Renormalization for critical orders close to 2N
aa r X i v : . [ m a t h . D S ] J a n RENORMALIZATION FOR CRITICAL ORDERS CLOSE TO N JUDITH CRUZ AND DANIEL SMANIA
Abstract.
We study the dynamics of the renormalization operator acting onthe space of pairs ( φ, t ) , where φ is a diffeomorphism and t ∈ [0 , , interpretedas unimodal maps φ ◦ q t , where q t ( x ) = − t | x | α +2 t −
1. We prove the so calledcomplex bounds for sufficiently renormalizable pairs with bounded combina-torics. This allows us to show that if the critical exponent α is close to an evennumber then the renormalization operator has a unique fixed point. Further-more this fixed point is hyperbolic and its codimension one stable manifoldcontains all infinitely renormalizable pairs. Introduction
The theory of renormalization were motivated by the conjecture of Feigenbaumand P. Coullet-C. Tresser which stated that the period-doubling operator, actingon the space of unimodal maps, has a unique fixed-point which is hyperbolic withan one-dimensional unstable direction.Lyubich [17] proved the Feigenbaum-Coullet-Tresser conjecture for unimodalmaps with even critical order asserting that the period-doubling fixed point is hy-perbolic, with a codimension one stable manifold (indeed Lyubich proved a far moregeneral result). An extension the results of Lyubich’s hyperbolicity to the space of C r unimodal maps with r sufficiently large were given by E. de Faria, W. de Meloand A. Pinto[8].All theses results on the uniqueness, hyperbolicity and universality of the fixedpoint of the renormalization operator are for unimodal maps whose critical exponentis an positive even integer. When the order is a non-integer positive integer, veryfew rigorous results are known. Our goal is to obtain some results in this case.Fix α > f = φ ◦ q t : [ − , → [ − , , where φ is orientation preserving diffeomorphism of the interval [ − , φ ( −
1) = − φ (1) = 1, and q t ( x ) = − t | x | α +2 t − . Note that q t preserves the interval [ − , t ∈ [0 , . Marco Martens [18] proved, based on real methods, the existence offixed points to the renormalization operator, for every periodic combinatorial type,acting on the class of unimodal maps mentioned above.It is not clear how to see the renormalization operator, acting on the class ofunimodal maps, as an analytic operator when the critical exponent α is not an evennatural number. In view of this problem we define a new renormalization operator,denoted by e R α , in a suitable space of pairs ( φ, t ) , where φ ◦ q t is a unimodal map. Date : September 15, 2018.2010
Mathematics Subject Classification.
Primary 37F25, 37E20, 37E05.
Key words and phrases. renormalization, unimodal, universality, hyperbolicity.Partially supported by CAPES..Partially supported by FAPESP 2008/02841-4, CNPq 310964/2006-7 and 303669/2009-8.
The advantage of dealing with the new renormalization operator is that it isa compact complex analytic operator when we endow the ambient space of pairs( φ, t ) with a structure of a complex analytic space. The complexification of therenormalization operator is done using a result of complex a priori bounds. Thenwe can see that the map α e R α is a real analytic family of operators. This allowus to use perturbation methods to solved the conjecture for the renormalizationoperator when the critical exponent α is close enough to an even natural number.So we stablished Theorem A. G iven a periodic combinatorics σ , if α is close enough to N thensome iterate of the renormalization operator associate with σ acting on the spaceof pairs ( φ, t ) , where φ is a real analytic map and t ∈ [0 , , has a hyperbolic fixedpoint with a codimension one stable manifold. Theorem B. F or α is close enough to N , the fixed point of the renormalizationoperator associate with σ is unique. Also we stablished the universality for infinitely renormalizable pairs
Theorem C. T he stable manifold of the fixed point contains all the pairs infinitelyrenormalizable with the combinatorics of the fixed point. The structure of the paper is as follows. In the Section 2 we first introducebasic notions on the renormalization of unimodal maps and unimodal pairs. Thenin the Section 3 we state our results on the hyperbolicity of the fixed point whenthe critical order is close enough to an even natural number. We present in theSection 4 the real and complex a priori bounds, the main tool in the proof of ourresults. In the Section 5 we introduce the composition operator, denoted by L ,which relates the new renormalization operator and the usual one. For even α, weconsider the usual renormalization operator as an operator acting on the space ofholomorphic functions in the Section 6. Also we show the relations between the tworenormalization operators when the critical exponent is an even natural number.In the last section we proceed to prove the main theorems.2. Preliminaries
Some notations.
Here the positive integers form the set of natural numberdenoted in the standard form by N . Let I be a bounded interval in the real line.The a -stadium set D a ( I ) is the set of points in the complex plane whose distanceto the interval I is smaller than a > . For sets V and W contained in the complexplane we say the subset V is compactly contained in W denoting by V ⋐ W. The Banach space C k ([ − , , R ) , k ∈ N , is the set of maps C k endowed with thesup norm | f | C k ([ − , = sup x ∈ [ − , {| f ( x ) | , | Df ( x ) | , . . . , | D k f ( x ) |} . We denote as Diff ([ − , that preserve the orienta-tion of the interval [ − , . This is an open subset of the Banach space C ([ − , , R ) . ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N Renormalization of unimodal maps.
In this part we present usual notionsof the renormalization operator, denoted by R . We follow the definitions and nota-tions as in A. Avila, M. Martens e W. de Melo [1]. Fix α > , the so-called criticalorder of the unimodal map. The parametric unimodal family q t : [ − , → [ − , , with t ∈ [0 ,
1] and critical exponent α, is defined by q t ( x ) = − t | x | α + 2 t − . The parameter t defines the maximum q t (0) = 2 t − . Let f = φ ◦ q t : [ − , → [ − , φ ∈ Diff ([ − , α > σ : J → J , where J is a finite set with q elements, endowed witha total order ≺ , is called a unimodal permutation with period q if it satisfies thefollowing condition. Embedding J in the real line preserving the order ≺ , thenthe graph of the permutation σ on R extends, by the union of the consecutivepoints of the graph of σ by segments, to the graph of a unimodal map. Moreoverthe period of σ is q .A collection I = { I , I , ..., I q } of closed intervals in [ − ,
1] is called a cycle fora unimodal map f if it has the following properties:(1) there exists a repelling periodic point p ∈ ( − ,
1) with I q = [ −| p | , | p | ] . (2) f : I i → I i +1 , i = 1 , , ..., q − , are difeomorphisms.(3) f ( I q ) ⊂ I with f ( p ) ∈ ∂I , the boundary of I . (4) the interiors of I , I , ..., I q are pairwise disjoint.Consider the collection J q = { , , ..., q } with the order relation ≺ defined by j ≺ i, j = i, iff inf I j < inf I i . Then the map σ : J q → J q σ ( i ) = i + 1 mod q, is a unimodal permutation. We say that σ = σ ( I ) is the combinatorics of the cycle I . As direct consequence of the definition of cycle I = { I , I , ..., I q } we have • I inheres an order from [ − , , • the map σ = σ ( I ) : I i I i +1 mod q on I is an unimodal permutation, • the orientation o I : I → {− , } is defined such that o I ( I i ) = 1 when f i ( p ) is the left extreme of I i and o I ( I i ) = − I is oriented. Definition 2.1.
A unimodal map f = φ ◦ q t is called renormalizable if it has a cycle.The first return map to I q will be, after a re-escaling, a unimodal map. The primerenormalization period of f is the smallest q > R such that for an unimodal renormalizablemap f = φ ◦ q t we have that R f is a unimodal map defined by R f ( z ) = 1 p f q ( pz ) ,z ∈ [ − , . The unimodal map R f is called the renormalization of f. JUDITH CRUZ AND DANIEL SMANIA
Renormalization of a pair.
Consider the set U = Diff ([ − , × [0 , , where an element ( φ, t ) ∈ U should be interpretated as the unimodal map f = φ ◦ q t : [ − , → [ − , , with critical exponent α > . The diffeomorphism φ is called the diffeomorphicpart of the unimodal map f . The metric on U is the product metric induced bythe norm of the sup on Diff ([ − , α N , it is convenient to consider unimodal maps as a pair ( φ, t ) . Definition 2.2.
A pair ( φ, t ) ∈ U is called renormalizable if f = φ ◦ q t is renormal-izable. The prime renormalization period of ( φ, t ) is the same of f. Let σ be an unimodal permutation and U σ = { ( φ, t ) ∈ U | f = φ ◦ q t has a cycle I com σ ( I ) = σ } . Let I ⊂ [ − ,
1] be an oriented interval. We consider the zoom operator Z I : Diff ([ − , → Diff ([ − , , which assign to the diffeomorphism φ : I → φ ( I ) , the diffeomorphism Z I ( φ ) :[ − , → [ − ,
1] defined by: Z I ( φ ) = A φ ( I ) ◦ φ ◦ A − I , where the transformation A J : J → [ − ,
1] is the unique affine, orientation pre-serving transformation carrying the closed interval J to the interval [ − , . Theintervals I e φ ( I ) have the same orientation.For ( φ, t ) ∈ U σ , we define the orientation preserving diffeomorphism φ : [ − , → [ − ,
1] as φ = Z φ − ( I ) ( φ ) . Here I = I q . On the other hand for each I i ∈ I , i = 0 , we define the orientationpreserving diffeomorphism q i : [ − , → [ − ,
1] e φ i : [ − , → [ − ,
1] by q i = Z I i ( q t )and φ i = Z q t ( I i ) ( φ ) , where q t ( I i ) and I i +1 are orientated in the same direction, this is the orientation o ( I i +1 ) defined by the cycle I . Furthermore, let t = | q t ( I q ) || φ − ( I ) | . Since f ( I q ) = φ ◦ q t ( I q ) ⊂ I , in the definition of the cycle I , we have that t ∈ [0 , . This is equivalent to q t (0) ∈ φ − ( I ) . Now we can define the renormalization operator. The σ -renormalization opera-tor, denoted by e R σ : U σ → U, is defined by the following expression e R σ ( φ, t ) = (( φ q − ◦ q q − ) ◦ ... ◦ ( φ ◦ q ) ◦ ( φ ◦ q ) ◦ φ , t ) . (2.1) ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N For each σ there exists a unique maximal factorization σ = < σ n , ..., σ , σ > such that e R σ = e R σ n ◦ ... ◦ e R σ ◦ e R σ . A unimodal permutation σ is called prime iff σ = < σ > . Obviously each permuta-tion in the maximal factorization is prime. So using primes unimodal permutationswe obtain a partition of the set of renormalizable pairs in U . Definition 2.3.
The renormalization operator denoted by e R : { renormalizable pairs } = [ σ prime U σ → U , is defined by e R | U σ = e R σ . We say that a pair ( φ, t ) ∈ U is N -times renormalizable iff e R n ( φ, t ) is defined forall 1 ≤ n ≤ N. And ( φ, t ) is infinitely renormalizable if it is N -times renormalizablefor all N ≥ . Definition 2.4.
The set of renormalization times { q n } n ∈ Λ , with Λ ⊂ N and where q n < q n +1 , is the set of integers q such that f is renormalizable of period q. We say that a pair N -times renormalizable ( φ, t ) ∈ U , for N big enough, hasbounded combinatorics by B > f = φ ◦ q t satisfies q n +1 /q n ≤ B, for all1 ≤ n < N. Iterating pairs.
Closely following the section 2 in [1] we observe that a se-quence of pairs in U , produced by applying any times the renormalization operator e R α , is such that each pair has as first component a decomposition of diffeomor-phism and the second a parameter carrying the information of the unimodal partof the unimodal map.Fix a N -times renormalizable pair f = ( φ, t ) ∈ U and let I n = { I n , I n , ..., I nq n } be the cycle corresponding to n -th renormalization, 1 ≤ n ≤ N. Each cycle will bepartitioned in sets I n = N [ k ≥ L nk . The level sets L nk , k ≥ , are defined by induction. Let I = L = { [ − , } . If I n +1 ∋ I n +1 i ⊂ I nj ∈ L nk and 0 / ∈ I n +1 i then I n +1 i ∈ L n +1 k +1 . If 0 ∈ I n +1 i then I n +1 i ∈ L n +10 . Observe that I n = n [ k ≥ L nk , for n ≤ N. First to I n ∈ I n , define the orientation preserving diffeomorphism φ n : [ − , → [ − ,
1] where φ n = Z φ − ( I n ) ( φ ) . Then for each I ni ∈ I n , i = 0 , define the diffeomorphisms, that preserve orientation, q ni : [ − , → [ − ,
1] and φ ni : [ − , → [ − ,
1] by q ni = Z I ni ( q t ) JUDITH CRUZ AND DANIEL SMANIA and φ ni = Z q t ( I ni ) ( φ ) , where q t ( I ni ) and I ni +1 are oriented in the same direction, this is with the orientation o ( I ni +1 ) defined by the cycle I n . Furthermore, define t n = | q t ( I n ) || φ − ( I n ) | . The above definitions describe the first component of e R n ( φ, t ) that consists of thecompositions of the diffeomorphisms q ni and φ ni : e R n ( φ, t ) = (( φ nq n − ◦ q nq n − ) ◦ ... ◦ ( φ n ◦ q n ) ◦ ( φ n ◦ q n ) ◦ φ n , t n ) . Precise statements of the main results
Let B V be the complex Banach space of holomorphic maps f defined in a neigh-borhood V with a continuous extension to V , endowed with the sup norm. Let A V be the set of holomorphic maps ϕ : V → C with continuous extension in V , where ϕ ( −
1) = − ϕ (1) = 1 . This is an affine subspace of B V . Denote by T V the complex Banach space of holomorphic maps ω ∈ B V of theform ω = ψ ( x n ) in a neighborhood of [ − , , with ω ( −
1) = ω (1) = 0 , endowedwith the sup norm. Let U V be the set of holomorphic maps f : V → C withcontinuous extension in V of the form f = ψ ( x n ) in a neighborhood of [ − , , with f ( −
1) = f (1) = − . Then U V is an affine space.Marco Martens [18, Theorem 2 .
2] showed that the new renormalization operator e R α , defined on the space of pairs whose first component are diffeomorphisms C in[ − ,
1] with critical exponent α > , has fixed points of any combinatorial type.From now on we fix a prime combinatorics σ with period smaller than B . Denoteby H α ( C, η, M ) , α > , the set of the pairs ( φ, t ) satisfying(1) φ ∈ A D η , where φ is univalent on D η (2) φ is real on the real line(3) | φ | C ([ − , ≤ C (4) ( φ, t ) is M -times renormalizable with combinatorics σ. A pair ( φ, t ) satisfying the first three above properties for some η, C is called aunimodal pair.
Theorem 3.1 (Complex bounds) . For all α > , there exists ε = ε ( B ) , δ = δ ( B ) > and C = C ( B ) > such for all α ∈ ( α − ε, α + ε ) the following holds:for all C > and η > there exists N = N ( C, η ) such that if ( φ, t ) ∈ H α ( C, η, M ) with M > N then e R nα ( φ, t ) ∈ H α ( C , δ , M − n ) for M > n ≥ N . Remark . Using methods of the proof of the Theorem 3.1 and in Sullivan[6] andMartens[18] it is possible to show that the first component of the C fixed points( φ ⋆ , t ⋆ ) of e R α are indeed analytic and univalent in a neighborhood of the interval[ − , . By Theorem 3.1 we have complex bounds for the diffeomorphic part of thisfixed point. Let δ < δ /
2. By Theorem 3.1 ( φ ⋆ , t ⋆ ) belongs to H α ( C , δ, ∞ ). Inthe case when α ∈ N , Sullivan[6] methods implies that there exists such fixedpoint.Fix δ such that 2 δ < δ . Define ˜ N = N ( C , δ ), where C , N is as in Theo-rem 3.1. Let ( φ, t ) ∈ H α ( C , δ, N + 1) with critical exponent α close enough to α N and let I e N = { I e N , I e N , ..., I e N q f N } be the cycle corresponding to e N -th renormaliza-tion. Then the maps φ e N = Z φ − ( I f N ) ( φ ) , q e N i = Z I f N i ( q t ) and φ e N i = Z q t ( I f N i ) ( φ ) , have univalent extensions in complex domains such that the composition( φ e N q f N − ◦ q e N q f N − ) ◦ ... ◦ ( φ e N ◦ q e N ) ◦ ( φ e N ◦ q e N ) ◦ φ e N is defined in D δ . Then it is possible to choose e γ > e R e N α has a extension to the ball e B (( φ, t ) , ˜ γ ) := { ( ψ, v ) ∈ A D δ/ × C , | ( ψ, v ) − ( φ ∗ , t ∗ ) | < e γ } , as a transformation ee R α : e B (( φ, t ) , e γ ) → A D δ × C , defined by the expression ee R α ( φ, t ) = (( φ e N q f N − ◦ q e N q f N − ) ◦ ... ◦ ( φ e N ◦ q N ) ◦ ( φ e N ◦ q e N ) ◦ φ e N , t f N ) , where t f N = | q t ( I e N ) || φ − ( I e N ) | . So ee R α is defined in an open neighborhood of H α ( C , δ, N + 1) in the space A D δ/ × C . The natural inclusion j : A D δ × C → A D δ/ × C is a linear compact operatorbetween Banach spaces. Definition 3.3.
The complex renormalization operator, denoted by e R , is definedby e R α = j ◦ ee R α . Note that e R α is a compact operator. Theorem 3.4 (Hiperbolicity) . Fix r ∈ N . There exists η > and a real analyticmap α → ( φ ∗ α , t ∗ α ) , where α ∈ (2 r − η, r + η ) , such that ( φ ∗ α , t ∗ α ) is a hyperbolic fixedpoint to the operator e R α , with codimension one stable manifold. The uniqueness of the fixed point to the operator e R α is showed in the followingresult. The proof will be postpone to the last section. Theorem 3.5 (Uniqueness of the fixed point) . Fix r ∈ N . For each α close to r, there exists an unique unimodal fixed point ( φ ∗ α , t ∗ α ) of e R α , and it belongs to H α ( C , δ , ∞ ) . We define the stable manifold of the fixed point ( φ ∗ α , t ∗ α ) , denoted by W s = W s ( φ ∗ α , t ∗ α ) , as the set W s := { ( φ, t ) : e R n ( φ, t ) → ( φ ∗ α , t ∗ α ) } By e R nα ( φ, t ) → n ( φ ∗ α , t ∗ α )we say that e R nα ( φ, t ) belongs to A D δ × C for n large enough, and e R nα ( φ, t ) convergesto ( φ ∗ α , t ∗ α ) in A D δ × C .Let V be a neighborhood of the fixed point ( φ ∗ α , t ∗ α ) of the operator e R α . Define N kV ( φ ∗ α , t ∗ α ) := { ( φ, t ) : e R j ( φ, t ) ∈ V , ≤ j ≤ k }} , JUDITH CRUZ AND DANIEL SMANIA where k ∈ N ∪ ∞ . Furthermore, we define N ∞ V ( φ ∗ α , t ∗ α ) := \ k ∈ N N kV ( φ ∗ α , t ∗ α ) , this is, the set of pairs such that all your iterates stay close to ( φ ∗ α , t ∗ α ) . So we areready to define the local stable manifold.
Definition 3.6.
For each V neighborhood of ( φ ∗ α , t ∗ α ) we define the correspondinglocal stable manifold to be W sV ( φ ∗ α , t ∗ α ) := W s ( φ ∗ α , t ∗ α ) ∩ N kV ( φ ∗ α , t ∗ α ) . As ( φ ∗ α , t ∗ α ) is a hyperbolic fixed point, we can choose V such that(3.1) W sV ( φ ∗ α , t ∗ α ) := N ∞ V ( φ ∗ α , t ∗ α ) . Other important result is the universality for infinitely renormalizable pairs.
Theorem 3.7 (Universality) . Fix r ∈ N . For α close to r we have that allunimodal pairs ( φ, t ) , infinitely renormalizable with combinatorics σ and order α ,belongs to the stable manifold of the unique, unimodal fixed point ( φ ∗ α , t ∗ α ) of ˜ R α ,this is [ C> [ η> H α ( C, η, ∞ ) ⊆ W s ( φ ∗ α , t ∗ α ) . Real and complex a priori bounds
We will present the main tool for the development of this work, the called com-plex bounds: there exists a complex domain V ⊃ [ − ,
1] such that for n big enoughthe first component of e R n ( φ, t ) , where ( φ, t ) ∈ U satisfying appropriated conditions,is well defined and univalent in V. The complex bounds has a lot applications in the study of the renormalizationoperator R r , r ∈ N . One of the most important applications is the convergence ofthe renormalization operator in the set of the infinitely renormalization maps andthe hyperbolicity of this operator in an appropriate space. Sullivan [6] introducedthis property for the infinitely renormalization maps with bounded combinatorics.Others related results about infinitely renormalizable unimodal maps with nobounded combinatorics, were given by Lyubich [15], Lyubich e Yampolsky [16],Graczyk e Swiatek [12], Levin e van Strien [10]. For multimodal analytic maps,infinitely renormalizable with bounded combinatorics Smania [4] proved “complexbounds”.We obtain complex bounds for the first component of the renormalization oper-ator e R α which is a univalent map. This tool is useful because it allow us to definethe complex renormalization operator e R α , where the critical exponent is α > . A main ingredient in the proof of the Complex Bounds’s Theorem is given inthe following lemma where we establish real bounds. We use this to obtain controlon the geometry of the cycles of pairs N -times renormalizables, for N enough big,with bounded combinatorics by a constant B > . For a proof of the real bounds see [19, Theorem 2 .
1, Chapter VI]. First fix thecritical exponent α > . ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N Lemma 4.1 (Real bounds) . [19] Let
B > be a constant. Then there exists < b < with the following property: for all C > , there exists N = N ( B, C ) ≥ such that if ( φ, t ) ∈ U is M -times renormalizable with bounded combinatorics by B with M > N, and | φ | C ([ − , ≤ C, we have that (1) if I n +1 i l ⊂ I nj , l = 1 , · · · , m n are the intervals of the ( n + 1) -th renormal-ization cycle, where N ≤ n < M, contained in the interval I nj of the n -thcycle then b < | I n +1 i l || I nj | < − b, (4.1) where l = 1 , · · · , m n , for all N ≤ n < M. (2) if J is a connected component of I nj \ S m n l =1 I n +1 i l then b < | J || I nj | < − b, (4.2) for all N ≤ n < M. Remark . Let α > δ > b < φ, t ) ∈ U , M -times renormalizableswith bounded combinatorics by B, for M > N big enough, with critical order e α, where | α − e α | < δ. So we can establish the following result.
Theorem 4.3.
Let
B > , α > , and C > . There exists δ > and ε = ε ( δ ) > , such that the following is satisfied: for each complex domain V containing theinterval [ − , there exists N = N ( B, α, C, V ) ≥ such that • if ( φ, t ) ∈ U is M -times renormalizable with bounded combinatorics by B, for M > N, with critical order e α, | α − e α | < δ and φ univalent map definedon V, and • | φ | C ([ − , ≤ C, Then for all N ≤ n < M the maps φ n = Z φ − ( I n ) ( φ ) , q nj = Z I nj ( q t ) and φ nj = Z φ − ( I nj +1 ) ( φ ) , where j = 1 , · · · , q n − , have univalent extensions for maps definedon a ε -stadium D ε . Proof.
Let δ > N ≥ S + a = { z ∈ C : | arg ( z ) | < πa } and S − a = { z ∈ C : | arg ( − z ) | < πa } , where a > . Suppose that the pair ( φ, t ) ∈ U satisfies the hypothesis of the theorem.We fix I n = { I n , I n , ..., I nq } the corresponding cycle to the n -th renormalization, N ≤ n ≤ M. We denote by x nj the boundary point of the interval I nj , j = 0 , nearestto the critical point. There is a level of renormalization between N and n such thatthe interval containing the critical point in this level contains the interval I nj . Sochoose the first k > I n − k ⊃ I nj , where N ≤ n − k < n. Then I nj is not containing in I n − k +10 . We have two cases:
Case I . First when k = 1. By Lemma 4.1 we obtain dist (0 , x nj ) | I nj | ≥ | I n | | I nj | = | I n | | I n − | · | I n − || I nj | > b · − b ) . Case II . For k >
1. We can see that I nj is contained in some interval I n − k +1 j ( n − k +1) ⊂ I n − k − I n − k +10 . Actually the interval I nj in contained in a nesting sequence ofintervals of deeper levels. So I nj ⊂ I n − j ( n − ⊂ · · · I n − k +2 j ( n − k +2) ⊂ I n − k +1 j ( n − k +1) and by theLemma 4.1 we have | I nj | < (1 − b ) k − | I n − k +1 j ( n − k +1) | ≤ (1 − b ) k − · ( | I n − k | − | I n − k +10 | ) . (4.3)From Eq.( 4.3) we obtain dist (0 , x nj ) | I nj | ≥ | I n − ( k − | | I nj | > − b ) k − · | I n − ( k − || I n − k | − | I n − k +10 | > b (1 − b ) k − In both cases we obtain dist (0 , x nj ) | I nj | > b − b ) . With this estimative we can define the diffeomorphisms φ n , q nj and φ nj , for j =1 , · · · , q n − , in a common domain in the complex plane. In fact, we know theprincipal branch of the logarithm function log is holomorphic on the set C \ { z ∈ R : z ≤ } . Let q + t : S + e α → C and q − t : S − e α → C be the univalent maps where q + t ( z ) = − te e α log z + 2 t − q − t ( z ) = − te e α log ( − z ) + 2 t − . We follow the proof defining a common domain to the maps q nj , for j = 1 , · · · , q n − , taking in mind two different domains for the critical exponent α > . Firstly when α ∈ (1 , . For I nj ⊂ S + e α (or I nj ⊂ S − e α ) applying the zoom operator Z I nj for thediffeomorphisms q + t | I nj (or q − t | I nj ). So we can define q nj = Z I nj ( q + t ) (or q nj = Z I nj ( q − t )) on the set ǫ -stadium D ǫ = { z ∈ C : dist ( z, [ − , < ǫ } , where ǫ = b − b ) . This set contains the interval [ − , . Now for α ≥ . We consider the distance a j of the boundary point x nj of the ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N interval I nj ⊂ S + e α (or I nj ⊂ S − e α ) to the boundary of the sector S + e α (or S − e α ). Asimples geometric calculus leads to the following relation a j = sin( π e α ) · dist (0 , x nj ) > sin( πα − δ ) · b − b ) | I nj | , doing a zoom of the diffeomorphisms q + t | I nj (or q − t | I nj ) and taking ǫ = sin( πα − δ ) · b − b ) , so we can define q nj = Z I nj ( q + t ) (or q nj = Z I nj ( q − t )) on a set ǫ -stadium D ǫ . Denote by ǫ nj the distance of the interval φ − ( I nj +1 ) to the boundary of V. It is clarethat ǫ nj ≥ dist ([ − , , ∂V )for all j = 0 , · · · , q n − . There exists N ≥ N and K > N ≤ n < M we have dist ([ − , , ∂V ) ≥ K | φ − ( I nj +1 ) | , where j = 0 , · · · , q n − . Then φ nj = Z φ − ( I nj +1 ) ( φ ) , for all j = 0 , · · · , q n − N ≤ n < M, are defined on D K = { z ∈ C : dist ( z, [ − , < K } . Choose ε < ǫ . We see that φ n , and the maps q nj , φ nj , for all j = 1 , · · · , q n − , aredefined on the ε -stadium D ε = { z ∈ C : dist ( z, [ − , < ε } , for all N ≤ n < M. (cid:3) Proposition 4.4.
Let ( φ, t ) ∈ U be as in the statement of the Theorem 4.3. Thereexists N > , L > , H > and b < such that the maps φ n = Z φ − ( I n ) ( φ ) ,q nj = Z I nj ( q t ) and φ nj = Z φ − ( I nj +1 ) ( φ ) , where j = 1 , . . . , q n − , with univalentextensions to maps defined on a domain D ε containing the interval [ − , , satisfying q n − X j =0 | φ nj − id | D ε ≤ Lb n , (4.4) and X I nj ∈ L nk | q ni − id | D ε ≤ H (1 − b ) k − , (4.5) for all N ≤ n < M. Proof.
Let R = dist ([ − , , ∂V ) . For the first estimative define K = K ( j, n ) = R | φ − ( I nj +1 ) | . There is N > N ≤ n < M we have 0 < ε < K/ φ nj = Z φ − ( I nj +1 ) ( φ ) , where j = 0 , · · · , q n − , are defined on a ball B (0 , K/ . From the Theorem A.1 | φ nj − id | D ε ≤ O ( 1 + εK ) . (4.6) On the other hand by the real bounds there exists constants L > b < q n − X j =1 | I nj | ≤ L b n , for all N ≤ n < M. As the diffeomorphism φ has bounded derivative those con-stants can be adjusted such that q n − X j =1 | φ − ( I nj ) | ≤ L b n , for all N ≤ n < M. So we have q n − X j =0 εK = (1 + ε ) q n − X j =0 | φ − ( I nj +1 ) | R ≤ (1 + ε ) R Lb n , this implies the Eq. (4.4).To obtain Eq. (4.5), we need analyze two cases: Case I . For k > I nj ⊂ I n − k \ I n − k +10 K = K ( j, n ) = | I n − k +10 | | I nj | se 1 < α < πα − δ ) . | I n − k +10 | | I nj | if α ≥ q nj = Z I nj ( q t ) , where j = 1 , · · · , q n − , are defined in the ball B (0 , K / . We have 1 + ε < K / | q nj − id | D ε ≤ O ( 1 + εK ) . (4.7)On the other hand we obtain X I nj ⊂ I n − k \ I n − k +10 εK ≤ (1 + ε ) X I nj ⊂ I n − k \ I n − k +10 | I nj || I n − k +10 |≤ ε ) b . (1 − b ) k − . Case II . For k = 1 . From Theorem 4.3, for the interval I nj ⊂ I n − , where N ≤ n − k < n < M, the univalent maps q nj = Z I nj ( q t ) are defined in the ǫ -stadium D ǫ , where ǫ > ε. Remember that q nj ( −
1) = − q nj (1) = 1 . So considering theunivalent maps q nj defined on D ǫ \ {− , } , by the Montel’s Theorem [13], forma normal family. Then there exists C > I nj ⊂ I n − , where N ≤ n − k < n < M, we have | q nj − id | D ε < C . ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N So from these two cases we conclude that there exists
H > X I nj ∈ L nk | q ni − id | D ε ≤ H (1 − b ) k − . (4.8) (cid:3) Now we are ready to give the proof of the Complex bounds’s Theorem.
Proof of the Theorem 3.1.
From Theorem 4.3 the maps φ n , q ni e φ ni , for all i =1 , · · · , q n − , have univalent extension on a set ε -stadium D ε containing the interval[ − , . Let D ε ⊃ D ε/ ⊃ [ − ,
1] be subsets strictly nested. Define for all j =1 , · · · , q n − nj = φ nq n − ◦ q nq n − ◦ · · · ◦ φ nq n − j and Ψ nj = φ nq n − ◦ q nq n − ◦ · · · ◦ φ nq n − j ◦ q q n − j . Define ρ ( φ nq n − ) := ε/ . We are going to construct by induction the domains to themaps Φ nj and Ψ nj , for j = 1 , · · · , q n − . First let j = 1 . By Lemma A.2 there existsa constant
K > q nq n − j : D ε → C and Φ nj : D ρ ( φ nqn − j ) → C there is a ρ ( q nq n − j )-stadium D ρ ( q nqn − j ) ⊂ D ρ ( Φ nj ) such that q nq n − j ( D ρ ( q nqn − j ) ) ⊂ D ρ (Φ nj ) . Moreover ρ ( q nq n − j ) ≥ e − k | q nqn − j − id | Dε ρ (Φ nj ) , where ρ ( q nq n − j ) and ρ (Φ nj ) are the distances between the boundary of D ρ ( q nqn − j ) and D ρ (Φ nj ) = D ρ ( φ nqn − j ) respectively to the interval [ − , . Notice that D ρ ( q nqn − j ) is the domain of definition of the map Ψ nj = Φ nj ◦ q nq n − j . So again by the Lemma A.2 there exists a constant
K > φ nq n − j − : D ε → C and Ψ nj : D ρ ( q nqn − j ) → C there is a ρ ( φ nq n − j − )-stadium D ρ ( φ nqn − j − ) ⊂ D ρ (Ψ nj ) such that φ q n − j − ( D ρ ( φ nqn − j − ) ) ⊂ D ρ (Ψ nj ) . Moreover ρ ( φ nq n − j − ) ≥ e − k | φ nqn − j − − id | Dε ρ (Ψ nj ) , where ρ ( φ nq n − j − ) e ρ (Ψ nj ) are thedistances between the boundary of D ρ ( φ nqn − j − ) and D ρ (Ψ nj ) = D ρ ( q nqn − j ) respectivelyto the interval [ − , . Here D ρ ( φ nqn − j − ) is the domain of definition of the mapΦ nj +1 = Ψ nj ◦ φ nq n − j − . Then for each j = 2 , · · · , q n − A and B. Finally we obtain a ρ ( φ n )-stadium D ρ ( φ n ) that is the domain of definitionof the map Ψ nq n ◦ φ n = φ nq n − ◦ q nq n − ◦ · · · ◦ φ n ◦ q n ◦ φ n where ρ ( φ n ) ≥ q n − Y i =1 e − K | q ni − id | Dε . q n − Y i =1 e − K | φ ni − id | Dε .ρ ( φ nq n − )(4.9)By Proposition 4.4 there exists L > , H > b < X I nj ∈I nk | q ni − id | D ε < H (1 − b ) k −
14 JUDITH CRUZ AND DANIEL SMANIA and X i | φ ni − id | D ε < Lb n , for all N < n < M and 1 ≤ k ≤ n. Follow from (4.9) that ρ ( φ n ) ≥ ε · e − K P nk =1 P Inj ∈ Lnk | q ni − id | Dε .e − K P i | φ ni − id | Dε = ε · e − KH b − KL , for all N < n < M. Taking δ = ε · e − KH b − KL we have that the first componentof the family e R n ( φ, t ) has a univalent extension on the domain D δ that not dependof n. Corollary 4.5.
For all
B > there exists δ ( B ) > , and N = N ( B, C, δ ) ≥ such that if ( φ, t ) ∈ H α ( C, δ / , M ) Then if
M > n ≥ N we have that the firstcomponent of e R n ( φ, t ) , is defined and univalent in a complex domain D δ . Corollary 4.6.
For each unimodal pair ( φ, t ) , infinitely renormalizable with boundedcombinatorics by B > , there exists N such that the sequence consisting of the firstcomponent of the pairs e R n ( φ, t ) , with n ≥ N , is a pre-compact family in D δ . Proof.
The diffeomorphic part of each e R n ( φ, t ) is a diffeomorphisms that preservethe interval [ − , . Actually this analytic diffeomorphism is a decomposition ofdiffeomorphism. By Theorem 3.1 the diffeomorphism part of each e R n ( φ, t ) , where n ∈ N , has univalent extension on a fix δ -stadium D δ containing the interval[ − ,
1] that no depend of n. Since each of those transformations fix − , followof the Montel’s Theorem that with the sup norm on the all holomorphic functionswe have that the first component of the pairs e R n ( φ, t ) form a pre-compact familyin D δ . (cid:3) Corollary 4.7.
There exists δ , C and N = N ( C , δ ) such that • We have e R N α ( H α ( C , δ , ∞ )) ⊂ H α ( C , δ , ∞ ) . • For all
C, η exists N = N ( C, η ) such that for all k ≥ we have e R N + kN α ( H α ( C, η, ∞ )) ⊂ H α ( C , δ , ∞ ) . Corollary 4.8.
A unimodal pair ( φ ∗ α , t ∗ α ) such that e R α ( φ ∗ α , t ∗ α ) = ( φ ∗ α , t ∗ α ) belongsto H α ( C , δ , ∞ ) . Composition transformation
When the critical order is an even natural number α = 2 r , r ∈ N , the relationbetween the new renormalization operator e R and the usual one R is given by the composition transformation , denoted by L, that we will define here. This allow usto transfer some results of the renormalization R to the new operator when thecritical exponent is an even number.Take ǫ > D ǫ ([0 , { z ∈ C : dist ( I, z ) < ǫ } . Now fix the critical exponent α = 2 r, with r ∈ N . Consider a unimodal map f = φ ◦ q t : [ − , → [ − , , with t ∈ [0 , , where q t : [ − , → [ − , , is q t ( x ) = − tx α + 2 t − , ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N with critical exponent α, and φ is a diffeomorphism C , that preserve the orien-tation, of the interval [ − , . Denote A t ( x ) = − tx + 2 t − . Then we can write f = ( φ ◦ A t )( x α ) . We know that a map τ : z z α is holomorphic in the complexplane without zero, and a map A t : z
7→ − tz + 2 t − , where t ∈ D ǫ ([0 , , is holomorphic in the complex plane. Notice that q t = A t ◦ τ. Let V ⊂ C be an open connected set containing the interval [ − , . Let e V beany open connected set containing the interval [ − ,
1] and compactly contained in V such that for every t ∈ D ǫ ([0 , ǫ , we have that q t ( e V ) is compactlycontained in V . Definition 5.1.
For ǫ small, define the complex analytic composition transforma-tion L : A V × D ǫ ([0 , → U e V as L ( φ, t )( z ) = φ ◦ q t ( z ) , for z in e V .
Remark . It is easy to see that if ( φ, t ) ∈ U then L ◦ e R ( φ, t ) = R ◦ L ( φ, t ) . Proposition 5.3 (Injectivity of L) . Let α > . If φ, e φ ∈ A V and t, e t ∈ C \ { } aresuch that φ ◦ q t = e φ ◦ q e t on [ − , , then φ = e φ and t = e t. Proof.
Suppose that φ ◦ q t = e φ ◦ q e t , on [ − , . Then for all y ∈ [0 ,
1] we obtain A t ( y ) = φ − ◦ e φ ◦ A e t ( y ) . Take y = − x + 2 e t − e t , where x ∈ [ − , . It is not difficult to verify t e t x + t e t − φ − ◦ e φ ( x ) . (5.1)Since φ − ◦ e φ (1) = 1 and φ − ◦ e φ ( −
1) = − t = e t and φ − ◦ e φ = id. (cid:3) By definition of L for each ( ω, v ) in T φ A V × C we have D L ( φ, t )( ω,
0) = ddu [( φ + uω ) ◦ q t ] | u =0 = ddu [ φ ◦ q t + u ( ω ◦ q t )] | u =0 = ω ◦ q t and by the chain rule D L ( φ, t )(0 , v ) = ddu [ φ ◦ q t + uv ] | u =0 = Dφ ( q t ) · ddu ( q t + uv ) | u =06 JUDITH CRUZ AND DANIEL SMANIA where ddu ( q t + uv ( x )) | u =0 = 2 v (1 − x r ) . So the derivative of L in ( φ, t ) ∈ A V × D ǫ ([0 , D L ( φ, t ) : F V × C → T e V given by D L ( φ, t )( ω, v )( x ) = ω ◦ q t ( x ) + Dφ ( q t ( x )) . v (1 − x r ) , (5.2)for all x in e V .
In the following propositions we will prove some properties of the differential
DL.
Proposition 5.4.
Let ( φ, t ) ∈ A V × (0 , . The operator D L ( φ, t ) is injective.Proof. Suppose that D L ( φ, t )( ω, v )( z ) = 0 , for all z ∈ e V .
Then from the Eq. (5.2)we have ω ◦ q t ( z ) = − Dφ ( q t ( z )) . v (1 − z r )(5.3) ω ◦ q t ( z ) = − Dφ ( q t ( z )) . vt ( q t ( z ) + 1)(5.4)From Eq. (5.4) we have that for every y ∈ [ − , q t (0)] ⊂ [ − , ω ( y ) = − Dφ ( y ) . vt ( y + 1) , (5.5)Since ω ( y ) and Dφ ( y ) . vt ( y + 1) , are analytic on [ − ,
1] we have that the Eq. (5.5)is satisfied for every y ∈ [ − , . If we take y = 1 we obtain ω (1) = − Dφ (1) vt (2)and since that ω (1) = 0 and Dφ (1) = 0 we have v = 0 . On the other hand thecondition v = 0 in (5.5) and the analycity of ω on V imply ω ( y ) = 0 , for all y ∈ V. So D L ( φ, t ) is injective. (cid:3) Lemma 5.5.
Let ( φ, t ) ∈ A V × (0 , . the image of the operator D L ( φ, t ) : F V × C → T e V is dense.Proof. It is no difficult to prove that the set of polynomial vector fields is densein T e V (see [11]). So will be sufficient to show that for all polynomial vector field e ω ∈ T e V , there exists ( ω, v ) ∈ F V × C such that D L ( φ, t )( ω, v ) = e ω. Since e ω is the form e ω = ψ ( x r ) , where ψ is a polynomial vector field in a neighbor-hood of [0 , e ω = β ◦ q t , where β = ψ ◦ A − t is a polynomialvector field. Take ω ( y ) = β ( y ) − Dφ ( y ) β (1)2 Dφ (1) (1 + y )and v = β (1) t Dφ (1) . (cid:3) ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N Remember that the continuous linear operator T from a Banach space E to aBanach space F is compact if, for each bounded sequence { x n } in E, the sequence { T x n } contains a convergence subsequence in F. Lemma 5.6.
Let ( φ, t ) ∈ A V × D ǫ ([0 , . The operator D L ( φ, t ) : F V × C → T e V is compact.Proof. Let { ( ω i , v i ) } ⊂ F V × C be a bounded sequence, this is there is a constant B > | ω i | V , | v i | < B, for all i > . By definition of L, we have that {D L ( φ, t )( ω i , v i ) } is a sequence ofanalytic vector fields on e V .
We took e V compactly contained in V such that q t ( e V ) iscompactly contained in V. Now we take a open subset U ⋑ e V compactly containedin V such that q t ( U ) is compactly contained in V. Then | ω i | V = sup x ∈ V | ω i ( x ) | ≥ sup x ∈ q t ( U ) | ω i ( x ) | = sup y ∈ U | ω i ◦ q t ( x ) | = | ω i ◦ q t | U . So ω i ◦ q t is bounded on U by B. Since Dφ is bounded in q t ( U ) ⊂ V, by the Eq. (5.2)there exists C > |D L ( φ, t )( ω i , v i ) | U < C for all i > . Since a uniformily bounded sequence of analytic maps in U is a normalfamily in U, all subsequences of {D L ( φ, t )( ω i , v i ) } has a convergence subsequenceon e V . (cid:3) Complex renormalization operators R and ˜ R Fixing the critical exponent α = 2 r where r ∈ N , we are going to consider therenormalization operator R α , defined in the Section 1, as an operator acting on thespace of holomorphic functions. In the last part of this section, we show that when α > D R α and D e R α coincides in the respective fixedpoints of the renormalization operators R α and e R α . Complex operator R α . Based in real methods Marco Martens [18] provedthe existence of the fixed points of the renormalization operators R α , of any combi-natorial type, acting in the space of smooth unimodal maps with critical exponent α > . From definition of the renormalization operator R α we have that it has afixed point, denoted by f ∗ , satisfying f ∗ ( z ) = 1 p f ∗ q ( pz ) , for some p ∈ ( − ,
1) such that ( f ∗ ) q ( p ) = p. Given an analytic function f : V → C , where V is an open set, define the openset D nV ( f ) := n − \ i =0 f − i V. Given a subset V ⊂ C and λ ∈ C , denote by λV := { λx : x ∈ V } . As a consequence of the complex bounds of Sullivan [6], fixing the critical expo-nent α = 2 r where r ∈ N , for all ε > N = N ( ε ) > D ε/ ⋐ D ε ⋐ D ε , satisfying: • f ∗ has a continuous extension to D ε which is holomorphic in D ε , and hasa unique critical point in e D ε . • we have p N D ε ⋐ D q N D ε/ ( f ∗ ) , in other words, we can iterate f ∗ : D ε/ → C at least q N times on a domain p N D ε . Now we can define the complex renormalization operator acting on the holo-morphic functions close enough to f ∗ . Observe that it is possible to choose γ = γ ( ε, N ) > f in the ball of center f ∗ and radius γ denoted by B ε/ ( f ∗ , γ ) := { f ∈ U D ε/ , | f − f ∗ | < γ } , the following is satisfied: • there exists an analytic continuation p f of the periodic point p of f ∗ , thisis f q ( p f ) = p f and p f ∼ p. • we have p f D ε ⋐ D q N ε/ ( f ) . We define the complex analytic operator b R r : B ε/ ( f ∗ , γ ) → U D ε as b R r ( f )( z ) := 1 p f f q N ( p f z ) . (6.1)So we define the complex analytic extension of the renormalization operator R r as R r := i ◦ b R r , where i : U D ε → U D ε is the inclusion. Note that i is a compact linear transforma-tion. Remark . Notice that we are free to choose ε > , N > , and γ > . In thesection 6.2 we will do a convenient to choose those constants.Edson de Faria, W. de Melo and A. Pinto [8], with the help of real and complexa priori bounds of Sullivan [6] and the result of hyperbolicity of Lyubich [17] (alsosee [5]), proved the hyperbolicity of the fixed point of the renormalization operatorwith respect an iterate of the renormalization operator acting on the space U D ε/ for some ε > . More precisely the Theorem 2.4 em [8] claims:
Theorem 6.2. [8]
For ε > small enough, there exists a positive number N = N ( ε ) > and γ > with the following property. The real analytic compactoperator R r : B ε/ ( f ∗ , γ ) → U D ε/ , defined by Eq. (6.1), has a unique hyperbolicfixed point f ∗ = φ ∗ ◦ q t ∗ ∈ B ( f ∗ , γ ) with codimension one stable manifold. Relating the complex operators R r and e R r . Now consider the criticalexponent α = 2 r, where r ∈ N . With the same notation of the Section 6.1, choose δ > , e N . Choose e γ such that ˜ R is defined in e B (( φ ⋆ , t ⋆ ) , ˜ γ ) := { ( φ, t )) ∈ A D δ/ × C , | ( φ, t ) − ( φ ∗ , t ∗ ) | < e γ } , where ( φ ∗ , t ∗ ) is the unique fixed point of ˜ R r . Such fixed point exists due Remark3.2. The uniqueness follows from the uniqueness of the fixed point of R and theinjectivity of L .Let ε be such that q t ∗ ( D ε ) ⊆ D δ/ . ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N And choose ε < ε , N and γ > N = N . f N and consider this iteration.Since ee R r is analytic there exists e C > ee R r ( e B δ/ (( φ ∗ , t ∗ ) , e γ )) ⊆ e B δ (( φ ∗ , t ∗ ) , e C e γ ) , for all e γ < e γ . Then there exists e γ such that if | ( φ, t ) − ( φ ∗ , t ∗ ) | A δ/ × C < e C e γ then φ ◦ q t is defined in D ε . In particular for each e γ ≤ e γ we have defined thefollowing composition transformations L : e B δ/ (( φ ∗ , t ∗ ) , e γ ) → U D ε , and L : e B δ/ (( φ ∗ , t ∗ ) , e C e γ ) → U D ε . Let e γ > e γ ≤ e γ we have L ( e B δ/ (( φ ∗ , t ∗ ) , e γ )) ⊆ B ε ( φ ∗ ◦ q t ∗ , γ ) . Choose e γ ≤ min { e γ , e γ , e γ } . So we stated the following results.
Proposition 6.3.
The following diagram commutes. e B δ/ (( φ ∗ , t ∗ ) , e γ ) ee R / / L (cid:15) (cid:15) L = k ◦ L $ $ j ◦ ee R = e R ' ' e B δ (( φ ∗ , t ∗ ) , e C e γ ) j / / e B δ/ (( φ ∗ , t ∗ ) , e C e γ ) L (cid:15) (cid:15) L = k ◦ L z z B ε ( φ ∗ ◦ q t ∗ , γ ) k (cid:15) (cid:15) U D ε k (cid:15) (cid:15) B ε/ ( φ ∗ ◦ q t ∗ , γ ) ˆ R / / i ◦ ˆ R = R U D ε i / / U D ε/ In particular we have that
R ◦ L = L ◦ e R on e B δ/ (( φ ∗ , t ∗ ) , e γ ) . For the respective tangent spaces we obtain
Proposition 6.4.
The following diagram commutes. F D δ/ × C D ee R ( φ ∗ ,t ∗ ) / / D L ( φ ∗ ,t ∗ ) (cid:15) (cid:15) D L ( φ ∗ ,t ∗ ) ! ! D e R ( φ ∗ ,t ∗ ) $ $ F D δ × C j / / F D δ/ × C D L ( φ ∗ ,t ∗ ) (cid:15) (cid:15) D L ( φ ∗ ,t ∗ ) } } T D ε k (cid:15) (cid:15) T D ε k (cid:15) (cid:15) T D ε/ D ˆ R φ ∗◦ qt ∗ / / DR φ ∗◦ qt ∗ : : T D ε i / / T D ε/ In particular we have that DR L ( φ ∗ ,t ∗ ) ·D L ( φ ∗ ,t ∗ ) = D L ( φ ∗ ,t ∗ ) ·D e R ( φ ∗ ,t ∗ ) on F D δ/ × C . Notice that, by Remark 5.2, L ( φ ∗ , t ∗ ) is fixed point of the operator R . An im-portant relation between the operators R and e R is the following result. Proposition 6.5.
Let ( φ ∗ , t ∗ ) ∈ A D δ/ × (0 , be the fixed point of the operator e R and L ( φ ∗ , t ∗ ) the corresponding fixed point of R . Then σ ( D e R ( φ ∗ ,t ∗ ) ) = σ ( DR L ( φ ∗ ,t ∗ ) ) . Proof.
Denote f ∗ = ( φ ∗ , t ∗ ) . Let λ = 0 be an eigenvalue of D e R f ∗ with eigenvector v = 0 , this is D e R f ∗ v = λv. By the relation of composition on the tangent spaces above we have DR L ( f ∗ ) · D L f ∗ = D L f ∗ · D e R f ∗ . Then DR L ( f ∗ ) · D L f ∗ ( v ) = D L f ∗ · D e R f ∗ ( v ) DR L ( f ∗ ) ( D L f ∗ v ) = λ ( D L f ∗ v )so λ is an eigenvalue of DR L ( f ∗ ) with D L f ∗ v = 0 from Proposition 5.4. Finally fromcompactness of the operators D e R f ∗ and DR L ( f ∗ ) it follows that σ ( D e R f ∗ ) ⊂ σ ( DR L ( f ∗ ) ) . Let w ∈ T D ε/ . Note that w ( x ) = P i a i x i , with a i = 0 if 2 r ∤ i . Let r v ≥ ǫ/ ψ ( x ) = P i a ri x i . The convergence radius for ψ, denoted by r ψ , is at least ( ε/ r since r ψ = 1lim sup i i p | a ri |≥ i ri p | a ri | ! r = r rw ≥ (cid:0) ε/ (cid:1) r , ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N so ψ is well defined in a neighborhood of 0. Note that w ( x ) = ψ ( x r ) for x in aneighborhood of 0. Indeed it is easy to see that ψ is defined in a θ (2 r, ε )-stadiumof the interval [0 ,
1] because w is defined in a neighborhood of the interval [ − , . Take µ (2 r, ε, t ) < δ/ µ (2 r, ε, t )-stadium D µ (2 r,ε,t ) by A − t ∗ is contained in the θ (2 r, ε )-stadium of the interval [0 , . Now define F : T D ε/ → F D µ (2 r,ε,t ) × C by F ( w ) = ( ω, b ) , (6.2)where ω ( y ) = ψ ◦ A − t ∗ ( y ) − Dφ ∗ ( y ) ψ ◦ A − t ∗ (1)2 Dφ ∗ (1) (1 + y )and b = ψ ◦ A − t ∗ (1) t ∗ Dφ ∗ (1) . Note that D L f ⋆ F ( w ) = w, where this equality holds in a complex neighborhood of [ − , λ = 0 be aneigenvalue of DR L ( f ∗ ) with eigenvector w = 0 , that is DR L ( f ∗ ) w = λw. on [ − , D L f ∗ · D e R f ∗ F ( w ) = DR L ( f ∗ ) · D L f ∗ F ( w )on a neighborhood of [ − , D L f ∗ · D e R f ∗ F ( w ) = DR L ( f ∗ ) w = λw = λ D L f ⋆ F ( w )on a neighborhood of [ − , D L f ∗ it follows that D e R f ∗ F ( w ) = λF ( w )in a neighborhood of [ − , F ( w ) ∈ F D η × C , for some η . It remainsto show that F ( w ) = ( ω, b ) belongs to F D δ/ × C . Indeed by the complex boundsthere exists N such that D ˜ R N f ⋆ : F D η × C → F D δ/ × C is well defined. In particular D ˜ R N f ⋆ F ( w ) = λ N F ( w ) belongs to F D δ/ × C , so F ( w )belongs to F D δ/ × C . We conclude that σ ( DR L ( f ⋆ ) ) ⊂ σ ( D e R f ∗ ). (cid:3) We denote by V β and e V λ the respective eigenspaces of the eigenvalues β ∈ σ ( DR L ( φ ∗ ,t ∗ ) ) and λ ∈ σ ( D e R ( φ ∗ ,t ∗ ) ) , this is V β = Ker ( DR L ( φ ∗ ,t ∗ ) − βId ) = { v : ( DR L ( φ ∗ ,t ∗ ) − βId ) v = 0 } e e V λ = Ker ( D e R ( φ ∗ ,t ∗ ) − λId ) = { ( ω, t ) : ( D e R ( φ ∗ ,t ∗ ) − λId )( ω, t ) = 0 } . Theses eigenspaces does not depending of the domains of definition of the mapssince to apply the renormalization operator ( R or e R ) are holomorphically improvingoperators. Proposition 6.6.
Let ( φ ∗ , t ∗ ) ∈ A D δ/ × (0 , be the fixed point of the operator e R . If λ ∈ σ ( D e R ( φ ∗ ,t ∗ ) ) \ { } then dim V λ = dim e V λ . Proof.
We have that V λ and e V λ are finite dimensional subspaces. Define the con-tinuous map T : e V λ → V λ by T ( v ) = D L ( φ ∗ ,t ∗ ) ( v ) . Since D L ( φ ∗ ,t ∗ ) is injective we have that dim Ker ( T ) =0 . Then dim e V λ = dim Ker ( T ) + dim Im ( T ) ≤ dim V λ . Also we can define thecontinuous injective map e T : V λ → e V λ by e T ( e w ) = F ( e w ) , where F is defined by the expression( 6.2). Then dim V λ ≤ dim e V λ . (cid:3) Proof of the main results
The results obtained in the Subsection 6.2 (the Propositions 6.5 and 6.6) andthe Theorem 6.2 [8] were crucial to establish a result of hyperbolicity for the newrenormalization operator e R r which is analog to the Theorem 6.2 [8], for the usualrenormalization R r . Proposition 7.1.
The transformation ( φ, t, α ) → e R α ( φ, t ) is complex analytic inthe variables ( φ, t ) and real analytic in the variable ( φ, t, α ) . Theorem 7.2.
Let α = 2 r ∈ N be the critical exponent. There exists a positivenumber N such that the operator e R r , as defined above, has a unique unimodal fixedpoint ( φ ∗ r , t ∗ r ) . Furthermore ( φ ∗ r , t ∗ r ) is hyperbolic with codimension one stablemanifold.Proof. By Martens [18], there exists a fixed point ( φ ∗ , t ∗ ) to the operator e R r . Since L ◦ e R r ( φ ∗ , t ∗ ) = R r ◦ L ( φ ∗ , t ∗ ) , see Remark 5.2, then φ ∗ ◦ q t ∗ is a fixed point to the operator R r . On the otherhand Sullivan [6] and Theorem 6.2[8] imply that the operator R r has a hyperbolicfixed point f ∗ = φ ∗ ◦ q t ∗ ∈ B ( f ∗ , γ ) with codimesion one stable manifold. ByProposition 6.5 and Proposition 6.6 we obtain that the fixed point ( φ ∗ , t ∗ ) of theoperator e R r is hyperbolic with codimension one stable maniflod. The uniquenessof the fixed point follows from Proposition 5.3. (cid:3) Proof of Theorem 3.4.
Define the operator F (( φ, t ) , α ) = e R α ( φ, t ) − ( φ, t ) . From Proposition 7.1 the operator F is complex analytic in the variables ( φ, t ) andreal analytic in the variables ( φ, t, α ) . We have D e R r ( φ ∗ r , t ∗ r ) − Id is invertible because ( φ ∗ r , t ∗ r ) is the hyperbolic fixed point of e R r . So we can con-clude the proof applying the Implicit Function Theorem.
Proof of Theorem 3.5.
Since ( φ ∗ r , t ∗ r ) is a hyperbolic fixed point, there exists aneighborhood V = B (( φ ∗ r , t ∗ r ) , η ) of ( φ ∗ r , t ∗ r ) such that for α ∼ r, there existsan unique fixed point of e R α in V . Therefore it only remains to verify that, for
ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N α ∼ r, all hyperbolic fixed points ( φ ∗ α , t ∗ α ) of e R α belong to V . In fact, we supposethat there exists a sequence α s → r, where s → ∞ , and fixed points ( φ ∗ α s , t ∗ α s ) of e R α s such that either | φ ∗ α s − φ ∗ r | D δ ≥ η or | t ∗ α s − t ∗ r | ≥ η. (7.1)From Corollary 4.7 ( φ ∗ α s , t ∗ α s ) ∈ H α s ( C , δ , ∞ ) . By definition of ee R α s we have that( φ ∗ α s , t ∗ α s ) = ee R α s ( φ ∗ α s , t ∗ α s )belongs to H α s ( C , δ , ∞ ) . Then we have ( φ ∗ α s , t ∗ α s ) is a pre-compact family on A D δ × C , in particular there is a subsequence ( φ ∗ α si , t ∗ α si ) converging to ( φ, t ) ∈H r ( C , δ , ∞ ) . Since e R α si ( φ ∗ α si , t ∗ α si ) = ( φ ∗ α si , t ∗ α si )taking s i → r , we conclude that ( φ, t ) ∈ H r ( C , δ , ∞ ) is fixed point of theoperator e R r . Then by the uniqueness of the fixed point to α = 2 r we have( φ, t ) = ( φ ∗ r , t ∗ r ) . This leads to a contradiction with (7.1).
Theorem 7.3 (Convergence) . If ( φ, t ) ∈ B δ (( φ ∗ , t ∗ ) , e γ ) is an unimodal pair, infin-itely renormalizable with combinatorics σ, then we have e R i r ( φ, t ) ∈ B δ (( φ ∗ , t ∗ ) , e γ ) , for all i large enough and e R i r ( φ, t ) → i ( φ ∗ , t ∗ ) in A D δ × C . Proof.
Observe that by the complex bounds, we have e R i r ( φ, t ) ∈ H r ( C .δ , ∞ ) , for all i large enough. Suppose that the statement of the theorem is false. So thereexists η > φ, t ) infinitely renormalizable with critical exponent 2 r such that | e R i j r ( φ, t ) − ( φ ∗ , t ∗ ) | A Dδ × C > η, (7.2)where i j → j ∞ . We have the first component of each par in the family { ee R r ◦ e R i j − r ( φ, t ) } j is defined and univalent in D δ . Then this familyis a pre-compact in A D δ × C , in particular there exists a convergence subsequence e R i j r ( φ, t ) → ( e φ, e t ) . The composition operator L satisfies L ◦ e R r = R r ◦ L, so it follows that R i j r ( φ ◦ q t ) = R i j r ◦ L ( φ, t ) → j L ( e φ, e t ) = e φ ◦ q e t . By Sullivan [6] ( also see [3]) the operator R r has an unique fixed point φ ∗ ◦ q t ∗ and furthermore R i r ( φ ◦ q t ) → i φ ∗ ◦ q t ∗ , for all φ ◦ q t infinitely renormalizable. So φ ∗ ◦ q t ∗ = e φ ◦ q e t . By Proposition 5.3 wehave that e φ = φ ∗ e e t = t ∗ . This leads to a contradiction with Eq. (7.2). (cid:3)
In the proof of the following result we use many tools and concepts of complexdynamic, as polynomial-like maps, quasiconformal maps and Sullivan’s pullbackargument see [19], [7], [2] and [3].
Theorem 7.4 (Equicontinuity) . Let ( φ, t ) ∈ H r ( C, δ , ∞ ) . Then there exists i such that for all ˜ γ > there exists ˜ η > with the following property. If ( ψ, v ) ∈H r ( C, δ , ∞ ) satisfies | ( ψ, v ) − ( φ, t ) | A Dδ × C < ˜ η then | ˜ R i r ( ψ, v ) − ˜ R i r ( φ, t ) | A Dδ × C < ˜ γ for all i ≥ i .Proof. Firstly we claim that there exists i such that for all γ > η > ψ, v ) ∈ H r ( C, δ , ∞ )satisfies | ( φ, t ) − ( ψ, v ) | A Dδ × C < η then |R i r ( ψ ◦ q v ) − R i r ( φ ◦ q t ) | D ε < γ for all i ≥ i . In fact, since a map f := φ ◦ q t is a unimodal analytic map ina neighborhood V of the interval [ − , r , by thecomplex bounds of Sullivan [6], for i big enough there exists a polynomial-likeextension R i r f : U f → U, where [ − , ⊂ U f ⋐ U . If ( ψ, v ) ∈ H r ( C, δ , ∞ ) is close to ( φ, t ), then g = φ ◦ q v is close to f , then R i r g has a polynomial-like extension R i r g : U g → U. Here we can choose U such that the disc U g is moving holomorphically with respectto g . In particular, by the theory of holomorphic motions, the pullback argumentof Sullivan (see [19]) and the no-existence of invariant lines fields with support onthe filled Julia set of f , there exists quasiconformal homeomorphisms h g : C → C such that(7.3) ( R i g ) ◦ h g ( x ) = h g ◦ ( R i f )( x )for all x ∈ U f . Furthermore the quasiconformality Q ( g ) of h g satisfies Q ( ψ ◦ q v ) → ψ, v ) → ( φ, t ) . Notice that since all the following renormalizations of f and g are conjugated byrescalings of the same conjugation h g , then the quasiconformality of those conju-gations are bounded by Q ( ψ ◦ q v ). Then since all conjugacies between the ith-renormalization of f and g , with i > i , fix -1 and 1 , it follows that theses con-jugacies converges uniformly to the identity on compact subsets of C when ( ψ, v )converges to ( φ, t ). ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N Since we proved complex bounds, the sequences R i r f e R i r g are bounded in U D ε ,it follows from Eq. (7.3) that if ( ψ, v ) is close enough to ( φ, t ) then |R i r ( ψ ◦ q v ) − R i r ( φ ◦ q t ) | D ε < γ for i ≥ i . So we have proved the claim.Suppose by contradiction that there exists a sequence ( ψ j , v j ) such that | ( φ, t ) − ( ψ j , v j ) | A Dδ × C → j | ˜ R i j r ( ψ j , v j ) − ˜ R i j r ( φ, t ) | A Dδ × C > ˜ γ, with i j ≥ i and i j → j ∞ .Notice, by the complex and real bounds, that˜ R i j r ( ψ j , v j ) = ( ψ j,i j , v j,i j ) , where ψ j,i j is univalent in D δ and fix 1 and − , and moreover0 < inf j v j,i j ≤ sup j v j,i j < . so we can suppose that the second coordinate of those pairs converge for same v ∞ ∈ (0 , ψ i,i j is univalent in D δ , taking a subsequence, if necessary, we cansuppose that ψ i,i j converges for some univalent map ψ ∞ in D δ . In particular R i j r ( ψ j ◦ q v j ) = L ( ˜ R i j r ( ψ j , v j )) → j ψ ∞ ◦ q v ∞ . On the other hand, by the claim that we proved at the beginning of the proof andthe Theorem 7.3 we obtain R i j r ( ψ j ◦ q v j ) → j φ ∗ ◦ q t ∗ . Then ψ ∞ ◦ q v ∞ = φ ∗ ◦ q t ∗ . By Proposition 5.3 we conclude that( ψ ∞ , v ∞ ) = ( φ ∗ , t ∗ ) . So(7.5) | ˜ R i j r ( ψ j , v j ) − ( φ ∗ , t ∗ ) | A Dδ × C → j . But Eq. (7.4) and Theorem 7.3 imply that | ˜ R i j r ( ψ j , v j ) − ( φ ∗ , t ∗ ) | A Dδ × C > ˜ γ/ j large enough. This contradicts Eq. (7.5). (cid:3) Let a > B ∞ a ( φ ∗ α , t ∗ α ) the set of infinitely renor-malizable pairs ( φ, t ) by e R α such that | ( φ, t ) − ( φ ∗ α , t ∗ α ) | A Dδ × C < a. Theorem 7.5.
For all γ > there exists N such that for all α ∼ r we have e R N α ( H α ( C , δ , ∞ )) ⊂ B ∞ γ ( φ ∗ r , t ∗ r ) . (7.6) Proof.
Let γ > . We claim there exists M such that e R M r ( H r ( C , δ , ∞ )) ⊂ B ∞ γ/ ( φ ∗ r , t ∗ r ) . (7.7)In fact, we suppose that there exists a sequence ( φ i , t i ) ∈ H r ( C , δ , ∞ ) such thatfor a subsequence j i → ∞ we have | e R j i r ( φ i , t i ) − ( φ ∗ r , t ∗ r ) | A Dδ × C > γ/ . (7.8)By Corollary 4.7 we obtain that ee R r ( φ i , t i ) ∈ H r ( C , δ , ∞ ) , in particular ee R r ( φ i , t i ) i is a pre-compact family in A D δ × C . Taking a subsequence, if necessary, we canassume without loss of generality that e R r ( φ i , t i ) = j ◦ ee R r ( φ i , t i ) → ( φ, t ) , where ( φ, t ) ∈ H r ( C , δ , ∞ ) . By Theorem 7.3 we have for γ > k > k > k | e R k r ( φ i , t i ) − ( φ ∗ r , t ∗ r ) | A Dδ × C < γ . And using the Theorem 7.4 there exists i > k > i > i and k > k we have that | e R k r ( e R r ( φ i , t i )) − e R k r ( φ, t ) | A Dδ × C < γ . Then for k > max { k , k } we obtain | e R k +12 r ( φ i , t i ) − ( φ ∗ r , t ∗ r ) | A Dδ × C < γ N = M + 1. We claim the N satisfies Eq. (7.6). Otherwise we could find asequence α s → r and ( φ s , t s ) ∈ H α s ( C , δ , ∞ ) such that | e R M +1 α s ( φ s , t s ) − ( φ ∗ r , t ∗ r ) | A Dδ × C > γ. (7.9)We have that the first component of the pairs from the family { ee R α s ( φ s , t s ) } s hasa complex univalent extension to D δ . So this family is pre-compact on A D δ × C . In particular there exists a subsequence ee R α si ( φ s i , t s i ) on A D δ × C that convergesto some ( e φ, e t ) ∈ H r ( C , δ , ∞ ) . From the above we have e R M r ( e φ, e t ) ∈ B ∞ γ/ ( φ ∗ r , t ∗ r ) . On the other hand e R M +1 α si ( φ s i , t s i ) → e R M r ( e φ, e t )in A D δ × C . So for i large enough we have | e R M +1 α si ( φ s i , t s i ) − ( φ ∗ r , t ∗ r ) | A Dδ × C < γ. This leads to a contradiction with the Eq. (7.9). (cid:3)
Proof of the Theorem 3.7.
Let V be a neighborhood satisfying the Eq. (3.1).Choose γ > B γ ( φ ∗ r , t ∗ r ) is contained in V . Claim I.
There exists γ such that for all α ∼ r we have B ∞ γ ( φ ∗ r , t ∗ r ) ⊂ W sV . ENORMALIZATION FOR CRITICAL ORDERS CLOSE TO 2 N In fact, let N be as in the Theorem 7.5. Choose γ < γ small enough such thatfor all i = 1 , .., N , e R i B ∞ γ ( φ ∗ r , t ∗ r ) ⊂ B ∞ γ ( φ ∗ r , t ∗ r ) ⊂ H α ( C , δ , ∞ ) . By the Theorem 7.5 we obtain e R i + kN B ∞ γ ( φ ∗ r , t ∗ r ) ⊂ B ∞ γ ( φ ∗ r , t ∗ r ) , for all k and i = 1 , ..., N . This proves the claim I.
Claim II.
If ( φ, t ) ∈ H α ( C, η, ∞ ), for some C > η >
0, there exists
N > e R Nα ( φ, t ) ∈ B ∞ γ ( φ ∗ r , t ∗ r )in the space A D δ × C . In fact, by Theorem 3.1 there exists N such that for α ∼ r and ( φ, t ) ∈ H α ( C, η, ∞ ) , e R N α ( φ, t ) ∈ H α ( C , δ , ∞ ) . And from the Theorem 7.5 for all γ > N such that e R N + N α ( φ, t ) ∈ B ∞ γ ( φ ∗ r , t ∗ r ) . So from the two claims we have proved [ C> [ η> H α ( C, η, ∞ ) ⊆ W s ( φ ∗ α , t ∗ α ) . Appendix A. Univalent maps
Here we show some results on the class of univalent functions in a domain U containing the interval [ − , Theorem A.1.
For some
K > and < ǫ < K/ the following statement holds:if φ is an univalent map defined in the ball B (0 , K ) satisfying φ ( −
1) = − , φ (1) = 1 and φ ([ − , ⊂ [ − , , then | φ − id | B (0 ,ǫ ) < O ( ǫK ) . The following lemma was established in [1] without a proof. This result is centralin the proof of the complex bounds (Theorem 4) for this reason we think that it isconvenient to present a proof of this.
Lemma A.2 ([1]) . Let
C > be a constant and E ⊃ E ⊃ [ − , be a domainsstrictly contained in the complex plane. There exists a constant K > such thatthe following is satisfied. Let φ : E → C and ψ : E ψ → C be univalent mapswhere E ψ ⊂ E and furthermore φ ([ − , − , , ψ ([ − , − , and | φ − id | E ≤ C. There exists a ρ ( φ ) -stadium D ρ ( φ ) ⊂ E ψ such that φ ( D ρ ( φ ) ) ⊂ E ψ . Moreover ρ ( φ ) ≥ e − K | φ − id | E ρ ( ψ ) , where ρ ( ψ ) is the distance between the boundary of E ψ to the interval [ − , . Inparticular, ψ ◦ φ is defined in D ρ ( φ ) . Proof.
Let D ρ ⊂ E be a ρ -stadium. The proof will be divided in two cases. Firstsuppose that there exists e C > ρ ( ψ ) > e C. So, if z ∈ ∂D ρ we havethe following dist ( φ ( z ) , [ − , ≤ dist ( φ ( z ) , z ) + dist ( z, [ − , ≤ | φ − id | E + ρ then φ ( D ρ ) ⊆ D | φ − id | E + ρ . Take ρ ( φ ) > ρ ( ψ ) = ρ ( φ ) + | φ − id | E . As ρ ( ψ ) > e C we have ρ ( φ ) ≥ (1 − e C | φ − id | E ) ρ ( ψ ) . Then we obtain
K > ρ ( φ ) ≥ e − K | φ − id | E ρ ( ψ ) . Finally we suppose that ρ ( ψ ) ≤ . Let x ∈ [ − , . Using the GeneralizationDistortion Theorem ( [14]) there exists
C > z, x ∈ D ρ ( ψ ) e − C | φ − id | E ≤ | Dφ ( z ) || Dφ ( x ) | ≤ e C | φ − id | E . On the other hand there exists x ∈ [ − ,
1] such that Dφ ( x ) = 1 and by the MeanInequality Theorem on E we have that for all z ∈ D ρ and x ∈ [ − , | φ ( z ) − φ ( x ) | ≤ e C | φ − id | E | z − x | . For all z ∈ ∂D ρ there exists x ∈ [ − ,
1] such that | z − x | = ρ. Then dist ( φ ( z ) , [ − , ≤ e C | φ − id | E ρ for all z ∈ D ρ . Take ρ ( φ ) = e − C | ( φ − id) | E ρ ( ψ ) . This finishes the proof. (cid:3)
Acknowledgment
We would like to thank C. Gutierrez, A. Messaoudi, W. de Melo and E. Vargasfor the very useful comments and suggestions.
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