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

1+

sequences that are C 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-

  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. 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.

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,

→ S

  i i i

  

i

with respect to the atlas A , with rotation number γ /(1 + γ ).

  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 SS = {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 = SS

,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

1+

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 = JJ is an arc in H with the property that JJ = { 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 ,rl −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

ι 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

  pg 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 pg r = i

  L i ) ( c ) pg l

  (0 g i i

  L L (0 i )

  J = pg ) 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

= A |

  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 iR 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

φ i

  1+ is C for every i ∈ N.

1.4 Markov maps

  

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

1+

  C circle diffeomorphism g . The Markov map M : HH +1 is given by

  i i i

  S (x) i f xR π g i i i +1

  M (x) = .

  ik −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 ) : HH 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 : HH 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 : HH , 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

  

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.

  

, 25, 1, (1961), 21-96. Transl. A.M.S., 2nd series, 46, 213-284.

  Nauk. Math.

  

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.

  

5. Lanford, O.: Renormalization group methods for critical circle mappings with general rota-

tion number. VIIIth International Congress on Mathematical Physics. World Sci. Publishing, Singapore, 532–536 (1987).

  

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).

  1+

  

10. Pinto, A. A. and Rand, D. A.: Train-tracks with C 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) 333-359.

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