico´ de Baganc¸a. Campus de Santa Apolonia,´ Ap. 1134, 5301-857 Braganc¸a, Portugal. e-mail: [email protected] Albert M. Fisher Departamento de Matematica,´ IME-USP, Caixa Postal 66281, CEP 05315-970 Sao˜ Paulo, Brasil. e-mail: afi[email protected] Alberto A. Pinto LIAAD-INESC Porto LA; Departamento de Matematica,´ Faculdade de Cienciasˆ da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007, Portugal; Departamento de Matematica´ e Centro de Matematica´ da Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal. e-mail: [email protected] D. A. Rand Warwick Systems Biology & Institute, University of Warwick, Coventry CV4 7AL, United Kingdom. e-mail: [email protected]

11 12 J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand conjugacy classes. A key object in this link is the smooth horocycle equipped with a hyperbolic Markov map.

2.2 Circle Difeomorphisms

Fix a natural number a N and let S be a counterclockwise oriented circle homeo- ∈ morphic to the circle S1 = R/(1 + γ)Z, where γ = ( a + √a2 + 4)/2 = 1/(a + − 1/(a + )). We note that if a = 1 then γ is the inverse of the golden number ··· (1 + √5)/2. A key feature of γ is that it satisfies the relation aγ + γ2 = 1. An arc in S is the image of a non trivial interval I in R by a homeomorphism α : I S. If I is closed (resp. open) we say that α(I) is a closed (resp. open) arc in → S. We denote by (a,b) (resp. [a,b]) the positively oriented open (resp. closed) arc in S starting at the point a S and ending at the point b S.A C1+ atlas A of S is ∈ ∈ a set of charts such that (i) every small arc of S is contained in the domain of some chart in A , and (ii) the overlap maps are C1+α compatible, for some α > 0. A C1+ circle is a triple (g,S,A ) where g : S S is a C1+α → diffeomorphism, with respect to the C1+α atlas A , for some α > 0, and g is quasi- symmetric conjugate to the rigid rotation r : S1 S1, with rotation number equal γ → to γ/(1 + γ). We denote by F the set of all C1+ circle diffeomorphisms (g,S,A ), with respect to a C1+ atlas A in S. In order to simplify the notation, we will denote the C1+ circle diffeomorphism (g,S,A ) only by g.

2.2.1 The Horocycle and Renormalization

pg S 0 SH SH H 0 U 3 S 3 g3(0) S SH 0 1

SH 2

S pg(0) ~ 2 g(0) = x

S1 g2(0)

Fig. 2.1 The horocycle H for the case a = 2. The junction ξ of the horocycle is equal to ξ = 2 3 πg(g(0)) = πg(g (0)) = πg(g (0)). 2 Anosov and Circle Diffeomorphisms 13

Let us mark a point in S that we will denote by 0 S, from now on. Let ∈ S0 = [0,g(0)] be the oriented closed arc in S, with endpoints 0 and g(0). For k k+1 k = 0,...,a let Sk = g (0),g (0) be the oriented closed arc in S, with endpoints k k+1 k a+1 g (0) and g (0) and such that Sk Sk 1 = g (0) . Let Sa+1 = g (0),0 be  ∩ − { } the oriented closed arc in S, with endpoints ga+1(0) and 0. We introduce an equi-   valence relation in S by identifying the a + 1 points g(0),...,ga+1(0) and form ∼ the oriented topological space H(S,g) = S/ . We call this oriented topological ∼ space the horocycle (see Figure 2.1) and we denote it by H = H(S,g). We consider the quotient topology in H. Let π : S H be the natural projection. The point g → ξ = π (g(0)) = = π (ga+1(0)) H is called the junction of the horocycle H. g ··· g ∈ For k = 0,...,a+1, let SH = SH (S,g) H be the projection by π of the closed arc k k ⊂ g Sk.A parametrization in H is the image of a non trivial interval I in R by a homeo- morphism α : I H. If I is closed (resp. open) we say that α(I) is a closed (resp. → open) arc in H.A chart in H is the inverse of a parametrization. A topological atlas B on the horocycle H is a set of charts ( j,J) , on the horocycle, 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 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 C1+ atlas B in H is a topological atlas B ∈ ∩ such that the overlap maps are C1+α and have C1+α uniformly bounded norms, for some α > 0.

pg S 0 SH U SH S 0 3 H 3 g3(0) S0

c SH 1 L J H 1 p (c) S S g 2 ~ 2 g(0) = pg(d) JL pg( ) R pg(0) 1 pg(J ) JR d 3 S 3 1 x g2(0)

j J a R i : I a R :

0 R

Fig. 2.2 The chart j : J R in case (ii). → 14 J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand

Let A be a C1+ atlas in S for which g is C1+. We are going to construct a C1+ H H atlas A in the horocycle that is the extended pushforward A = (πg) A of the atlas A in S. ∗ If x H ξ then there exists a sufficiently small open arc J in H, containing x, ∈ \{1 } such that πg− (J) is contained in the domain of some chart (I,i) in A . In this case, 1 H we define (J,i πg− ) as a chart in A . If x = ξ and J is a small arc containing ◦ 1 S 1 ξ, then either (i) πg− (J) is an arc in or (ii) πg− (J) is a disconnected set that 1 consists of a union of two connected components. In case (i), πg− (J) is connected and we define J,i π 1 as a chart in A H . In case (ii), π 1(J) is a disconnected ◦ g− g− JL JR JR = [gl( ),d) set that is the union of two connected arcs k and l of the form l 0 and JL = (c,gk(0)], respectively, for some k,l 0,...,a+1 with k = l (see Figure 2.2). k ∈ { } 6 Let (I,i) A be a chart such that I (c,d). We define j : J R as follows, ∈ ⊃ → i π 1(x), if x π ([gl(0),d)) j(x) = ◦ g− ∈ g . i gl k π 1(x), if x π ((c,gk(0)])  ◦ − ◦ g− ∈ g We call the atlas determined by these charts, the extended pushforward atlas of A H and, by abuse of notation, we will denote it by A = (πg) A . Let g = (g,S,A ) be a C1+ circle diffeomorphism with∗ respect to a C1+ atlas A in S. Let RS = SH SH be projection by π of the oriented close arc S S of S. 0 ∪ a+1 g 0 ∪ a+1 Let i S : S S S RS be the natural inclusion and let RA be the restriction, R 0 ∪ a+1 ⊂ → A RS, of the C1+ atlas A to RS. The renormalization of g = (g,S,A ) is the triple | (Rg,RS,RA ), where Rg : RS RS is the map given by (see Figure 2.3), → i S ga+1 = Rg i S, f or Rg i S(S ), R ◦ ◦ R | R 0 i S g = Rg i S, f or Rg i S(S )  R ◦ ◦ R | R a+1 For simplicity of notation, we will denote the renormalization of a C1+ circle diffeomorphism g, (Rg,RS,RA ), only by Rg. We recall that F denotes the set of all C1+ circle diffeomorphisms (g,S,A ) with respect to a C1+ atlas A in S. Lemma 2.1. The renormalization Rg of a C1+ circle diffeomorphism g F is a C1+ ∈ circle diffeomorphism, i.e. the map R : F F given by R(g) = Rg is well defined. → In particular, the renormalization Rrγ of the rigid rotation is the rigid rotation rγ . The proof of Lemma 2.1 is in [21]. The marked point 0 S determines a marked point 0 in the circle RS. Since Rg ∈ is homeomorphic to a rigid rotation, there exists h : S RS, with h(0) = 0, such → that h conjugates g and Rg.

Definition 2.1. If h : S RS is C1+, we call g a C1+ fixed point of renormalization. → We will denote by R the set of all C1+ circle diffeomorphisms g F that are C1+ ∈ fixed points of renormalization.

We note that the rigid rotation rγ , with respect to the atlas Aiso, is an affine fixed point of renormalization. Hence, r R. γ ∈ 2 Anosov and Circle Diffeomorphisms 15

S 0

4 S g (0) 3 g3(0) S0

g S ~ 2 g(0) =

S1 g2(0)

pg pg

H H RS S S H H = 0 U 3 RS S S = 0 U 3 Rg|iRS(S0)

SH SH 1 1

SH SH 2 2

pg(0) pg(0) 4 pg(g (0))

x x Rg|iRS(Sa+1)

Fig. 2.3 The renormalization Rg = (Rg,RS,RA ) of a C1+ circle diffeomophism g = (g,S,A ).

2.2.2 Markov Maps

Let g = (g,S,A ) be a C1+ circle diffeomorphism, with respect to a C1+ atlas A , and Rg = (Rg,RS,RA ) its renormalization. Let H = H(S,g) and RH = H(RS,Rg) be the horocycles determined by the C1+ circle diffeomorphisms g and Rg, respect- ively. Let A H and A RH be the atlas in the horocycles H and RH, that are the ex- tended pushforwards of the atlases A and RA , respectively. Let i : S H be the H → natural inclusion. Let h : S RS be the homeomorphism that conjugates g and Rg sending the → marked point 0 of g in the marked point 0 of Rg. Definition 2.2. The Markov map M associated to g F is the map M : H H g ∈ g → defined by

1 1 iH h− iRS iH− (x), i f x iH (S0 Sa+1) Mg(x) = 1 ◦ ◦ k ◦ 1 ∈ ∪ . i h i g i− (x), i f x i (S ), k = 1,...,a  H ◦ − ◦ RS ◦ − ◦ H ∈ H k

We observe that, in particular, the rigid Markov map Mrγ is an affine map with H 4 2 2 respect to the atlas Aiso. Noting that h(g (0)) = (Rg) (0) = g then Mg is as rep- 16 J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand resented in Figure 2.4. We observe that the identification in H of g(0) with g2(0) makes the Markov map Mg a local homeomorphism.

r (0) g

r2 (0) g

r3 (0) g 0

g2 g-g2 g2 g g r (0) r4 (0) 0 r3 (0) r2 (0) r (0) g g g g g

H Fig. 2.4 The Markov map Mrγ with respect to the atlas Aiso.

1+ Lemma 2.2. Let g F . The Markov map Mg associated to g is a C local diffeo- ∈ H morphism with respect to the atlas A = (πg) A if, and only if, the diffeomorphism g is a fixed point of renormalization. ∗

The proof of Lemma 2.2 is in [21].

2.3 Anosov Diffeomorphisms

Fix a positive integer a N and consider the Anosov automorphism G : T T ∈ a → given by G (x,y) = (ax + y,x), where T is equal to R2/(vZ wZ) with v = (γ,1) a × and w = ( 1,γ). Let π : R2 T be the natural projection. Let A and B be the − → 0 0 rectangles [0,1] [0,1] and [ γ,0] [0,γ] respectively. A M of × − × Ga Ga is given by A = π(A0) and B = π(B0) (see Figure 2.5). The unstable of Ga are the projection by π of the vertical lines in the plane, and the stable manifolds of Ga are the projection by π of the horizontal lines in the plane. A C1+ Anosov diffeomorphism G : T T is a C1+α diffeomorphism, with α > 0, → 1+α such that (i) G is topologically conjugate to Ga; (ii) the has a C uniformly hyperbolic splitting into a stable direction and an unstable direction (see [39]). We denote by G the set of all such C1+ Anosov diffeomorphisms with an absolutely continuous with respect to the Lebesgue measure. 2 Anosov and Circle Diffeomorphisms 17

A B

G 2(A) A

G (B) 2 v 2 g g B w

2 g p(0,0) g

Fig. 2.5 The Anosov automorphism G : T T. 2 →

If h is the between Ga and G, then a Markov partition MG of G is given by h(A) and h(B). Let d = dρ be the distance on the torus T, determined by a Riemannian metric ρ. We define the map Gι = G if ι = u, or G = G 1 if ι = s. For ι s,u and x T, we denote the local ι-manifolds through ι − ∈ { } ∈ x by ι n n W (x,ε) = y T : d(G− (x),G− (y)) ε, for all n 0 . ∈ ι ι ≤ ≥ By the Stable  Theorem (see [39]), these sets are respectively contained in the stable and unstable immersed manifolds

ι n ι n W (x) = Gι W Gι− (x),ε0 n 0 [≥  which are the image of C1+α immersions κ : R T, for some 0 < α 1 and ι,x → ≤ some small ε0 > 0. An open (resp. closed) ι-leaf segment I is defined as a subset of ι W (x) of the form κι,x(I1) where I1 is an open (resp. closed) subinterval (non-empty) in R. An ι-leaf segment is either an open or closed ι-leaf segment. The endpoints of an ι-leaf segment I = κι,x(I1) are the points κι,x(u) and κι,x(v) where u and v are the endpoints of I1. The interior of an ι-leaf segment I is the complement of its boundary. A map c : I R is an ι-leaf chart of an ι-leaf segment I if c is a → homeomorphism onto its image. 18 J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand 2.3.1 Spanning Leaf Segments

One can find a small enough ε0 > 0, such that for every 0 < ε < ε0 there is δ = δ(ε) > 0 with the property that, for all points w,z T with d(w,z) < δ, W u(w,ε) ∈ and W s(z,ε) intersect in a unique point that we denote by

[w,z] = W u(w,ε) W s(z,ε). ∩ A rectangle R is a subset of T which is (i) closed under the bracket, i.e. x,y R ∈ ⇒ [x,y] R, and (ii) proper, i.e. it is the closure of its interior in T. If `u and `s are ∈ respectively unstable and stable closed leaf segments intersecting in a single point then we denote by [`u,`s] the set consisting of all points of the form [w,z] with w `u ∈ and z `s. We note that [`u,`s] is a rectangle. Conversely, given a rectangle R, for ∈ each x R there are closed unstable and stable leaf segments of T, `u(x,R) W u(x) ∈ ⊂ and `s(x,R) W s(x) such that R = [`u(x,R),`s(x,R)]. The leaf segments `u(x,R) ⊂ and `s(x,R) are called, respectively, unstable and stable spanning leaf segments.

R s s l (x,R) l (z,R)

u z l (x,R) x

u l (w,R) w q(w)=[w,z]

Fig. 2.6 A basic stable holonomy θ : `u(x,R) `u(z,R). →

2.3.2 Basic Holonomies

Suppose that x and z are two points inside any rectangle R of T. Let `s(x,R) and `s(z,R) be two stable spanning leaf segments of R containing, respectively, x and z. We define the map θ : `s(x,R) `s(z,R) by θ(w) = [w,z] (see Figure 2.6). Such → maps are called the basic stable holonomies. They generate the pseudo-group of all stable holonomies. Similarly, we can define the basic unstable holonomies. 2 Anosov and Circle Diffeomorphisms 19 2.3.3 Lamination Atlas

The stable lamination atlas L s = L s(G,ρ), determined by a Riemannian metric ρ, is the set of all maps e : I R, where e is an between the induced → Riemannian metric on the stable leaf segment I and the Euclidean metric on the reals. We call the maps e L s the stable lamination charts. Similarly, we can define ∈ the unstable lamination atlas L u = L u(G,ρ). By Theorem 2.1 in [27], the basic unstable and stable holonomies are C1+ with respect to the lamination atlas L s.

2.3.4 Circle Diffeomorphisms

Let G G be a C1+ Anosov diffeomorphism with an invariant measure absolutely ∈ continuous with respect to the Lebesgue measure and topologically conjugate to the Anosov automorphism Ga by the homeomorphism h. For each Markov rectangle s R, let tR be the set of all unstable spanning leaf segments of R. Thus, by the local s s product structure, one can identify tR with any stable spanning leaf segment ` (x,R) S s s of R. We form the space G by taking the disjoint union th(A) th(B), where h(A) and h(B) are the Markov rectangles of the Markov partition MFG, and identifying s s two points I t and J t if (i) R = R0, (ii) the unstable leaf segments I and J are ∈ R ∈ R0 6 unstable boundaries of Markov rectangles, and (iii) int(I J) = /0.Topologically, ∩ 6 S S S the space G is a counterclockwise oriented circle. Let π G : R M R G be the ∈ G → natural projection sending x R to the point `u(x,R) in S . ∈ G F S S S S Let I be an arc of G and I a leaf segment such that π G (I) = I . The chart i : I R in L = L s(G,ρ) determines a circle chart iS : IS R for IS given by → → iS πS = i. We denote by A = A (G,ρ) the set of all circle charts iS determined ◦ G G by charts i in L = L s(G,ρ). Given any circle charts iS : IS R and jS : JS R, 1 → 1 → 1 the overlap map jS i−S : iS(IS JS) jS(IS JS) is equal to jS i−S = j θ i , ◦ ∩ → ∩ ◦ ◦ ◦ − where i = iS πS : I R and j = jS πS : J R are charts in L , and ◦ G → ◦ G → 1 1 θ : i− (iS(IS JS)) j− ( jS(IS JS)) ∩ → ∩ is a basic stable holonomy. By Theorem 2.1 in [27], there exists α > 0 such that, for 1 1 1+α all circle charts iS and jS in AG, the overlap maps jS i−S = j θ i− are C 1+α ◦ ◦ ◦ diffeomorphisms with a uniform bound in the C norm. Hence, AG = A (G,ρ) is a C1+ atlas. Suppose that I and J are stable leaf segments and θ : I J is a holonomy map → such that, for every x I, the unstable leaf segments with endpoints x and θ(x) cross ∈ once, and only once, a stable boundary of a Markov rectangle. We define the arc ro- tation map θ˜ : πS (I) πS (J), associated to θ, by θ˜ (πS (x)) = πS (θ(x)) (see G G → G G G G Figure 2.7). By Theorem 2.1 in [27] there exists α > 0 such that the holonomy θ : I J is a C1+α diffeomorphism, with respect to the C1+ lamination atlas s → 1+ L (G,ρ). Hence, the arc rotation maps θ˜G are C diffeomorphisms, with respect to the C1+ atlas A (G,ρ). 20 J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand

A

B j q i q(x) j i J 1= ( 1) j i 0=q( 0) 2=q( 2)

p(0,0)

I i i 1 i g (x) 0 0S G x 2 g g g G G G B

x g x l0 l(q( ))= G(l( ))

i g i g i l( 0)= G(l0) l(q( 1))= G(l( 1))

A

Fig. 2.7 The arc rotation map g = θ˜ : πS (I) πS (J). We note that S = πS (I) = πS (J) and G G G → G G G S `(x) = π G (x) is the unstable spanning leaf segment containing x.

1+ Lemma 2.3. There is a well-defined C circle diffeomorphism gG, with respect to 1+ S ˜ the C atlas AG = A (G,ρ), such that gG π G (I) = θG, for every arc rotation map ˜ | θG. In particular, if Ga is the Anosov automorphism, then gGa is the rigid rotation rγ , with respect to the isometric atlas Aiso = A (Ga,E), where E corresponds to the Euclidean metric in the plane.

The proof of Lemma 2.3 is in [21].

2.3.5 Train-Tracks and Markov Maps

Roughly speaking, train-tracks are the optimal leaf-quotient spaces on which the stable and unstable Markov maps induced by the action of G on leaf segments are local homeomorphisms. Let G G be a C1+ Anosov diffeomorphism. Let h be the homeomorphism that ∈ s conjugates G with Ga. We recall that, for each Markov rectangle R, tR denotes the set of all unstable spanning leaf segments of R and, by the local product structure, s s one can identify tR with any stable spanning leaf segment ` (x,R) of R. We form s s the space TG by taking the disjoint union th(A) th(B), where h(A) and h(B) are the F 2 Anosov and Circle Diffeomorphisms 21

Markov rectangles of the Markov partition M and identifying two points I ts and G ∈ R J ts if (i) the unstable leaf segments I and J are unstable boundaries of Markov ∈ R0 rectangles and (ii) int(I J) = /0.This space is called the stable train-track and it is ∩ denoted by TG.

Let πTG : R M R TG be the natural projection sending the point x R to ∈ G → ∈ the point `u(x,R) in .A topologically regular point I in is a point with a F TG TG unique preimage under πTG (i.e. the preimage of I is not a union of distinct unstable boundaries of Markov rectangles). If a point has more than one preimage by πTG , then we call it a junction. Hence, there is only one junction. s A chart i : I R in L = L (G,ρ) determines a train-track chart iT : IT R for → TG TG → IT given by iT πTG = i. We denote by A = A (G,ρ) the set of all train-track ◦ s charts iT determined by charts i in L = L (G,ρ). Given any train-track charts iT : R R TG 1 IT and jT : JT in A , the overlap map jT iT− : iT (IT JT ) jT (IT JT ) → 1 → 1 ◦ ∩ → ∩ is equal to j i− = j θ i , where i = i π : I R and j = j π : J R T ◦ T ◦ ◦ − T ◦ TG → T ◦ TG → are charts in L , and

1 1 θ : i− (i (I J )) j− ( j (I J )) T T ∩ T → T T ∩ T is a basic stable holonomy. By Theorem 2.1 in [27] there exists α > 0 such that, for TG 1 all train-track charts iT and jT in A (G,ρ), the overlap maps jT iT− = j θ 1 1+α ◦ 1+α ◦ ◦ i− have C diffeomorphic extensions with a uniform bound in the C norm. T 1+α Hence, A G (G,ρ) is a C atlas in TG. The (stable) Markov map M : T T is the mapping induced by the action G G → G of G on unstable spanning leaf segments, that it is defined as follows: if I T , ∈ G MG(I) = πTG (G(I)) is the unstable spanning leaf segment containing G(I). This map MG is a local homeomorphism because G sends short stable leaf segments homeomorphically onto short stable leaf segments. A stable leaf primary cylinder of a Markov rectangle R is a stable spanning leaf segment of R. For n 1, a stable leaf n-cylinder of R is a stable leaf segment I such ≥ that (i) GnI is a stable leaf primary cylinder of a Markov rectangle R (I) M ; 0 ∈ G (ii) Gn (`u(x,R)) R (I) for every x I, where `u(x,R) is an unstable spanning leaf ⊂ 0 ∈ segment of R. For n 1, an n-cylinder is the projection into T of a stable leaf n-cylinder ≥ G segment. Thus, each Markov rectangle in T projects in a unique primary stable leaf segment in TG. Given a topological chart (e,U) on the train-track TG and a train-track segment C U, we denote by C e the length of e(C). We say that MG has bounded geometry ⊂ 1+ | | in a C atlas A , if there is κ1 > 0 such that, for every n-cylinder C1 and n-cylinder 1 C with a common endpoint with C , we have κ− < C / C < κ , where the 2 1 1 | 1|e | 2|e 1 lengths are measured in any chart (e,U) of the atlas such that C1 C2 U. We note 1+ ∪ ⊂ that MG has bounded geometry, with respect to a C atlas A , if, and only if, there are κ > 0 and 0 < ν < 1 such that C κ νn, for every n-cylinder and every 2 | |e ≤ 2 e B. ∈ 1+ By Section 4.3 in Pinto-Rand [25], we obtain that MG is C and has bounded TG geometry in A (G,ρ). In [21] it is proved that MG is the Markov map MgG , as in 22 J. P. Almeida, A. M. Fisher, A. A. Pinto and D. A. Rand

Definition 2.2. Hence, gG is a fixed point of renormalization. Futhermore, in [21] it is proved that the map G g induces an one-to-one correspondence between C1+ → G conjugacy classes of Anosov diffeomorphisms with an invariant measure absolutely continuous with respect to the Lebesgue measure and C1+ conjugacy classes of circle diffeomorphisms that are fixed points of renormalization. We recall that G denotes the set of C1+ Anosov diffeomorphisms G that are topologically conjugate to the linear Anosov automorphism Ga and have an invariant measure absolutely continuous with respect to the Lebesgue measure. 1+ 1+ Theorem 2.1. For every G G , the C circle diffeomorphism gG is a C fixed ∈ 1+ point of renormalization, with respect to the C atlas AG = A (G,ρ). The proof of Theorem 2.1 is in [21].

Acknowledgments

Previous versions of this work were presented in the International Congresses of Mathematicians ICM 2006 and 2010, EURO 2010, ICDEA 2009 and in the celebra- tion of David Rand’s 60th birthday, achievements and influence, University of War- wick. We are grateful to Dennis Sullivan and Flavio´ Ferreira for a number of very fruitful and useful discussions on this work and for their friendship and encour- agement. We thank LIAAD-INESC Porto LA, Calouste Gulbenkian Foundation, PRODYN-ESF, POCTI and POSI by FCT and Ministerio´ da Cienciaˆ e da Tecnolo- gia, and the FCT Pluriannual Funding Program of the LIAAD-INESC Porto LA and of the Research Centre of Mathematics of University of Minho, for their financial support. A. Fisher would like to thank FAPESP, the CNPQ and the CNRS for their financial support.

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