aa r X i v : . [ m a t h . D S ] F e b THE FULL RENORMALIZATION HORSESHOE FORMULTIMODAL MAPS:THE CONTINUITY OF THEANTI-RENORMALIZATION OPERATOR FOR MULTIMODALMAPS
YIMIN WANG
Abstract.
In this paper, we consider the renormalization operator R for mul-timodal maps. We prove the renormalization operator R is a self-homeomorphismon any totally R -invariant set. As a corollary, we prove the existence of thefull renormalization horseshoe for multimodal maps. Introduction
Renormalization has been an important idea and tool in dynamical systems.Feigenbaum’s renormalization conjecture says that a certain renormalization opera-tor has a hyperbolic fixed point. In fact, the original case considered by Feigenbaumis the periodic doubling case [5]. And such a conjecture was also formulated by Coul-let and Tresser independently from Feigenbaum. For the periodic doubling case,Lanford [9] proved the existence of the hyperbolic fixed point with computer assis-tant , Sullivan [21] and McMullen [13] proved the uniqueness of the fixed point andthe exponential contraction of R . Finally, Lyubich [10] considered the renormal-ization operator R on the space QG of quadratic-like germs, he defined a complexstructrue on QG and then proved the hyperbolicity of the renormalization horse-shoe. In [11], Lyubich proved the set of infinitely renormalizable real polynomialshas Lebesgue measure zero. It implies his famous result: a typical real polynomialis either regular or stochasitc. Avila and Lyubich[1] generalize the result to ana-lytic unimodal case by introducing a method of path holomorphic structure andcocycles. There are also parallel results about the renormalization conjecture forcritical circle maps, see [23, 24, 25].In [18], Smania introduced multimodal maps of type N and proved that deeprenormalizations of infinitely renormalizable multimodal maps are multimodal mapsof type N for some positive integer N . Let I = [ − , f : I → I is called a multimodal map of type N , if there exists unimodal maps f , · · · , f N − with following properties:(1) f j : I → I is a unimodal map fixing − f = f N − ◦ · · · ◦ f ;(3) 0 is a quadratic critical point of f j such that f j (0) ≥ f ′′ j (0) < f , f , · · · , f N − ) a unimodal decomposition of f . For convenience,we will also assume that f is even, i.e. f ( x ) = f ( − x ) for all x ∈ I . Since we Date : February 25, 2021. concern about the infinitely renormalizable case, such an assumption will not losegenerality.A multimodal map f of type N is called renormalizable if there exists a periodicinterval J of period p of f such that f p | J is affinely conjugate to a multimodal mapof type N . There is a canonical way to normalize f p | J to be a multimodal mapof type N , and we call the normalized map R f the renormalization of f and thesmallest integer p is called the renormalization period of f . We say f is infinitelyrenormalizable with bounded combinatorics if f is infinitely renormalizable and therenormalization period p k of R k f is bounded.Let I be the set of all the infinitely renormalizable real-analytic multimodal mapsof type N equipped with the C -topology. Then the renormalization operator R for multimodal maps of type N induces a dynamical system R : I → I . In [19, 20],Smania considered the sub-dynamical sysetem : R| I p : I p → I p , where I p is theset of infinitely renormalizable multimodal maps of type N with combinatoricsbounded by p . He proved that the ω -limit set Ω p of the renormalization operator R| I p is compact and R| Ω p : Ω p → Ω p is topologically conjugate to a full shift offinite elements.In this paper, we prove the renormalization operator of multimodal maps of type N has a full horseshoe: Theorem A.
Let I be the set of all the infinitely renormalizable multimodal mapsof type N and Σ be the set of all the renormalization combinatorics for multimodalmaps of type N . Then there exists a precompact subset A ⊂ I such that the restric-tion R| A of R is topologically conjugate to a two-sided full shift on Σ Z . See section 3 for a definition of the renormalization combinatorics.To prove the full renormalization horseshoe for multimodal maps of type N ,there are two main difficulties. One is to prove infinitely renormalizable multimodalmaps has complex bounds, which has been done by Shen[16]. Since such complexbounds has been built, we can modify the argument of Avila-Lyubich[1] to get asemi-conjugacy desired in Thereom A: Theorem B.
Let I and Σ be as in the assumptions of Theorem A. Then thereexists a precompact subset A ⊂ I such that R ( A ) = A and a continuous bijection h which gives a topological semi-conjugacy between R| A and a two-sided full shifton Σ Z . Another difficuliy is to prove the inverse h − of the semi-conjugacy in Theorem Bis continuous. For the uniformly bounded combinatorics case, the proof is easy.However, it is not trivial to deal with the unbounded combinatorics case. To thisend, we prove the following dichotomy: Key Lemma.
Let { f k } be a sequence of bi-infinitely renormalizable multimodalmaps of type N which is precompact under C -topology. If the renormalizationperiods p k of f k tend to infinity, then each limit of R f k is either a polynomial ofdegree n or with bounded real trace. It is worth mentioning that Avila-Lyubich proved this theorem in the unimodalcase[1], which inspired us so much. Even in that case, the proof is nontrivial andcomplicated.As a corollary of the Key Lemma, we prove
HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 3
Theorem C.
For any totally R -invariant precompact subset A ′ ⊂ I , the restriction R − | A ′ of the anti-renormalization operator R − to A ′ is continuous. Let us now describe the organization of the paper.The proof of Theorem A will be postponed to section 5. In section 2, we recallsome background of real box maps and use the distortion results for real box mapsto prove some compactness lemmas which are the indispensable tools in the proofof the Key Lemma. We recall the definition of renormalization combinatorics insection 3 and then prove Theorem C. From section 4 to the end of this paper, we willuse the idea of path holomorphic space together with Theorem C, following Avila-Lyubich, to show the existence of the full renormalization horseshoe for multimodalmaps of type N . We study the complexification of the multimodal maps of type N and its renormalization operator in section 4, the external and inner structurewill be discussed there. The path holomorphic structure on each hybrid leaf willbe defined in section 5 and we modify the argument of Avila and Lyubich [1] toshow that the renormalization operator contracts exponentially fast along the real-symmetric hybrid leafs by virtue of the complex bounds. Acknowledgement.
The author would like to thank his supervisor Weixiao Shenfor advice and helpful discussions on this problem.2.
Renormalization Operator and infinitely renormalizable maps
In this section, we will first introduce the definition of renormalization of multi-modal maps of type N and recall some results about the real bounds for real boxmaps. Then we will prove the Main Theorem: Key Lemma.
Let { f k } be a sequence of bi-infinitely renormalizable multimodalmaps of type N which is precompact under C -topology. If the renormalizationperiods p k of f k tend to infinity, then each limit of R f k is either a polynomial ofdegree n or with bounded real trace. The extended maps and renormalization.
Fix a multimodal map f oftype N and a unimodal decomposition ( f , f , · · · , f N − ) of f . Let I N = { ( x, j ) | x ∈ I , ≤ j < N } , following Smania [19], we define the extended map F of f : F : I N −→ I N ( x, j ) ( f j ( x ) , j + 1 mod N ) . Clearly, the extended map of f is not unique since f can have several unimodaldecompositions. The extended map is a real box map. Definition 2.1.
A closed interval J ∋ is called a k -periodic interval of an ex-tended map F if it satisfies: (1) F k ( J × { } ) ⊂ J × { } , (2) J × { } , F ( J × { } ) , · · · , F k − ( J × { } ) are closed intervals with disjointinteriors, (3) for every ≤ j ≤ N − , there exists exactly one ≤ m < k such that × { j } ∈ F m ( J × { } ) , (4) k > N .Let p = k/ N , we also say J is a p -periodic interval of f . YIMIN WANG
Let F be an extended map of a multimodal map f of type N . Consider amaximal k -periodic interval J of F , i.e., F k ( ∂J × { } ) ⊂ ∂J × { } (If F has a k -periodic interval, then it must have a maximal k -periodic interval). Then thereexists a canonical affine transformation A : J → I such that A (0) = 0 and A ◦ f k/ N ◦ A − : I → I is a multimodal map of type N . To see this, for every0 ≤ j < N , let m j be the integer such that 0 × { j } ∈ F m j ( J × { } ) and J j be the interval such that J j × { j } is the symmetrization of F m j ( J × { } ) withrespect to (0 , j ). There is a periodic point z j of f on the boundary of J j , let A j : J j → I be the affine transformation such that A j (0) = 0, A j ( z j ) = − A j : J j × { j } → I , ˜ A j ( x, j ) = ( A j ( x ) , j ) for all 0 ≤ j < N . For convenience, wemake a convention that m n = m = 0, J N = J , A N = A , ˜ A N = ˜ A . Then forevery 0 ≤ j ≤ N − F j := ˜ A j +1 ◦ F m j +1 − m j ◦ ˜ A − j : I × { j } → I × { j + 1 mod N } is a unimodal map with critical point 0 and ˜ A ◦ F k ◦ ˜ A − = F N − ◦ · · · ◦ F ◦ F ,it implies A ◦ f k/ N ◦ A − : I → I is also a multimodal map of type N . If f doesnot have a periodic interval with period strictly smaller that k/ N , then we call A ◦ f k/ N ◦ A − is the real renormalization of f and denote it by R f . Clearly, therenormalization of f does not depend on the unimodal decomposition of f . If R f isagain renormalizable, then we will say f is twice renormalizable . If this procedurecan be done infinitely many times, then f will be called infinitely renormalizable .In this paper, we mainly concern about the infinitely renormalizable maps. Definition 2.2.
A multimodal map f of type N is called anti-renormalizable, ifthere exists a renormalizable multimodal map g of type N such that R g = f . Since Smania had proved the renormalization operator R is an injection [20,Proposition 2.2], the anti-renormalization operator is also well-defined. Similar toinfinitely renormalizable maps, we can define infinitely anti-renormalizable maps,and a multimodal map is called bi-infinitely renormalizable if it is both infinitelyrenormalizable and anti-infinitely renormalizable.2.2. Background in real box maps.
Throughout Section 2, we will assume f is a bi-infinitely renormalizable multimodal map of type N with renormalizationperiod p > F of f . Set c j = (0 , j ) ∈ I × { j } for all0 ≤ j ≤ N −
1. Assume ( α,
0) is the F n - fixed point closest to c and ( − α,
0) isthe reflection of ( α,
0) with respect to c . Set I := ( α, − α ) × { } . Let ( β,
0) be the preimage of F − n ( α,
0) closest to the point ( − ,
0) and define a set E := I × { } \ { ( β, , ( − β, , ( α, , ( − α, } . Definition 2.3.
An open subset B of I N is called nice if S k ≥ F k ( ∂B ) ∩ B = ∅ . The concept of nice interval was first introduced by Martens [12]. For a nicesymmetric interval B , we denote D B the first entry domain of B under the iteratesof F , that is, D B = { x ∈ I N | F k ( x ) ∈ B for some integer k ≥ } . HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 5
For any x ∈ D B , the minimal positive integer k = k ( x ) such that F k ( x ) ∈ B iscalled the first entry time of x . The first entry map to B is defined as: R B : D B −→ Bx F k ( x ) ( x ) . The restriction of R B to D B ∩ B is called the first return map . For x ∈ D B ,we shall use L x ( B ) to denote the connected component of D B containing x . Let L x ( B ) = L x ( B ), and for any positive integer j ≥
2, let L jx ( B ) = L x ( L j − x ( B )),whenever it makes sense.For a symmetric nice interval c ∈ I ⊂ I N , the scaling factor of I is defined as: λ I := | I ||L c ( I ) | . For a nice open set K ∩ ω ( c ) = ∅ (where ω ( c ) is the ω -limit set of c ), let M ( K )be the collection of intervals which are pullbacks of components of K . Shen[16]defined the limit scaling factor of K as:Λ K := sup I λ I , where the supremum is taken over all symmetric nice intervals in M ( K ).We say B ⋑ B ⋑ B ⋑ · · · is a nest if there exists x ∈ B such that B n +1 = L x ( B n ) for all n ∈ N .A sequence of nice symmetric intervals B ⋑ B ⋑ · · · ⋑ B L ∋ c is called acentral cascade with respect to c , if B j +1 = L c ( B j ) for all 1 ≤ j ≤ L − R B j ( c ) ∈ B j +1 for each 1 ≤ j ≤ L −
2. Such a central cascade is called maximal if R B L − displays a non-central return, i.e., R B L − ( c ) / ∈ B L . We say that the centralcascade is of saddle node type if R B | B has all the critical points in B L , and doesnot have a fixed point. Remark 2.1. If B ⋑ B ⋑ · · · ⋑ B L is a maximal central cascade, then B j ⋑ B j +1 ⋑ · · · ⋑ B L is also a maximal central cascade for all ≤ j ≤ L − . By a chain we mean a sequence of open intervals { G s } ks =0 such that G s +1 is acomponent of F − ( G s ) for every 0 ≤ s ≤ k −
1. The order of the chain is thenumber of the integers s with 0 ≤ s < n such that G s intersects Crit( F ) and theintersection multiplicity is the maximal number of the intervals G s (0 ≤ s ≤ n )which have a non-empty intersection.We shall need the following known results: Theorem 2.1 ([22, Theorem A]) . There exists < λ = λ ( k F k C ) with the follow-ing property. Let us consider a nest B ⋑ B ⋑ B ⋑ · · · . If R B k does not displaya central return, then λ B k +1 = | B k +1 || B k +2 | > λ. Real bounds for S -unimodal maps were proved earlier by Martens [12].We say an open interval I is a δ -neighborhood of an interval J , which is denotedby (1 + 2 δ ) J , if J ⋐ I and each component I \ J has length equal to δ | J | . Lemma 2.1 ([15, Proposition 2.2]) . For any p, q ∈ N and any δ > , there existsa constant δ = δ ( δ, p, k F k C ) > such that the following holds. Let G = { G j } sj =0 YIMIN WANG and G ′ = { G ′ j } sj =0 be chains such that G j ⋐ G ′ j for all ≤ j ≤ s . Assume theorder of G ′ is at most p and { j | G ′ j ⋑ G s } ≤ q. If (1 + 2 δ ) G s ⊂ G ′ s , then (1 + 2 δ ) G ⊂ G ′ . Moreover, δ → ∞ as δ → ∞ . For two intervals I and J , we briefly say J is geometrically deep inside I if thereis a large δ such that (1 + 2 δ ) J ⊂ I . Remark 2.2. If F q | I : I → I ′ is a first return map and J ′ is geometrically deepinside I ′ , then it follows immediately from Lemma 2.1 that J := ( F q | I ) − ( J ′ ) isgeometrically deep inside I . All the central cascades have been proved to be essentially saddle-node in thefollowing sense by Shen[16]:
Lemma 2.2 ([16, Proposition 5.1, Theorem 5.4]) . For any δ > and ρ > , thereexists b = b ( δ, ρ, k F k C ) and η = η ( δ, ρ, k F k C ) > with the following property.Consider a central cascade B ⋑ B ⋑ · · · ⋑ B L , assume B ⊃ (1 + 2 δ ) B , Λ B < ρ and L > b . Then (1) the central cascade B b ⋑ B b +1 ⋑ · · · ⋑ B L ′ is of saddle-node type for some L − b < L ′ ≤ L ; (2) for any x ∈ B L ′ , we have | R B ( x ) − x | ≥ | B | b ;(3) for each ≤ j ≤ L − , we have the Yoccoz equality: ηk < | B j \ B j +1 || B | < η k where k = min( j, L − j ) . Admissible intervals and transition maps.
We say an interval T is an admissible interval if T ∈ M ( E ). An admissible interval T ′ is called a pullback of T if T ′ ∈ M ( T ). More precisely, we say T ′ is a k -pullback of T if F k ( T ′ ) ⊂ T and F k ( ∂T ′ ) ⊂ ∂T .For an admissible interval T , let A T : T → int( I ) be an orientation-preservingaffine homeomorphism. Let T ′ be a k -pullback of T , then the transition map of T and T ′ is defined as G T,T ′ := A T ◦ F k ◦ A − T ′ : int( I ) → int( I ) . Let { G s } ks =0 be the chain from T ′ to T , that is, a sequence of open intervals suchthat G k = T , G = T ′ and G s +1 is a component of F − ( G s ) for every 0 ≤ s ≤ k − c ∈ Crit( F ), if c / ∈ G s for every 1 ≤ s ≤ k −
1, then we saythe transition map G T,T ′ is short (with respect to c ) , otherwise G T,T ′ is called long(with respect to c ) . For a long transition map, let 1 ≤ m < · · · < m ℓ ≤ k − G m j ∋ c , set T ℓ +1 − j = G m j , then we have a canonicaldecomposition (with respect to c ): G T,T ′ = G T,T ◦ G T ,T ◦ G T ℓ − ,T ℓ ◦ G T ℓ ,T ′ . We consider the principal nest: I := ( α, − α ) × { } ∋ c , I = L c ( I ) , · · · , I n = L c ( I n ) , · · · , HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 7 and I ∞ = T I n is a periodic interval since f is renormalizable with respect to 0.To describe the geometric properties of F , we need the following definition. Definition 2.4.
For each admissible interval I , let U ( I ) be the union of all thecomponents of D I which intersect I ∩ ω ( c ) . We say I has C -bounded geometry if (1) ((1 + 2 C − ) I − (1 − C − ) I ) ∩ ω ( c ) = ∅ ; (2) for each component J of I \ ∂U ( I ) , | J | > C − | I | .An admissible interval which satisfies condition (1) is called C -nice. Remark 2.3. As f is infinitely renormalizable, for any c ∈ Crit( F ) , ω ( c ) = ω ( c ) = P ( F ) , where P ( F ) is the postcritical set of F . Let m (0) = 0 and let m (1) < m (2) < · · · < m ( κ ) be all the non-central returnmoments, i.e., R I m ( k ) − displays a non-central return. The integer κ is called the height of F . Lemma 2.3.
For any q > and ρ > , there exists C = C ( ρ, k F k C ) > and C = C ( ρ, q, k F k C ) > with the following properties. If Λ I < ρ , then (1) 1 + C − < λ I m ( k ) < ρ for all ≤ k ≤ κ ; (2) I is C -nice; (3) for any t ∈ N with inf k | t − m ( k ) | ≤ q , I t has C -bounded geometry.Proof. Theorem 2.1 implies (1). Statements (2) and (3) follows from [16, Proposi-tion 5.10] and [16, Theorem 5.5] respectively. (cid:3)
Given an interval J ⊂ R , let C J := C \ ( R \ J ) denote the plane slit along tworays. Following Shen[16], we define the Epstein class as following. For any C > η >
0, the class K ( C, η ) consists of diffeomorphisms φ : I → I with followingproperties:(1) the C / -norm of φ is at most C ;(2) φ − | int( I ) extends to a real symmetric (1 + η )-qc map from C int( I ) into itself.For any u ∈ [ − / , / Q u ( z ) = u ( z −
1) + z . For any v ∈ (0 , P v ( z ) = v ( z −
1) + 1. Let SE ( C, η, M ) denote the set of all functions Φ : I → I whichcan be written as Φ = ψ m ◦ φ m ◦ · · · ψ ◦ φ for some m ≤ M , where for each 1 ≤ j ≤ m , φ j ∈ K ( C, η ); and ψ j = Q u j forsome u j ∈ [ − / , /
2] or ψ j = P v j for some v j ∈ [1 /C, I → I is in the Epstein class if Φ ∈ SE ( C, , M ) for some C >
M > Remark 2.4.
For any
C > , η > and N > , SE ( C, η, M ) is compactin C -topology. If Φ k ∈ SE ( C, /k, M ) converges to Φ in C -topology, then Φ ∈ SE ( C, , M ) . Lemma 2.4. If F k : J → F k ( J ) is a diffeomorphism, then F − k : F k ( J ) → J extends to a conformal map from C F k ( J ) into C J .Proof. Since F is infinitely anti-renormalizable, there exists a sequence { H j } ∞ j =1 ofinfinitely renormalizable extended maps with following properties: • there exist positive integers m , m , · · · and multi-intervals J , J , · · · suchthat H m j j | J j is affinely conjugate to F ; • each fiber of J i has length less than λ − j with λ > j = 1 , , · · · ,where λ is given by Theorem 2.1. YIMIN WANG
Let Ψ j be the affine conjugacy between F and H m j j | J j . Then H km j j : Ψ j ( J ) → Ψ j ( F k ( J )) is a diffeomorphism, by [16, Proposition 5.7], H − km j j : Ψ j ( F k ( J )) → Ψ j ( J ) can extend to a exp( O ( λ − j ))-qc map from C Ψ j ( F k ( J )) into C Ψ j ( J ) . Use theaffine conjugacy, we conclude F − k : F k ( J ) → J can extend to a exp( O ( λ − j ))-qcmap Φ j from C F k ( J ) into C J for all j ∈ N . By the compactness of normalized K -qcmaps, Φ j converges uniformly to a conformal map Φ from C F k ( J ) into C J . Clearly,Φ | F k ( J ) = F − k . (cid:3) Lemma 2.5.
There exists C = C ( δ, M ) > with the following property. Let { G ′ s } ks =0 and { G s } ks =0 be chains satisfying: • G ′ s ⊃ G s for all ≤ s ≤ k and G ∩ ω ( c ) = ∅ ; • the multiplicity of { G ′ s } ks =0 is at most M ; • G ′ k ⊃ (1 + 2 δ ) G k and | f k ( G ) | ≥ δ | G k | .For any ≤ s ≤ k , let γ s : G s → I be the orientation-preserving affine homeomor-phism. Then the map γ k ◦ F k ◦ γ − : I → I belongs to the Epstein class SE ( C, , M N ) .Proof. We will use a similar argument in the proof of Lemma 2.4. Since F isinfinitely anti-renormalizable, there exists a sequence { H j } ∞ j =1 of infinitely renor-malizable extended maps with following properties: • there exist positive integers m , m , · · · and multi-intervals J , J , · · · suchthat H m j j | J j is affinely conjugate to F ; • each fiber of J i has length less than λ − j with λ > j = 1 , , · · · ,where λ is given by Theorem 2.1.Let Ψ j be the affine conjugacy between F and H m j j | J j . Put b G ′ j,s = Ψ j ( G ′ s ) and b G j,s = Ψ j ( G s ) for all 0 ≤ s ≤ k . Then for any j ∈ N , two chains { b G ′ j,s } ks =0 and { b G ′ j,s } ks =0 satisfy the following conditions: • b G ′ j,s ⊃ b G j,s for all 0 ≤ s ≤ k and b G j, ∩ Ψ j ( ω ( c )) = ∅ ; • the multiplicity of { b G ′ j,s } ks =0 is at most M ; • b G ′ j,k ⊃ (1 + 2 δ ) b G j,k and | f k ( b G j, ) | ≥ δ | b G j,k | . • | b G ′ j,k | < λ − j .By [16, Proposition 5.8], there is a constant C > η >
0, there exists j ∈ N such that b γ j ,k ◦ ( H m j j | J j ) k ◦ b γ − j , : I → I belongs to SE ( C, η, M N ), where b γ j ,k = γ k ◦ Ψ − j and b γ j , = γ ◦ Ψ − j .Thus γ k ◦ F k ◦ γ − : I → I belongs to SE ( C, η, M N ). As η is arbitrary, γ k ◦ F k ◦ γ − : I → I belongs to the Epstein class SE ( C, , M N ). (cid:3) Lemma 2.6.
There exists a constant C ′ = C ′ ( C ) > with the following prop-erty. Let c ∈ Crit( F ) and c ∈ T be a C -nice admissible interval. If T ′ = L c ( T ) and | R T ( T ′ ) | ≥ C − | T | , then T ′ is C ′ -nice and the transition map G T,T ′ ∈SE ( C ′ , , N ) . HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 9
Proof.
Let { G s } ks =0 be the chain from T ′ to T and { G ′ s } ks =0 be the chain such that G ′ k = (1 + 2 C − ) T and G ′ ⊃ T ′ .By [16, Lemma 3.8], the chain { G ′ s } ks =0 has order at most N and multiplicity atmost 4. As | F k ( G ) | = | R T ( T ′ ) | > C − | T | = C − | G k | . It follows Lemma 2.5 thatthere exitsts C ′ > G T,T ′ ∈ SE ( C ′ , , N ).Now we prove T ′ is C ′′ -nice for some C ′′ >
0. To this end, let { ˜ G s } ks =0 bethe chain such that ˜ G k = (1 + C − ) T and ˜ G ⊃ T ′ . A straightforward way toprove T ′ has C ′′ -nice property is using Lemma 2.1, which is left to the readers.We will use another way, by using the compactness of the Epstein class, which alsoworks for subsequent Lemmas. By a similar argument in the previous paragraph,we can conclude γ k ◦ F k ◦ γ − : I → I is in the Epstein class SE ( C ′ , , N ), where γ k : ˜ G k → I and γ : ˜ G → I are orientation-preserving affine homeomorphisms.We shall prove T ′ is well inside ˜ G , that is, ˜ G contains a definite neighborhood of T ′ . For otherwise, there exists a sequence { F j } ∞ j =0 of extended maps with followingproperties: • for any j ∈ N , T ( F j ) is a C -nice admissible interval of F j ; • for any j ∈ N , there exists x j ∈ T ( F j ) and z j ∈ ∂ ˜ G ( F j ) such that | x j − z j || T ′ ( F j ) | − → j → ∞ ; • for any j ∈ N , | F k j j ( x j ) − F k j j ( z j ) | > C − | T ( F j ) | .Let γ j, : ˜ G ( F j ) → I and γ j,k j : ˜ G k j ( F j ) → I be the orientation-preservinghomeomorphisms and b x j = γ j, ( x j ), b z j = γ j, ( z j ). Without loss of generality, wecan assume b x j → b x ∈ I , b z j → b z ∈ I and Φ j := γ j,k j ◦ F k j j ◦ γ − j, converges to some b Φ in C -topology. Then by the properties of { F j } we have • | b x j − b z j | → j → ∞ , and then b x = b z ; • | Φ j ( b x j ) − Φ j ( b z j ) | > C − C C = 12(1 + C ) .Then by the uniform convergence, it follows 0 = | b Φ( b x ) − b Φ( b z ) | > C ) , which isridiculous. Thus T ′ is well inside ˜ G , which implies there exists C ′′ > /C ′′ ) T ′ \ T ′ ∩ ω ( c ) = ∅ since ˜ G \ T ′ ∩ ω ( c ) = ∅ .A similar argument shows that T ′ \ (1 − /C ′′ ) T ′ ∩ ω ( c ) = ∅ . Hence, T ′ is C ′′ -nice. Enlarge C ′ so that C ′ > C ′′ , and we are done. (cid:3) Corollary 2.1.
There exists a constant C ′ = C ′ ( C, d ) > with the followingproperty. Let c ∈ Crit( F ) and c ∈ T be a C -nice admissible interval. If T ′ = L d c ( T ) and R T ◦ · · · ◦ R L d − c ( T ) ( T ′ ) has length at least C − | T | , then T ′ is C ′ -nice and thetransition map G T,T ′ ∈ SE ( C ′ , , d N ) .Proof. It follows immediately by Lemma 2.6 and induction. (cid:3)
Compactness for transition maps.Lemma 2.7.
For any θ > , there exists a constant ξ = ξ ( θ, k F k C ) > such thatthe following holds. Let B, B ′ ∈ M ( I ) , assume B ′ = L x ( B ) for some x ∈ B and B ⊃ (1 + 2 ξ ) B ′ . Then L c ( B ) ⊃ (1 + 2 θ ) L c ( B ′ ) . Proof.
See [16, Proposition 4.1]. (cid:3)
Lemma 2.8.
Let T , T , · · · , T L be a maximal central cascade with respect to some c ∈ Crit( F ) . Assume there exists a positive integer C > such that • T is C -nice with limit scaling factor Λ T < C ; • T ⊃ (1 + 2 C − ) T and min( ℓ, L − ℓ ) ≤ C ; • | T | ≤ C | R ℓT ( T ℓ ) | .Then there exists a positive integer C ′ = C ′ ( C, k F k C ) such that the transition map G T ,T ℓ belongs to the Epstein class SE ( C ′ , , C ′ N ) .Proof. Let b = b ( C − , C, k F k C ) and L − b < L ′ ≤ L be as in Lemma 2.2. Since T is C -nice and | T | ≤ C | R ℓT ( T ℓ ) | , by Corollary 2.1, if ℓ ≤ max(3 b, C ) is not large, thenthere exists a positive integer C ′ > max(3 b, C ) such that G T ,T ℓ ∈ SE ( C ′ , , ℓ N ) ⊂SE ( C ′ , , C ′ N ).Now we suppose that L ≥ ℓ > max(3 b, C ). By Lemma 2.2, T b , T b +1 , · · · , T L ′ isof saddle-node type and for any x ∈ T L ′ , | R T ( x ) − x | ≥ b | T | . Clearly, x, R T ( x ) , · · · , R bT ( x ) lie in order. Thus, we have R bT ( T ℓ ) ⊂ T ℓ − b \ T L ′ .Let J be the component of T ℓ − b \ T L ′ containing R bT ( T ℓ ). Then R ℓ − bT | J maps J diffeomorphicly onto a component b J of T b \ T L ′ − ℓ +2 b since all the critical points of R T b are contained in T L ′ . By Lemma 2.4, ( R ℓ − bT | J ) − : b J → J can extend to a con-formal mapping from C b J onto C J . We shall prove that the diffeomorphism R ℓ − bT | J has uniformly bounded distortion, which implies γ ◦ R ℓ − bT | J ◦ γ − ∈ K ( C ′′ ,
1) forsome C ′′ > γ : J → I and γ : b J → I are orientation preserving affinehomeomorphisms.Since L ′ − ℓ + 2 b ≤ L − ℓ + 2 b ≤ C + 2 b , by Lemma 2.2 or Yoccoz’s Lemma, thereexists C = C ( C, b ) > C | T | ≤ | T L ′ − ℓ +2 b − − T L ′ − ℓ +2 b | . By Corollary 2.1, there exists C ′ > G T ,T b lies in a compact set SE ( C ′ , , b N ) and T b is C ′ -nice. Thus we can extend R ℓ − bT | J to be a diffeomor-phism onto a C -neighborhood of b J , where C = min(1 /C ′ , /C ). By real Koebeprinciple, R ℓ − bT | J has uniformly bounded distortion.Finally, since G T ,T b and G T ℓ − b ,T ℓ both lie in SE ( C ′ , , b N ), we conclude G T ,T ℓ = G T ,T N ◦ γ ◦ R ℓ − N ′ − NT ◦ γ − ◦ G T ℓ − N ′ ,T ℓ belongs to ∈ SE ( C ′ , , C ′ N ) by enlarging C ′ . (cid:3) The following lemmas will play an important role in the proof of the Key Lemma.
Lemma 2.9.
Let c ∈ Crit( F ) . Assume c ∈ T is an admissible interval with | T | < Λ |L c ( I ∞ ) | for some Λ > and c ∈ T ′ is a k -pullback of T with k ≤ N p . Let { G s } ks =0 bethe chain from T ′ to T . If G s ℓ = T , T , · · · , T ℓ = G s is a central cascade in thedecomposition of the chain { G s } ks =0 with following properties: (1) if T , T , · · · , T L is a maximal central cascade, then either L = ℓ or T ℓ +1 = G s for all ≤ s < s ; HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 11 (2) there exists positive integer
C > such that T is C -nice with scaling factor λ T > C − and | T | < C | F s ℓ ( I ∞ ) | , (3) either c / ∈ G s for s ≤ s ≤ s ℓ or T , T , · · · , T ℓ is a cascade with respect to c ,then there exists a positive integer C ′ = C ′ (Λ , C, k F k C ) such that • T ℓ is C ′ -nice with λ T ℓ > C ′− ; • | T ℓ | < C ′ | F s ( I ∞ ) | ; • G T ,T ℓ ∈ SE ( C ′ , , C ′ N ) .Proof. First, we prove there exists ρ = ρ (Λ , k F k C ) such that the limit scalingfactor Λ T of T is less than ρ . Fix some constant θ > Λ and let ξ = ξ ( θ, k F k C ) begiven by Lemma 2.7. We claim that Λ T ≤ ρ := 2 ξ + 1. For otherwise, there existsa critical point c ′ and an admissible interval I ∋ c ′ such that I ⊃ (1 + 2 ξ ) L c ′ ( I ).By Lemma 2.7,Λ L c ( I ∞ ) ⊂ (1 + 2 θ ) L c ( I ∞ ) ⊂ (1 + 2 θ ) L c ( L c ′ ( I )) ⊂ L c ( I ) ⊂ T, which contradicts with the assumption.Let us now consider the maximal central cascade T , T , · · · , T L . We shall provethere exists q = q (Λ , C, k F k C ) such that Q := min( ℓ, L − ℓ ) ≤ q .Set J i := F i ( I ∞ ) for all i ≥
1. Let b = b ( C − , ρ, k F k C ), L − b < L ′ ≤ L and η = η ( C − , ρ, k F k C ) be given by Lemma 2.2. Without loss of generality, we mayassume L > ℓ > b . Case 1. J s ⊂ T L ′ . It follows from Lemma 2.2 that T b , T b +1 , · · · , T L ′ is a centralcascade of saddle-node type and for any x ∈ T L ′ , | R T ( x ) − x | > b | T | . Note that R bT ( x ) / ∈ T L ′ for any x ∈ J s . Indeed, for such x ∈ J s ⊂ T L ′ , x, R T ( x ) , · · · R bT ( x ) lie in order. So if R bT ( x ) ∈ T L ′ , then | R bT ( x ) − x | ≥ N ′ × b | T | = | T | , which is a contradiction. Thus we obtain J s ℓ = R ℓ − T ( J s ) = R ℓ − − bT ◦ R bT ( J s ) ⊂ R ℓ − − bT ( T L ′ − b \ T L ′ ) = T L ′ − ℓ +1 \ T L ′ − ℓ +1+ b . Hence, C − < | J s ℓ || T | ≤ | T L ′ − ℓ +1 \ T L ′ − ℓ +1+ b || T | . By Lemma 2.2, we have | T L ′ − ℓ +1 \ T L ′ − ℓ +1+ b || T | ≍ bηQ . So Q is bounded in terms of C and k F k C . Case 2.
Set M = max { b, [ √ Cη ] + 1 } . If J s ⊂ T ℓ + m \ T ℓ + m +1 for some m > M with ℓ + m < L ′ , then J s ℓ = R ℓ − T ( J s ) ⊂ T m +1 \ T m +2 . It follows from Lemma 2.2,that C − < | J s ℓ || T | ≤ | T m +1 \ T m +2 || T | ≤ max { η ( m + 1) , η ( L − m − } . As η ( m + 1) ≤ ηM < C − , we have ( L − m − < Cη , and then ℓ ≤ L − m < p Cη + 1 . Case 3.
Now suppose J s ⊂ T ℓ + m \ T ℓ + m +1 for some m ≤ M . Take a maximalinteger s ≤ r < s + N p such that J r ⊂ T , where p is the renormalization periodof f . Such an r exists since J s ℓ ⊂ T . Clearly, R T ( J r ) = J s + N p = J s . Claim. J r ⊂ T \ T .Let u = s ℓ − s ℓ − , then R T | T = F u | T . If the claim fails, i.e., J r ⊂ T = G s ℓ − ,then T ℓ +1 is the component of F − u ( T ℓ ) containing J r and u + r = N p + s . If u ≤ s ,then G s − u is the component of F − u ( T ℓ ) containing J r . This implies T ℓ = G s − u ,which contradicts with condition (1). Thus, u > s . However, it is also impossible.Indeed, if u > s , then r = N p + s − u < N p . Then G s ℓ − + N p − r ⊃ J N p ∋ c since J r ⊂ G s ℓ − . A direct computation shows that s ℓ − + N p − r = s ℓ − + s − u = s ℓ − − s − ( s ℓ − s ℓ − ) = s − s ℓ ∈ [0 , s ] ∩ N . By condition (3), T , T , · · · , T ℓ should be a central cascade with respect to c .Then by the definition of r , r ≥ N p since J N p ⊂ T . This is a contradiction, as u + r = N p + s and u > s . Hence, the claim follows.Let D ⊃ J r be the component of the first return domain to T and let F v | D : D → T be the corresponding return map. By the definition of r , we have v = N p + s − r .Set D ′ := ( F v | D ) − ( T ℓ + m \ T ℓ + m +1 ), by the Markov property, F v ( L J r ( D )) mustbe contained in T ℓ + m \ T ℓ + m +1 . It follows D ′ ⊃ L J r ( D ). We claim (1+2 ξ ) L J r ( D ) D . For otherwise, by Lemma 2.7, L c ( D ) ⊃ (1 + 2 θ ) L c ( L J r ( D )). So1Λ ≤ | I ∞ || T | ≤ |L c ( L J r ( D )) ||L c ( D ) | <
11 + 2 θ < . This is absurd. By Lemma 2.2, we have | T ℓ + m \ T ℓ + m +1 || T | ≍ { ℓ, L − ℓ } ) = 1 Q . It follows from Lemma 2.1 that there exists δ = δ ( Q ) such that (1 + 2 δ ) L J r ( D ) ⊂ D and δ ( Q ) → ∞ as Q → ∞ . As δ ≤ ξ , there exists q = q ( ξ, C, k F k C ) = q (Λ , C, k F k C ) such that min( ℓ, L − ℓ ) = Q ≤ q . Thus we are done.Hence, T satisfies the following properties:(1) T is C -nice with limit scaling factor Λ ≤ ρ ;(2) T ⊃ (1 + 2 C − ) T ;(3) C | R ℓT ( T ℓ ) | ≥ C | F s ℓ ( I ∞ ) | > | T | ;(4) Q = min( ℓ, L − ℓ ) ≤ q .By Lemma 2.8, there exists a positive integer C ′ > G T ,T ℓ ∈ SE ( C ′ , , C ′ N ).As | T | < C | F s ℓ ( I ∞ ) | and T is C -nice, it follows easily there exists C ′′ = C ′′ (Λ , C, k F k C )such that | T ℓ | < C ′′ | F s ( I ∞ ) | and T ℓ is C ′′ -nice from the fact that G T ,T ℓ lies in acompact set. The argument is similar to that we used in the proof of Lemma 2.6.We only prove | T ℓ | < C ′′ | F s ( I ∞ ) | . Indeed, if there does not exist such C ′′ , thenthere exists a sequence ( F i ) of extended maps with following properties: • { G i := G T ( F i ) ,T ℓi ( F i ) } converges uniformly to some map G ; • there exist x ( i ) , x ( i ) ∈ F s ℓi i ( I ∞ ( F i )) such that b x ( i ) − b x ( i ) → i → ∞ ,where b x ( i ) = A T ℓi ( F i ) ( x ( i )) and b x ( i ) = A T ℓi ( F i ) ( x ( i )); HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 13 • | F s ℓi − s i i ( x ( i )) − F s ℓi − s i i ( x ( i )) | = | F s i i ( I ∞ ( F i )) | > C − | T ( F i ) | , in otherwords, | G i ( b x ( i )) − G i ( b x ( i )) | > C − .Without loss of generality, we can assume b x ( i ) → b x ∈ [ − , | G i ( b x ( i )) − G i ( b x ( i )) | > C − , take a limit, we obtain 0 = | G ( b x ) − G ( b x ) | > C − . This is absurd.As Q = min( ℓ, L − ℓ ) is uniformly bounded, by (3) of Lemma 2.2, λ T ℓ is uniformlybounded both below and above. Hence, enlarge C ′ to be large enough, then theconclusions of this lemma follow. (cid:3) Lemma 2.10 (Long transition maps) . Assume c ∈ T is an admissible intervalwith | T | < Λ |L c ( I ∞ ) | for some Λ > and T ′ ∋ c is a k -pullback of T with k ≤ N p . If there exists apositive integer C > such that (1) T is C -nice with λ T > C − ; (2) | T | < C | F k ( I ∞ ) | .Then there exists a positive integer C ′ = C ′ (Λ , C, k F k C ) > such that G T,T ′ ∈SE ( C ′ , , C ′ N ) .Proof. First, a similar argument in the proof of Lemma 2.9 shows that there exists ρ = ρ (Λ , k F k C ) such that the limit scaling factor Λ T of T is less than ρ .Let { G s } ks =0 be the chain from G = T ′ to G k = T . Statement ( M ). Let Y = G y , Y ′ = G y ′ (0 ≤ y ′ < y ≤ k ) be two symmetricadmissible intervals in the chain such that (1) Y is C -nice with λ Y > C − ; (2) | Y | < C | F y ( I ∞ ) | .If F ) T ( y − S s = y ′ +1 G s ) ≤ M , then there exists a positive integer C ′ = C ′ (Λ , C, k F k C ) > such that • Y ′ is C ′ -nice with λ Y ′ > C ′− ; • | Y ′ | < C ′ | F y ′ ( I ∞ ) | ; • G Y,Y ′ ∈ SE ( C ′ , , C ′ N ) .Proof of Statement (0) . By Lemma 2.4, the diffeomorphism F − ( y − y ′ − : F y − y ′ ( Y ′ ) → F ( Y ′ )can extend to a conformal map from C F y − y ′ ( Y ′ ) into C F ( Y ′ ) . Since Y is C -nice, F y − y ′ − | F ( Y ′ ) can be extended to a diffeomorphism onto (1 + 2 C − ) Y . By Koebe’sdistortion theorem, F y − y ′ − | F ( Y ′ ) has uniformly bounded distortion, which impliesthat G Y,Y ′ ∈ SE ( C ′ , ,
2) for some C ′ >
0. Then it follows easily | Y ′ | < C ′ | F y ′ ( I ∞ ) | by the compactness of SE ( C ′ , , C ′′ > λ Y ′ > C ′′− . Assume thecritical point contained in Y ′ is c ′ . Let B ′ = L c ′ ( Y ′ ) and r B ′ be the return timefrom B ′ to Y ′ , then r B ′ > y − y ′ . For otherwise, G y ′ + r B ′ ⊃ Y ′ contains a criticalpoint, a contradiction. Thus, F y − y ′ ( B ′ ) lies in a first return component B to Y .Since λ Y > C − and Λ Y ≤ Λ T < ρ , we have ρ − | Y | < |L ˆc ( Y ) | < CC + 2 | Y | , where ˆc is the critical point in Y . Thus, there exists b C > b C − | Y | < | B | < (1 − b C − ) | Y | . As F y − y ′ − | F ( Y ′ ) has uniformly bounded distortion, thereexists C ′′ > C ′′− ) | B ′ | < | Y ′ | .Enlarge C ′ so that C ′ > C ′′ , the conclusion of Statement (0) follows. Statement ( M − ⇒ Statement ( M ) . Case 1. c ∈ y − S s = y ′ +1 G s . Let y ′ < v < y and y ′ < v ′ < y be the largest andsmallest integer such that c ∈ G v ′ ⊂ G v respectively. Consider the canonicaldecomposition of G G v ,G v ′ with respect to c , that is, G v = T ⋑ · · · ⋑ T L ⋑ T · · · ⋑ T L ⋑ · · · ⋑ T ℓ ⋑ · · · ⋑ T ℓL ℓ = G v ′ , where T j , · · · , T jL j is a central cascade satisfying condition (1) in Lemma 2.9, j =1 , · · · , ℓ . As | T | < Λ |L c ( I ∞ ) | , by Theorem 2.1 and Lemma 2.7, we know ℓ isbounded in terms of Λ and k F k C . Let s ji be the integer such that T ji = G s ji forall 1 ≤ j ≤ ℓ , 1 ≤ i ≤ L j . By Statement (0), there exists a positive integer e C > T is e C -nice with λ T > e C − ;P2(1). | T | < e C | F s ( I ∞ ) | ;P3(1). G Y,G v ∈ SE ( e C , , e C N ).It follows from Lemma 2.9 thatH1(1). T L is b C -nice with λ T L > b C − ;H2(1). | T L | < b C | F s L ( I ∞ ) | ;H3(1). G T ,T L ∈ SE ( b C , , b C N ).By Statement ( M − e C > T is e C -nice with λ T > e C − ;P2(2). | T | < e C | F s ( I ∞ ) | ;P3(2). G T L ,T ∈ SE ( e C , , e C N ).By induction, we can obtain two finite sequences of positive integers { e C j } ℓj =2 and { b C j } ℓj =2 such that for all 2 ≤ j ≤ ℓ , the following holds.P1( j ). T j is e C j -nice with λ T j > e C − j ;P2( j ). | T j | < e C j | F s j ( I ∞ ) | ;P3( j ). G T j − L ,T j ∈ SE ( e C j , , e C j N );H1( j ). T jL j is b C j -nice with λ T jLj > b C − j ;H2( j ). | T jL j | < b C j | F s jLj ( I ∞ ) | ;H3( j ). G T j ,T jLj ∈ SE ( b C j , , b C j N ).Consider the transition map G G v ′ ,Y ′ , by Statement ( M − e C ℓ +1 such that • Y ′ is e C ℓ +1 -nice with λ Y ′ > e C − ℓ +1 ; • | Y | < e C ℓ +1 | F y ′ ( I ∞ ) | ; • G G v ′ ,Y ′ ∈ SE ( e C ℓ +1 , , e C ℓ +1 N ). HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 15
Let C ′ = e C ℓ +1 + Σ ℓj =1 ( b C j + e C j ), then the conclusions of Statement ( M ) follow. Case 2. c / ∈ y − S s = y ′ +1 G s . Let y ′ < v < y be the largest integer such thatCrit( F ) ∩ G v = ∅ . Assume the critical point lying in G v is c and let y ′ ≤ v ′ ≤ v be the smallest integer such that G v ′ ∋ c . Consider the canonical decomposition of G G v ,G v ′ with respect to c , that is, G v = T ⋑ · · · ⋑ T L ⋑ T · · · ⋑ T L ⋑ · · · ⋑ T ℓ ⋑ · · · ⋑ T ℓL ℓ = G v ′ , where T j , · · · , T jL j is a central cascade satisfying condition (1) in Lemma 2.9, j =1 , · · · , ℓ . Then the proof is essentially the same as that of Case 1. (cid:3) Lemma 2.11.
Let T be a component of E and T ′ ∋ c be an ( N p − b ) -pullbackof T for some ≤ b ≤ N p . If there exists Λ > such that | I | < Λ | I ∞ | , then thereexists a positive integer C ′ = C ′ (Λ , b, k F k C ) such that G T,T ′ ∈ SE ( C ′ , , C ′ N ) .Proof. First, we show there exists a constant Λ ′ > E ≤ Λ ′ . Fixsome constant θ > Λ and let ξ = ξ ( θ, k F k C ) be given by Lemma 2.7. ThenΛ E ≤ Λ ′ := 2 ξ + 1. Indeed, if Λ E > Λ ′ then there exists a critical point c and anadmissible interval I ∋ c such that I ⊃ (1 + 2 ξ ) L c ( I ). By Lemma 2.7,Λ I ∞ ⊂ (1 + 2 θ ) L c ( I ∞ ) ⊂ (1 + 2 θ ) L c ( L c ( I )) ⊂ L c ( I ) . By Markov property, L c ( I ) ⊂ I , thus | I | ≥ Λ | I ∞ | , which contradicts with theassumption.Recall that F is bi-infinitely renormalizable with renormalization period largethan 2 N . As Λ E ≤ Λ ′ , by [16, Theorem 5.6], | T | > C − and T has C -boundedgeometry for some C >
0. Let { G s } N p − bs =0 be the chain from T ′ to T and s be thelargest integer such that G s ∩ Crit( F ) = ∅ . Put T = G s . We shall prove thereexists a positive integer C > G T,T ∈ SE ( C , , C N );(2) T is C -nice with λ T > C − (3) | T | < C |L c ( I ∞ ) | , where c is the critical point in T ;(4) | T | < C | F s ( I ∞ ) | .Then the conclusion will follow easily. Indeed, by Lemma 2.10, (2),(3) and (4)implies there exists a positive integer C ′′ > G T ,T ′ ∈ SE ( C ′′ , , C ′′ N ).Let C ′ = C + C ′′ , then G T,T ′ ∈ SE ( C ′ , , C ′ N ).Now we prove (1). To this end, we first show there exists b C > | F N p − b ( I ∞ ) | > b C − . We claim | I | > e C − for some e C = e C ( k F k C ) >
0. Forotherwise, there exists a sequence { F j } of extended maps such that • F N j converges to a map H in C -topology; • ( F N j ) ′ ( α ( F j ) , < − α ( F j ) , → (0 ,
0) as j → ∞ .This implies − ≤ H ′ (0 ,
0) = 0, which is absurd. Let x ∈ ∂I ∞ be a periodic pointof F , let b x , b z ∈ F N p − b ( I ∞ ) such that F b ( b x ) = x and F b ( b z ) = c . By the MeanValue Theorem, there exists b ζ ∈ T such that | b x − b z || ( F b ) ′ ( b ζ ) | = | x − c | > e C − .As | ( F b ) ′ ( b ζ ) | is uniformly bounded above in terms of k F k C , | b x − b z | has uniformlower bound. So there exists b C > | F N p − b ( I ∞ ) | > b C − .Put u = N p − b − s . By Lemma 2.4, the diffeomorphism F − ( u − : T → F ( T )can extend to a conformal map form C T into C F ( T ) . Since T is C -nice, by realKoebe principle, F u − has uniformly bounded distortion. This implies γ ◦ F u − ◦ γ : I → I belongs to K ( C ,
1) for some C >
1, where γ : T → I and γ : F ( T ) → I are orietation-preserving homeomorphism. Thus, G T,T ∈ SE ( C , , I ′ = T , I ′ = L c ( I ′ ) , · · · , I ′ j +1 = L c ( I ′ j ) , · · · . Let m ′ (1) < m ′ (2) < m ′ ( κ ′ ) be all the non-central return moments, i.e., R I ′ m ′ ( i ) displaysa non-central return. Note that κ ′ is bounded in terms of Λ and k F k C . Indeed,for any 1 ≤ j ≤ κ ′ , let V j = L c ( I ′ m ′ ( j )+1 ) and U j = L c ( I ′ m ′ ( j )+2 ), then V j +1 ⊂ U j . By Theorem 2.1 and Lemma 2.7, there exists λ ′ > λ ′ U j ⊂ V j .Hence, if κ ′ is not bounded, then | I ∞ | / | I | should be small, a contradiction. SinceΛ T ≤ Λ E < Λ ′ , by Lemma 2.2, | I ′ m ′ ( κ ′ )+1 | ≍ | I ′ | . We claim |L c ( I ∞ ) | ≍ | I ′ m ′ ( κ ′ )+1 | .For otherwise, by Lemma 2.7, I ∞ = L c ( L c ( I ∞ )) will be geometrically deep inside L c ( I ′ m ′ ( κ ′ )+1 ) ⊂ I . This is a contradiction. So |L c ( I ∞ ) | ≍ | I ′ m ′ ( κ ′ )+1 | ≍ | I ′ | = | T | . (cid:3) Let us now recall the definition of Kozlovski-Shen-vanStrien’s enhanced nest (see[7, Section 8] ).
Lemma 2.12.
Let T ∋ c be an admissible interval. Then there is a positive integer ν with f ν ( c ) ∈ T such that the following holds. Let T ′ be the component of f − ν ( T ) containing c and let { G j } νj =0 be the chain from T ′ to T . Then (1) { ≤ j ≤ ν − G j ∩ Crit( F ) = ∅} ≤ N ; (2) T ′ ∩ ω ( c ) is contained in the component of f − ν ( L f ν ( c ) ( T )) .Proof. See [7, Lemma 8.2]. (cid:3)
For an open set B and a point x ∈ B , we use Comp x ( B ) to denote the componentof B containing x . For each admissible interval T ∋ c , let ν = ν ( T ) be the smallestinteger with the properties specified by Lemma 2.1. Following Kozlovski-Shen-vanStrien, we define G ( T ) = Comp c ( f − ν ( L f ν ( c ) ( T ))) , H ( T ) = Comp c ( f − ν ( T )) . Let T be an admissible interval and T ′ be a k -pullback of T for some k >
0. Considerthe chain { G s } ks =0 from T ′ to T , we call T ′ is a kid of T if G s ∩ Crit( F ) = ∅ for1 ≤ s ≤ k − B , let ˆ L x ( B ) denote the component of D B ∪ B containing x . Definition 2.5.
Given an admissible interval T ∋ c , by a successor of T , we meanan admissible interval of the form ˆ L c ( ˆ T ) , where ˆ T is a kid of ˆ L c ′ ( T ) for some c ′ ∈ Crit( F ) . Since F is (infinitely) renormalizable, each admissible interval T has a smallestsuccessor and we denote it by Γ( T ).Then we can define Kozlovski-Shen-vanStrien’s enhanced nest (briefly, KSS nest)as following: let E = I and for each k ≥ L k = G ( E k ) ,M k, = H ( E k ) ,M k,j +1 = Γ( M k,j ) for 0 ≤ j ≤ N − , HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 17 E k +1 = M k, N = Γ N ( H ( G ( E k ))) . For each j ≥
0, let r j be the first return time from L c ( E j ) to E j . Since F is renormalizable, there is a smallest nonnegative integer χ such that r χ = N p .Let m j be the integer such that F m j ( E j +1 ) ⊂ E j and F m j ( ∂E j +1 ) ⊂ ∂E j for0 ≤ j ≤ χ − Proposition 2.1.
There exists
C > such that E j is C -nice for ≤ j ≤ χ , m j +1 ≥ m j and r j +1 ≥ m j for ≤ j ≤ χ − .Proof. See Proposition 8.1 and Lemma 8.3 in [7]. (cid:3)
Fact 1. E χ ⊂ I m ( κ ) − .Proof. It follows easily from the fact that the return time r χ = N p to E χ is strictlylarger than the return time to I m ( κ ) − . (cid:3) Corollary 2.2. N p ≥ max(0 , χ − P j =0 m j ) .Proof. By the definition, N p = r χ > r χ − . It follows easily Proposition 2.1 that r χ − ≥ m χ − ≥ m χ − > m χ − ≥ m χ − + 2 m χ − ≥ · · · ≥ χ − X j =0 m j . (cid:3) Now we are ready to prove the Key Lemma.
Proof of the Key Lemma.
Without loss of generality, we may assume that the renor-malization sequences R f k converges to g ∞ . For all k ∈ N , let F k be the ex-tended map of f k and Φ k be the orientation-preserving linear map such thatΦ k ( I ∞ ( F k )) = [ − , κ k = κ ( F k ) the heightof F k , I k ∞ = I ∞ ( F k ) for all k ∈ N . Case 1.
If sup k | I m ( κ k ) | / | I k ∞ | → ∞ , F N p k k | I m ( κ )+1 : I m ( κ )+1 → I m ( κ ) can beextended to a polynomial-like map g k : U k → V k such that mod( V k \ J ( g k )) → ∞ as k → ∞ . Thus g ∞ is a polynomial of degree 2 N . Case 2.
Assume sup k κ ( F k ) = ∞ and sup k | I m ( κ k ) | / | I k ∞ | < Λ for some Λ >
0. Inthis case, we may assume κ k → ∞ . Since Φ k ( I k ∞ ) = [ − ,
1] for all k , we obtain2 ≤ | Φ k ( I m ( κ k ) ) | ≤ . Then for each j ∈ N , Φ k ( I m ( κ k − j ) ) are bounded intervals (the bound depends on j ). In particular, Φ k ( E χ ( F k ) ) are bounded intervals since E χ ( F k ) ⊂ I m ( κ k ) − , andso are Φ k ( E χ ( F k ) − ).For every j, k ∈ N , let T k,j be the component of F − N p k k ( I m ( κ k − j ) ) containing c , by Corollary 2.2, T k,j ⊂ E χ ( F k ) − . Thus sup j,k | Φ k ( T k,j ) | < ∞ . Passing to asubsequences(use diagonal argument) we may assume Φ k ( I m ( κ k − j ) ) and Φ k ( T j,k )converge respectively to closed intervals D j and D ′ j for all j ∈ N . Moreover, wehave sup j | D ′ j | < ∞ since sup j,k | Φ k ( T k,j ) | < ∞ . By Theorem 2.1, | D j | → ∞ , and then S D j = R . It follows from Lemma 2.10 that g ∞ has an analytic proper extension g ∞ : D ′ j → D j for every j ∈ N . Hence g ∞ has a maximal analytical extension to S j D ′ j which is a bounded interval. Case 3.
Now we assume sup k κ ( F k ) < ∞ , sup k | I m ( κ k ) | < Λ | I k ∞ | for some Λ > p k → ∞ as k → ∞ . In this case, we have sup k | I ( F k ) | / | I k ∞ | < b Λ for some b Λ > i ∈ N , let T k,i be the component of F − ik ( E ( F k )) containing c . Sincefor any 0 ≤ j ≤ i , F N p k k ( ∂T k, N p k − j ) ⊂ {± , ± α ( F k ) , ± β ( F k ) } × { } =: A k , thereexists a k ∈ A k such that F − N p k k ( a k ) ∩ T k, N p k − i ) ≥ i . Thus there are at leastmax( i − ,
0) critical points of F N p k k in T k, N p k − i .Similar to Case 2, we can assume Φ k ( T k, N p k − i ) converges to D i for every i ∈ N .Also we can assume Φ k ( I ( F k )) converges to a bounded interval D ∞ , then S i D i ⊂ D ∞ . It follows from Lemma 2.11 that g ∞ has an analytic extension to D i for all i ∈ N . If g ∞ has an analytic extension to some Ω ⋑ S i D i , then g ∞ must be aconstant function since g ′∞ has infinitely many zeros in S i D i . Hence, the real traceof g ∞ is contained in D ∞ . (cid:3) The continuity of the anti-renormalization operator R − In this section, we will recall the definition of renormalizaiton combinatoricsintroduced by Smania[19, section 2] and prove Theorem C.Let J and J ′ be two disjoint intervals which are contained in N − S j =0 I × { j } , we say J ≺ J ′ if there exists j such that J, J ′ ⊂ I × { j } and J lies to the left of J ′ . Definition 3.1.
Denote by h A, A
Crit , π, P, m i the combinatorial data which con-tains • A = N − S j =0 B j where B j is a collection of disjoint intervals contained in I ×{ j } with B j = m for ≤ j ≤ N − ; • A Crit = { J ∈ A | J ∋ (0 , j ) for some ≤ j ≤ N − } and A Crit ∩ B j ) = 1 for all ≤ j ≤ N − ; • π : A → A is a bijection with the following property: if c ∈ A Crit , then a ≺ b ≺ c implies π ( a ) ≺ π ( b ) ≺ π ( c ) and c ≺ b ≺ a implies π ( a ) ≺ π ( b ) ≺ π ( c ) ; • For any a ∈ A there exists c ∈ A Crit so that π j ( c ) = a , for some j ≥ ; • (0 , ∈ P ∈ A Crit . Definition 3.2.
Two combinatorial data σ = h A, A
Crit , π, P, m i and ˜ σ = h ˜ A, ˜ A Crit , ˜ π, ˜ P , ˜ m i are equivalent if there exists a bijection φ : A → ˜ A such that • φ ( A Crit ) = ˜ A Crit ; • for any x, y ∈ A , x ≺ y if and only if φ ( x ) ≺ φ ( y ) ; • φ ◦ π = ˜ π ◦ φ ; • φ ( P ) = ˜ P . We use M = M ( σ ) to denote the equivalence classes of σ and let Σ ′ be the setof all the combinatorics.Now we are going to define the product of two combinatorics. HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 19
Let M and M be two combinatorics, their product M ∗ M is definded asfollowing:Assuming σ s = h A s , A Crits , π s , P s , m s i is a representative of M s ( s = 1 , J si := π is ( P s ) for s = 1 , ≤ i < m s N . Choose a family { ψ i } m N − i =0 of orientation preserving homeomorphisms such that(1) ψ i : J i → I × { i mod N } for 0 ≤ i < m N ;(2) if J i ∈ A Crit , then ψ i (0 , i mod N ) = (0 , i mod N ).If i = j mod N , then we define I i,j := ψ − i ( J j ). Then we can define a newcombinatorial data σ = h A, A
Crit , P, π, m m i such that • A = { I i,j | i = j mod N , ≤ i < m N , ≤ j < m N } ; • A Crit = { J ∈ A | J ∋ (0 , j ) for some 0 ≤ j ≤ N − } ; • π : A → A such that π ( I i,j ) = ( I i +1 ,j +1 mod ( m N ,m N ) , if I i,j ∈ A Crit I i +1 ,j mod ( m N ,m N ) , otherwise • P = I , ∋ (0 , σ is a combinatorial data and its equivalence class M := [ σ ] does notdepend on the choices in the above construction. Finally, M is defined to be theproduct M ∗ M .A combinatoric M is said to be primitive if it does not have decomposition M = M ∗ M . Let Σ ⊂ Σ ′ be the set of all the primitive combinatorics. Definition 3.3.
Let f be a multimodal map of type N and consider an extendedmap F induced by a decomposition ( f , · · · , f N − ) of f . If P is a maximal periodicinterval for F of period k , then we can associate the following combinatorial data σ = h A, A
Crit , π, P, k/ N i• A = { F i ( P ) : 0 ≤ i < k } ; • A Crit = { F i ( P ) : c ∈ F i ( P ) for some critical point c of F } ; • π : A → A is defined by π ( F i ( P )) = F i +1 mod N ( P ) .If M = [ σ ] is primitive, then we say f is renormalizable with renormalizationcombinatoric M . We say f is a multimodal maps of type N with combinatorics ( M k ) k ≥ ∈ Σ N if f is infinitely renormalizable and R k f is renormalizable with renormalizationcombinatoric M k for all k ≥ M ∈ Σ N there exists a real polynomial in I with combinatorics M (see [19, section 2.1 and section 5.1]). By Kozlovski-Shen-vanStrien’s combinatorial rigidity theorem [8], such a real polynomial is unique. Lemma 3.1.
Assume M m → M in Σ N . If for every m ∈ N , f m is an infinitelyrenormalizable multimodal map of type N with combinatorics M m , then any limitpoint g of { f m } is infinitely renormalizable with combinatorics M .Proof. Without loss of generality, we may assume f m → g . Let J km be the restric-tive interval of the k -th pre-renormalization. As M m → M , M ml = M l for allsufficiently large m and l ≤ k . Thus the periods of J km are the same, and so wecan assume J km converges to a periodic interval 0 ∈ J k for g . For otherwise, 0 willbe a supperattracting periodic point of g . It follows that the post-critical set of g is contained in a solenoidal attractor. Thus g only has repelling periodic point. Hence the restrictive intervals and the post-critical set move continuously, whichimplies g is infinitely renormalizable with combinatorics M . (cid:3) Lemma 3.2.
If a sequence { f k } of infinitely renormalizable multimodal maps oftype N converges to an infinitely renormalizable multimodal map f of type N , then R f k converges to R f .Proof. Let J = [ − a, a ] be the renormlization interval of f , then one of {− a, a } is arepelling periodic point of f . Without loss of generality, we assume a is a repellingperiodic point of f with period p . Since f k converges to f , there exists a k suchthat a k is a repelling periodic point of f k with period p and a k converges to a . Itfollows easily f pk | [ − a k ,a k ] is affine conjugate to a multimodal maps of type N by theuniform convergency. Thus the renormalization period b p k of f k is at most p for all k large.Without loss of generality, we can assume the renormalization period of f k is q ≤ p for all k ∈ N and the corresponding restrictive interval [ − b a k , b a k ] convergesto [ − b a, b a ]. Since f k converges to f in C -topology, f q ([ − b a, b a ]) ⊂ [ − b a, b a ]. Clearly − b a = b a , for otherwise b a = 0 will be a supperattracting periodic point of f , whichis impossible as f is infinitely renormalizable. Thus, [ − b a, b a ] is a periodic interval of f . Note that f q | [ − b a, b a ] is a pre-renormalization of f . Hence, R f k converges to R f . (cid:3) Theorem C.
For any totally R -invariant precompact subset A ′ ⊂ I , the restriction R − | A ′ of the anti-renormalization operator R − to A ′ is continuous.Proof. For any sequence { f k } ⊂ A ′ converging to f ∈ A ′ , we prove R − f k → R − f as k → ∞ . Let N p k be the renormalization period of f k for all k ∈ N . Claim sup k p k < ∞ . Proof of the Claim : Arguing by contradiction, we may assume p k → ∞ . Thenby the Key Lemma, we know that either f = lim k →∞ R ( R − f k )is a polynomial of degree 2 n or it has bounded real trace. But both of these two casesare impossible. Indeed, every polynomial of degree 2 n cannot be anti-renormalizabledue to the degree. On the other hand, f is infinitely anti-renormalizable, let g − k = R − k f . By Theorem 2.1, we know g − k can be extended to [ − λ k , λ k ] for all k ∈ N .Since f is affinely conjugate to an iteration of g − k for all k ∈ N , f cannot havebounded real trace.Since the renormalization periods N p k of R − f k are bounded, then by a similarargument in the proof of Lemma 3.1, we can conclude all the limit point g of {R − f k } is renormalizable and R g = lim k n R ( R − f k n ) = f . Thus R − f k convergesto R − f . (cid:3) The following corollary follows immediately from Lemma 3.2 and Theorem C.
Corollary 3.1.
For any totally R -invariant precompact subset A ′ ⊂ I , the restric-tion R| A ′ of the renormalization operator R to A ′ is a self-homeomorphism. HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 21 Polynomial-like extension for multimodal maps of type N In order to prove Theorem B, we will use the complex method, that is, we extend f to the complex plane and use the tools in complex dynamics. In this section, wewill first recall some definitions and results in polynomial dynamics. Then we proveeach hybrid leaf is homeomorphic to a standard model space E N , which is simplyconnected.We say P is a polynomial of type N if there exists quadratic polynomials P j = a j z − a j − j = 0 , · · · , N −
1) such that P = P N − ◦ · · · ◦ P . And ( P , · · · , P N − )is called a quadratic decomposition of P .A polynomial-like map f : U → V of degree d is a holomorphic proper map ofdegree d where U ⋐ V are quasidisks. The filled Julia set of f is K ( f ) := { z | f n ( z ) ∈ U, ∀ n ≥ } and the boundary of K ( f ) is call the Julia set of f . The idea of polynomial-likemap was first introduced by Douady and Hubbard [4].In this paper, we will consider a special kind of polynomial-like maps which iscalled polynomial-like map of type N . Definition 4.1.
A polynomial-like map f is called a polynomial-like map of type N if there exists quasidisks U = U ⋐ U ⋐ · · · ⋐ U N − ⋐ U n = V and holomor-phic branched double covering f j : U j → U j +1 with a unique critical point z = 0( j = 0 , · · · , N − such that f = f N − ◦ · · · ◦ f : U → V is a polynomial-likerepresentatives of f . We will normalize the polynomial-like map of type N so that 0 , − ∈ U and f ( −
1) = − f , the corresponding polynomial-like germ [ f ] is anequivalence class of polynomial-like maps ˜ f such that K ( ˜ f ) = K ( f ) and ˜ f = f nearthe filled Julia set K ( f ). The modulus of a polynomial-like germ is defined as:mod f = sup mod( V \ U ) , where the supremum is taken over all polynomial-like representatives ˜ f : U → V of f . In this paper, we will not distinguish the notion of a polynomial-like map and itscorresponding polynomial-like germ, and let C N be the family of all the normalizedpolynomial-like germs of type N with connected Julia set. For any δ > C N ( δ ) isused to denote the set of f ∈ C N with mod f ≥ δ . Note that mod f = ∞ if andonly if f is exactly a polynomial.4.1. Topology and Complex analytic structure.
Given a Jordan disk, let B U be the Banach space of functions which are holomorphic on U and continuous upto the boundary ∂U and denote k · k U the L ∞ -norm of B U .Now we define the topology of C N as following. We say f k converges to f in C N if and only if there exists quasidisk W ⋑ K ( f k ) such that f k , k = 1 , , . . . and f arewell defined on W for all sufficiently large k and k f k − f k W → k → ∞ . Wesay K ⊂ C N is closed if for every sequence { f k } ⊂ K , the limit points of { f k } alsobelong to K .Two polynomial-like germ f and g are called hybrid equivalent , if there existsquasiconformal map h : C → C such that h ◦ f = g ◦ h near K ( f ) and ¯ ∂h = 0almost everywhere on K ( f ). For b = ( b , · · · , b N − ) ∈ C N , let P b := ( b N − z − b N − − ◦ · · · ◦ ( b z − b − H ( b ) := { f | f is hybrid equivalent to P b } and let H ( b ) be the component of ˜ H ( b )containing P b . Such an H ( b ) is called a hybrid leaf and it has a natural topologyinduced from the topology of C N . A hybrid leaf is called real symmetric if it containsa real map. Let H ( b , ǫ ) denote the set of all the f ∈ H ( b ) with mod f ≥ ǫ , then H ( b , ǫ ) is a precompact set and any precompact subset K of H ( b ) is contained insome H ( b , ǫ ) (see [14, section 5]). Theorem 4.1.
For every polynomial-like map f of type N , there exists a polynomialof type N hybrid equivalent to f .Proof. The proof is based on quasiconformal surgery. See [19, Proposition 4.1], [6,Theorem A] and also [17]. (cid:3)
On the contrary, we have
Theorem 4.2.
If a polynomial-like map f : U → V is hybrid equivalent to somepolynomial P of type N , then f is a polynomial-like map of type N .Proof. Let ( P , · · · , P N − ) be a quadratic decomposition of P and h be a hybridconjugacy between f and P . Select quasidisks U ⋐ U ⋐ · · · ⋐ U N such that U = U and U N = V . Let W N = h ( V ), W N − = P − N − ( W N ), W N − = P − N − ( W N − ), · · · , W = P − ( W ) = h ( U ). Choose quasiconformal mappings ϕ j : U j → W j suchthat ϕ ( U j − ) = W j − and ϕ j (0) = 0 for all 1 ≤ j < N . Then h − ◦ P N − ◦ ϕ N : U N − → U N is a quasiregular map. Thus, we can choose a quasiconformal map ψ N − : U N − → U N − such that ψ N − (0) = 0 and f N − := h − ◦ P N − ◦ ϕ N ◦ ψ N − : U N − → U N is a holomorphic proper map. Similarly, there exists quasiconformal map ψ N − : U N − → U N − such that ψ N − (0) = 0 and f N − := ψ − N − ◦ ϕ − N ◦ P N − ◦ ϕ N − ◦ ψ N − : U N − → U N − is a holomorphic proper map. By induction, there exist quasiconformal maps ψ j : U j → U j such that ψ j (0) = 0 and f j := ψ − j +1 ◦ ϕ − j +2 ◦ P j ◦ ϕ j +1 ◦ ψ j : U j → U j +1 are holomorphic proper maps (1 ≤ j ≤ N − f := ψ − ◦ ϕ − ◦ P ◦ h , then f N − ◦ · · · ◦ f ◦ f = h − ◦ P N − ◦ · · · ◦ P ◦ P ◦ h = f . Now we only need to checkwhether f is holomorphic. Denote f N − ◦ · · · ◦ f ◦ f by G , by differentialing weobtain a.e. z ∂f ¯ ∂z = ∂G∂w ∂f ¯ ∂z + ∂G ¯ ∂w ¯ f ¯ ∂z = G ′ ∂f ¯ ∂z . It follows from Weyl’s Lemma that f is holomorphic and we are done. (cid:3) Modifying Lyubich’s argument of complex analytic variety in [10]and [11], wecan define the complex analytic structure of H ( ) as following. Set B U := { f ∈B U | f ( j ) (0) = 0 , j = 1 , , · · · , N − } , then it is also a Banach space under the L ∞ -norm. If f : U → V is a polynomial-like representative of f ∈ H ( ), it is easyto see g ∈ B U ( f, ǫ ) has a polynomial-like restriction on a quasidisk slightly smallerthan U for ǫ sufficiently small (where B U ( f, ǫ ) is an ǫ -neighborhood of f in B U .) HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 23
Thus, we have a natural continuous inclusion J U,f,ǫ : B U ( f, ǫ ) → H ( ) and B U ( f, ǫ )is called a Banach slice of H ( ) centered at f . For convenience of notion, we use S U to stand for a Banach slice without specifying f and ǫ . Roughly speaking, J U : S U → H ( ) can be understood as the local chart of H ( ) and Lyubich [10, 11]proved: Lemma 4.1 (Lyubich) . The family of local charts J U satisfies the following prop-erties: P1 Countable base and Compactness. There is a countable family of BanachSlices S i = S U i such that for any f ∈ H ( ) , the Banach Slice S U centeredat f is compactly contained in some S i . P2 Lifting of analyticity. If W ⊂ U , then the inclusion map J U,W : S U → B W is complex analytic. Moreover, let U ⋑ V . Let us consider a locally boundedmap φ : V → B V defined on a domain V in some Banach space. Assumethat the map J V,W ◦ φ : V → B W is analytic, then the map J V,U ◦ φ : V → B U is also analytic. P3 Density. If W ⊂ U , then B U is dense in B W . A space with properties P − P is called a complex analytic variety, then H ( )is a complex analytic variety.A map φ : D → H ( ) is called analytic if for any z ∈ D , there exists a smalldisk D ( z, δ ) and a Banach slice S U such that φ ( D ( z, δ )) ⊂ S U and the restriction φ | D ( z,δ ) : φ ( D ( z, δ )) → S U is analytic in the Banach sense. Clearly, φ is an analyticimplies that φ is continuous.4.2. External maps of polynomial-like germs.
A real analytic circle map g : T → T is called expanding if there exists k ≥ | D g k ( z ) | > z ∈ T where D denotes the derivative with respect to z .Let E N be the family of real analytic expanding circle covering maps g : T → T ofdegree 2 N normalized so that g (1) = 1. E N is simply connected ([1, Lemma 2.1]).Since g is real analytic and expanding, it can be extended to be a holomorphiccovering g : U → V of degree 2 N where U ⋐ V are annular neighborhood of T .Similarly, we can define the modular of expanding circle maps as:mod g = sup mod( V \ U ∪ D ) , where the supremum is taken over all extensions g : U → V of g .We will use the Inductive limit topology of E N (see [10, Appendix 2]) . In thistopology, a sequence g k ∈ E N converges to g ∈ E N if there exists a neighborhood W of T such that all the g k admit a holomorphic extension to W , and g k → g uniformly on W , i.e. sup z ∈ W | g k ( z ) − g ( z ) | → f ∈ C N , consider the B¨ottcher coordinate φ f : C \ K ( f ) → C \ D , then g = φ f ◦ f ◦ φ − f is well defined in a small outer neighborhood of T , by Schwarzreflection principle, g can be extended to a holomorphic expanding map of degree 2 N in a neighborhood of T . Such a map g is unique up to a rotation conjugation, thus itcan be normalized so that g ∈ E N and called an external map of f . Unfortunately,such an external map g may not be unique. However, we can construct a canonicalexternal map g ∈ E N from a polynomial-like germ f ∈ C N . Indeed, we prove thefollowing theorem. Theorem 4.3.
For every b ∈ C N , there exists a homeomorphism I b : E N → H ( b ) .Moreover, mod I b ( g ) = mod g for all g ∈ E N .Proof. Firstly, given g ∈ E N , we choose a continuous path { g t } connecting g = z N and g = g . Then we can construct a continuous family of K -q.c maps h t : C \ D → C \ D with Beltrami differential ν t continuously depending on t such that h = idand h t ◦ g = g t ◦ h t near the unit circle T in the following way:Since { g t } is compact, there exist representatives g t : W t \ D → W t \ D where W t , W t are quasidisks with mod W t \ W t ≥ ǫ for some ǫ >
0. By Gr¨otzsch’sextremal problem, we know W t contains a Euclid disk D (0 , r ) with r >
1. As g t isuniformly expanding near T , we can choose 1 < r < r such that g − t ( D (0 , r )) ⋐ D (0 , r ) for all t . Let γ t ≡ ∂ D (0 , r ) and γ t = g − t ( γ t ), define h t ≡ id outside D (0 , r ) and we can lift h t to ∂ D (0 , r /d ) such that g t ◦ h t = g . Then by extension,we can obtain K -q.c maps h t : A → A t where A t is the annulus bounded by γ t and γ t . Moreover, h t depends continuously on t . Finally, we can lift h t to C \ D by respecting the dynamics so that the Beltrami differentials ν t of h t satisfying k ν t k≤ k = K − K +1 for all t .Now we are going to construct a continuous path { f t } ⊂ H ( b ) from the path { g t } . Consider the B¨ottcher coordinate ξ b : C \ D → C \ K ( P b ), we define Beltramidifferentials µ t on C such that µ t = ( ξ b ) ∗ ν t on C \ K ( P b ) and µ t = 0 on K ( P b ).By the Measurable Riemann Mapping Theorem, we can obtain a continuous path { q t } of K -q.c maps such that ¯ ∂q t = µ t ∂q t and q t fixes 0, 1. Then Q t := q t ◦ P b ◦ q − t defines a polynomial-like map of degree 2 N which is hybrid to P b and Q = P b .By Theorem 4.2, Q t is affinely conjugate to a map in H ( b ). Thus we can usea continuous family of affine transformations { A t } to normalize Q t so that f t := A t ◦ Q t ◦ A − t ∈ H ( b ). Let φ t = A t ◦ q t , then f t = φ t ◦ P b ◦ φ − t . Lemma 4.2.
For every t ∈ [0 , , g t is an external map of the polynomial-like map f t .Proof. Let ψ t : C \ D → C \ K ( f t ) be the continuous family of Riemann mapsnormalized so that ψ = ξ b and ˜ g t = ψ − t ◦ f t ◦ ψ t : T → T fixes 1, hence ˜ g t is anexternal map of f t .To see the existence of ψ t , we consider another normalized family of B¨ottchercoordinates ˆ ψ t : C \ D → C \ K ( f t ) such that ˆ ψ t ( ∞ ) = ∞ , ˆ ψ ′ t ( ∞ ) >
0. It isa continuous family due to the semi-upper continuity of K ( f t ) and the semi-lowercontinuity of J ( f t )( see [10, Lemma 4.15] for an example). Then G t | γ = ˆ ψ − t ◦ f t ◦ ˆ ψ t is continuous in t where γ is a Jordan curve in C \ D . By the Schwarz reflection andMaximum Principle, G t : T → T is a continuous family. Let z ( t ) be a fixed pointof G t so that z ( t ) is continuous and z (0) = 1. Then ψ t := ˆ ψ t ◦ e i Arg z ( t ) : C \ D → C \ K ( f t ) is continuous in t and and ˜ g t = ψ − t ◦ f t ◦ ψ t : T → T fixes 1.It remains to show that g t = ˜ g t for all t ∈ [0 , σ t := ψ − t ◦ φ t ◦ ξ b : C \ D → C \ D is a quasiconformal map conjugating g to ˜ g t whose Beltramidifferential coincides with that of h t . It follows that λ t := σ t ◦ h − t : C \ D → C \ D is a rotation conjugating g t to ˜ g t . Since 1 is the fixed point of g t , λ t (1) must beone of the fixed point of ˜ g t by the conjugacy. Let t = sup { t | λ x (1) = 1 , x ≤ t } ,we claim t = 1. Indeed, λ = id implies λ (1) = 1, thus 0 ≤ t ≤ λ t (1) = 1. Hence t = 1, for otherwise, there exists ˜ t slightlylarge than t such that λ x (1) = 1 for all x ≤ ˜ t , which contradicts with the definitionof t . We conclude that λ t = id and hence g t = ˜ g t for all t ∈ [0 , HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 25 (cid:3)
Let I b ( g ) := f and we should check that the polynomial-like map f we con-structed above does not depend on the choice of the path { g t } .To this end, we first show that once the connecting path g t is chosen, the path f t does not depend on the choice of h t . Suppose ˜ h t is another K -q.c conjugationfrom g = z N to g t , let ˜ f t and ˜ φ t be the corresponding polynomial-like map andhybrid conjugation. Then η t = ˜ φ t ◦ φ − t is a hybrid conjugation from f t to ˜ f t near K ( f t ). Let ψ t : C \ D → C \ K ( f t ) and ˜ ψ t : C \ D → C \ K ( ˜ f t ) be the Riemannmappings in the above construction of the externmal maps respectively.Let ˜ η t = ˜ ψ − t ◦ η t ◦ ψ t , then ˜ η t (1) is a fixed point of g t and g t ◦ ˜ η t = ˜ η t ◦ g t on T .Indeed, ˜ η t is a K -q.c map defined on U \ D , then it can be extended to U \ D . It iscontinuous in t and ˜ η (1) = 1, and hence ˜ η t (1) = 1. Lemma 4.3.
Let g : T → T be an expanding circle map , h is an automorphism of T such that h ◦ g = g ◦ h and h (1) = 1 , then h = id .Proof. It is trivial when g = m d ( x ) = dx mod 1, where d = 2 N . It is well-knownany expanding circle map is quasisymmetricly conjugate to dx mod 1 (see [10, thepoof of Lemma 3.8]), so the conclusion follows easily. (cid:3) Lemma 4.4 ([2, Lemma 2.1]) . Let S be a hyperbolic Riemann surface with boundary γ and H : S → S be a K -q.c map homotopic to the identity rel the boundary, then d S ( x, H ( x )) ≤ C ( K ) where d S is the hyperbolic distance and C ( K ) is a constantonly depend on K . By the above two Lemmas, we conclude that ˜ η t = id on T andd ˜ U \ K ( ˜ f t ) ( η t ( x ) , ˜ ψ t ◦ ψ − t ( x )) = d ˜ ψ − t ( ˜ U \ K ( ˜ f t )) ( ˜ ψ − t ◦ η t ( x ) , ψ − t ( x ))= d U \ K ( f t ) ( ˜ ψ − t ◦ η t ◦ ψ t ( z ) , z )= d U \ K ( f t ) (˜ η t ( z ) , z ) ≤ C ( K )where x = ψ t ( z ) and η t ( x ) → K ( ˜ f t ), ˜ ψ t ◦ ψ − t ( x ) → K ( ˜ f t ) as z → K ( f t ). Henced Euclid ( η t ( x ) , ˜ ψ t ◦ ψ − t ( x )) → z → K ( f t ). By [4, Lemma 2], we obtain aquasiconformal map: Q t ( x ) = ( ˜ ψ t ◦ ψ − t ( x ) , x ∈ C \ K ( ˜ f t ) η t ( x ) , x ∈ K ( ˜ f t )It follows from Weyl’s lemma that Q t is conformal, and thus Q t is affine. But˜ f = f = P b and Q = id, by the continuity, we conclude that ˜ f t = f t for all t ∈ [0 , f = f does not change whether the path { g t } is alternated. Given two paths { g t } and { ˜ g t } connecting z d and g , by the simplyconnectedness of E N , we can choose a homotopy g st with g t = g t and g t = ˜ g t .For every s ∈ [0 , f s be the polynomial-like map corresponding to the path { g st } , then f s are hybrid equivalent and g is the external map of f s . By a similarargument as above, we can show that there exists a continuous family of affinetransformations Λ s such that Λ = id and Λ s ◦ f ◦ Λ − s = f s . Hence, by thecontinuity, f s = f for all s ∈ [0 , I b ( g ) implies the continuity of I b . Let us now prove that I b is a bijection. For every f ∈ H ( b ), choose a path { f t } ⊂ H ( b ) to connect P b and f . We will prove that { f t } has a unique lift in E N ,and this implies that I b is a bijection. Consider a continuous family of conformalmappings ϕ t : C \ D → C \ K ( f t ) such that ϕ t ( ∞ ) = ∞ and ϕ ′ t ( ∞ ) >
0. Then˜ g t := ϕ − t ◦ f t ◦ ϕ t : T → T is an analytic expanding map of degree 2 N for all t ∈ [0 ,
1] and ˜ g = z N . We can choose a continuous family of conformal maps A t : C \ D → C \ D so that g t := A − t ◦ ˜ g t ◦ A t : T → T belongs to E N and A = id.It is easy to check that I b ( g t ) = f t by the definition, and so { g t } is a lift of { f t } . If { f t } has another lift { ˆ g t } , then ˆ g = z N since P b has a unique preimage z z N .Let ψ t : C \ D → C \ K ( f t ) and ˆ ψ t : C \ D → C \ K ( f t ) be the conformal mapsso that f t ◦ ψ t = ψ t ◦ g t , ψ = ξ b and f t ◦ ˆ ψ t = ˆ ψ t ◦ ˆ g t , ˆ ψ = ξ b respectively. Set η t := ψ − t ◦ ˆ ψ t , we get η t ◦ ˆ g t = g t ◦ η t and η = id. Thus η t (1) is one of the fixedpoint of g t . But η (1) = 1, so by continuity, η t (1) = 1 for all t ∈ [0 , η t can only be the identity.Hence ψ t = ˆ ψ t , and this implies g t = ˆ g t for all t ∈ [0 , { g t } is theunique lift of { f t } .For every ǫ >
0, consider the restriction I b | E N ( ǫ ) : E N ( ǫ ) → H ( b , ǫ ) of I b to E N ( ǫ ). It is a continuous bijection, so by the compactness of E N ( ǫ ), it is a home-omorphism. If f n → f in H ( b ), then { f n } ∞ n =1 ∪ { f } is a compact subset of H ( b ),thus it is contained in some H ( b , ǫ ). Hence g n := I − b ( f n ) ∈ E N ( ǫ ) for all n ∈ N .Since E N ( ǫ ) is compact, every subsequence of g n has a limit point, and by thecontinuity and the bijectivity of I b , the limit point must be I − b ( f ). This implies I − b ( f n ) converges to I − b ( f ), thus I − b is continuous. (cid:3) By Theorem 4.3, for every f ∈ C N , there exists a unique b such that f ∈ H ( b )and we denote it by χ ( f ).4.3. Complex renormalization for multimodal maps of type N.
Let us nowdefine the complex renormalization for multimodal maps of type N .We say a multimodal map f of type N has a polynomial-like extension if thereexists quasidisks U , U , · · · , U N such that f : U → U N is a polynomial-like map, I ⊂ U j and f j : U j → U j +1 is holomorphic proper for all 0 ≤ j ≤ N − f j : I → I ) N − j =0 is a unimodal decomposition of f . Definition 4.2.
A multimodal map f of type N is called complex renormalizableif it is real renormalizable and both itself and its real renormalization R f havepolynomial-like extensions. The germ of the polynomial-like extension of R f willbe called the complex renormalization of f . In [16], Shen proved the complex bounds for all the infinitely renormalizable realanalytic box map without critical points of odd order:
Theorem 4.4 ([16, Theorem 3’]) . There exists ǫ > with the following property.If F is a compact family of infinitely renormalizable multimodal maps of type N ,then there exists K > such that for any k > K and f ∈ F , R k f has an polynomial-like extension R k f : U → V with mod V \ U ≥ ǫ . Hence, all the maps in I are actually infinitely complex renormalizable (see also[16, Theorem 3]), so from now on we don’t distinguish the terminology of real HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 27 renormalization and complex renormalization for multimodal maps. We mentionhere that Clark, Trejo and vanStrien [3] proved the complex bounds for all theinfinitely renormalizable real analytic box map recently.For polynomial-like germs of type N , one can still easily define the pre-renormalizationjust as a first return map. However, there is not a canonical way to normalize sucha first return map to the normalized form. A usual way to do this, is to use the ex-ternal marking. Nevertheless, we can define the renormalization for polynomial likegerm f which is in a hybrid leaf of a complex renormalizable multimodal map f ∗ oftype N as following: Choose a path { f t } t ∈ [0 , in this hybrid leaf to connect f ∗ and f , let h t be the hybrid conjugacy between f ∗ = f and f t , then f t = h − t ◦ f ∗ ◦ h t .Suppose the renormalization period of f ∗ is p , then f pt = h − t ◦ f p ∗ ◦ h t restricting tosome small region is a pre-renormalization for f t . Finally, there exists a continuousfamily { Λ t } of affine maps such that Λ t ◦ f pt ◦ Λ − t ∈ H ( R f ∗ ) for all t ∈ [0 ,
1] suchthat Λ ◦ f p ∗ ◦ Λ − = R f ∗ and we define R f := Λ ◦ f p ◦ Λ − . As H ( f ∗ ) is homeo-morphic to E N , it is simply connected, so the definition of R f dose not depend onthe choice of the path { f t } .5. Path holomorphic structure on hybrid leaves
In this section, we will use the method of path holomorphic space developedin [1] by Avila and Lyubich. Following Avila-Lyubich [1], we define the pathholomorphic structure on all the real-symmetric hybrid leaves. Under the pathholomorphic structure, the renormalization operator between two hybrid leaves iscontracting with respect to the corresponding Carathe´odary metric. Use Avila andLyubich’s idea of cocycles, one can transfer the beau bounds (uniform a priori com-plex bounds) for real maps to the beau bounds for entire hybrid leaves of real maps.Altogether the contracting property for the renormalization operator and the beaubounds, we show the exponential contraction of the renormalization operator alongthe real-symmetric leaves. Some proofs in this section are similar to the unimodalcase, so we will skip these proofs. For details, we refer the readers to section 3 − Definition 5.1.
Let X be a topological space, a path holomorphic structure H ol ( X ) on X is a family of continuous paths Γ = { γ | γ : D → X continuous } such that (1) Γ contains all constant maps; (2) for any γ ∈ Γ and holomorphic map φ : D → D , the composition γ ◦ φ belongs to Γ .A topological space X equipped with a path holomorphic structure is called a pathholomorphic space. Every element in H ol ( X ) will be called a holomorphic path. For two path holomorphic space X and Y , we say Φ : X → Y is a path holomorphicmap if Φ maps each holomorphic path in X to a holomorphic path in Y . We denoteby H ol ( X , Y ) the set consisting of all the path holomorphic map from X to Y . Definition 5.2 (Holomorphic path in H ( )) . Let { f λ } λ ∈ D be a continuous path in H ( ) , we say { f λ } λ ∈ D is a holomorphic path if there exists a holomorphic motion h λ ( z ) : D × C → C such that (1) h = id ; (2) f λ ◦ h λ = h ◦ f on K ( f ) ; (3) ¯ ∂h λ = 0 a.e on K ( f ) . Definition 5.3 (Locally holomorphic path) . Let { f λ } λ ∈ D be a continuous path in H ( ) , we say { f λ } λ ∈ D is a locally holomorphic path if for any λ ∈ D , there existsa disk D ( λ , r ) and a holomorphic motion h λ ( z ) : D ( λ , r ) × C → C such that (1) h λ = id ; (2) f λ ◦ h λ = h λ ◦ f λ on K ( f λ ) ; (3) ¯ ∂h λ = 0 a.e on K ( f λ ) . Note that { f λ } λ ∈ D is a holomorphic path if and only if it is a locally holomor-phic path. Let H ol ( H ( )) be the set of all the holomorphic paths in H ( ), thenH ol ( H ( )) is a path holomorphic structure on H ( ).The following lemma explains the relation between path holomorphic structureand analytic structure in H ( ). Lemma 5.1.
A map φ : D → H ( ) is a holomorphic path in H ( ) if and only if itis analytic.Proof. If φ : D → H ( ) is analytic, then for any λ ∈ D , there exist a sufficientlysmall round disk D ( λ , r ) and a Banach slice S U = B U ( ˆ f , ǫ ) such that φ ( D ( λ , r )) ⊂S U . As r is sufficiently small, we can assume ǫ is also sufficiently small so that thereexist quasidisks U f slightly smaller than U and a quasidisk V such that f : U f → V is polynomial-like for every f ∈ φ ( D ( λ , r )). Then we can easily construct ananalytic family of quasiconformal maps { h λ : V \ U φ ( λ ) → V \ U φ ( λ ) } λ ∈ D ( λ ,r ) whichrespects the dynamics on the boundaries. Let µ λ be the Beltrami differential of h λ and pull µ λ back by φ ( λ ), then µ λ can be extended to V \ K ( φ ( λ )). Finally, define µ λ = 0 on C \ V ∪ K ( φ ( λ )) and by the Measurable Riemann Mapping Theorem,we obtain an analytic family of quasiconformal maps H λ such that H λ ◦ φ ( λ ) = φ ( λ ) ◦ H λ on U φ ( λ ) and ¯ ∂H λ = 0 a.e. on K ( φ ( λ )). It follows that φ ( D ) is aholomorphic path in H ( ).(Indeed, it is actually a Beltrami path.)Vice versa, if φ : D → H ( ) is holomorphic path, set f λ = φ ( λ ) for all λ ∈ D , thenfor any λ ∈ D there exist a disk D ( λ , r ) ⊂ D and a holomorphic motion h λ ( z ) : D ( λ , r ) × C → C such that h λ = id, f λ ◦ h λ = h λ ◦ f λ on K ( f λ ) and ¯ ∂h λ = 0a.e on K ( f λ ). Choose r sufficiently small, we can assume there exist quasidisks U and V such that f λ : U → V is polynomial-like and S λ ∈ D ( λ ,r ) K ( f λ ) ⋐ U . Since φ is a continuous map, φ ( D ( λ , r )) is contained in some Banach slice S U once r issmall. It remains to prove that for every z ∈ U , f λ ( z ) is holomorphic in λ . As f λ is hybrid conjugate to z d , int( K ( f λ )) is a quasidisk. Since f is a holomorphicfunction of ( λ, z ) ∈ D ( λ , r ) × int( K ( f λ )), Hartog’s Theorem implies that f λ ( z ) isin fact a holomorphic function of ( λ, z ) through D ( λ , r ) × U . (cid:3) Definition 5.4 (Beltrami path) . For every b ∈ C , a path { f λ } λ ∈ D ( λ ,r ) ⊂ H ( b ) is called a Beltrami path if there exists a holomorphic motion h λ : C → C over D ( λ , r ) , based on λ , that provides a hybrid conjugacy between f λ and f λ . The proof of Lemma 5.1 has implied the following corollary:
Corollary 5.1.
A continuous path { f λ } λ ∈ D ⊂ H ( ) is a holomorphic path if andonly if it is a Beltrami path. Now we are going to use the homeomorphism I b ◦ I − to define the path holo-morphic structure on the hybrid leaf H ( b ) for each b . HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 29
Definition 5.5 (Path holomorphic structure on H ( b )) . For every b ∈ C , a con-tinuous path { f λ } λ ∈ D ⊂ H ( b ) is a holomorphic path if {I b ◦ I − ( f λ ) } λ ∈ D is aholomorphic path in H ( ) . Let h D ( · , · ) be the hyperbolic metric on D and let d D ( · , · ) := e h D − e h D + 1 . By con-vexity, d D is a metric on D .For each hybrid leaf H ( b ), following Avila-Lyubich [1], we define d H ( b ) ( f , f ) = sup φ ∈ Hol( H ( b ) , D ) d D ( φ ( f ) , φ ( f )) , for any f , f ∈ d H ( b ) . It is a well-defined Carathe´odory metric on the path holo-morphic space H ( b ). (See [1, Theorem 4.2 and Lemma 4.1].)Since the homeomorphism I b ◦ I − and the renormalization operator R mapBeltrami paths to Beltrami paths, we obtain: Corollary 5.2.
For every b ∈ C , a continuous path { f λ } λ ∈ D ⊂ H ( b ) is a holomor-phic path if and only if it is a Beltrami path. Lemma 5.2.
For every b , b ∈ C , the renormalization operator R : H ( b ) →H ( b ) is path holomorphic. Recall that I is the set of all the infinitely renormalizable multimodal maps oftype N . A hybrid leaf H ( b ) is called real-symmetric if it contains a polynomial-like extension for some multimodal maps of type N . Let b I = S f ∈I H ( χ ( f )), where H ( χ ( f )) is the real-symmetric hybrid leaf containing f . A family F ⊂ b I is said tohave beau bounds if there exists ǫ > δ > n δ so that mod( R n f ) ≥ ǫ for all n ≥ n δ and any f ∈ F with mod( f ) ≥ δ . ByTheorem 4.4, I has beau bounds. Let us restate the following two theorems in [1]to our situation. For more details, we refer the readers to section 6 − Theorem 5.1. [1, Theorem 6.2]
There exists ǫ > with the following property.For any γ > and δ > there exists N = N ( γ, δ ) such that for any two maps f, ˜ f ∈ C N ( δ ) ∩ b I in the same real-symmetric hybrid leaf we have R k f, R k ˜ f ∈ H b k ( ǫ ) , and d H b k ( R k f, R k ˜ f ) < γ, k ≥ N, where b k = χ ( R k f ) .Proof. For each real-symmetric hybrid leaf H ( b ), we can associate a cocycle G = G b with values in H ol ( H ( ) , H ( )) as following: for each b ∈ C , let Ψ b := I ◦ I − b : H ( b ) → H ( ) and we define G m,n (Ψ b ( f )) := Ψ b n − m ( R n − m ( f )) , where b n − m = χ ( R n − m ( f )). Let G be the set of all such cocycles which correspondto real-symmetric hybrid leaves. Similar to the unicritical case, one can show that G satisfies the hypotheses of [1, Theorem 6.3] (see [1, Lemma 6.4,Lemma 6.5] foran example). Then the Theorem follows from [1, Theorem 6.3]. (cid:3) Theorem 5.2. [1, Theorem 5.1]
Let
F ⊂ C N be a family of infinitely renormalizablemaps with beau bounds which is forward invariant under renormalization. If F is a union of entire hybrid leaves then there exists λ < such that whenever f, ˜ f ∈ F are in the same hybrid leaf, we have d H b k ( R k f, R k ˜ f ) ≤ Cλ k , k ∈ N , where b k = χ ( R k f ) and C > only depends on mod f and mod ˜ f . Combine these two theorem, we conclude that
Corollary 5.3.
For any two f, ˜ f ∈ b I in the same real-symmetric hybrid leaf, thenthere exists C > and < λ < such that for sufficiently large k ∈ N , d H b k ( R k f, R k ˜ f ) ≤ Cλ k . where b k = χ ( R k f ) and C > only depends on mod f and mod ˜ f . Now we are going to prove Theorem B.
Theorem B.
Let I and Σ be as in the assumptions of Theorem A. Then thereexists a precompact subset A ⊂ I and a topological semi-conjugacy between R| A and a two-sided full shift on Σ Z .Proof. The proof is similar to Avila and Lyubich’s, but for completeness we give aproof here. There exists ǫ > b I . For every real-symmetric polynomial P b ∈ b I , we have mod R k P b ≥ ǫ . Given M = ( M k ) k ∈ Z , for any k <
0, there is a unique real-symmetric polynomial P k with combinatorics ( M k ) k ≥ k . Then for any l ≥ k , set f l,k = R l − k P k . Clearly, f l,k is infinitely renormalizable with combinatorics ( M k ) k ≥ l and mod f l,k ≥ ǫ . Bythe precompactness of C N ( ǫ ), we may select a subsequence k ( j ) → −∞ such that f l,k ( j ) converges to some f l ∈ C N ( ǫ ). It follows from Lemma 3.1 that f l is infinitelyrenormalizable with combinatorics ( M k ) k ≥ l . Using the diagonal procedure (goingbackwards in l ), we ensure that R f l − = f l . Then f is a bi-infinitely renormalizablemap with combinatorics M and mod R k f ≥ ǫ for all k ∈ Z .Let us now prove that the f we constructed in the above paragraph is unique. If˜ f is another bi-infinitely renormalizable map with combinatorics M and mod R k ˜ f ≥ ǫ for all k ∈ Z . By the combinatorial rigidity, R k f and R k ˜ f are in the samereal-symmetric hybrid leaf for all k ∈ Z . It follows from Theorem 5.2 thatd H b k ( R k f , R k ˜ f ) = d H b k ( R l R k − f , R l R k − l ˜ f ) ≤ Cλ l , l ∈ N , and let l → + ∞ we get R k f = R k ˜ f for all k ∈ Z . In particular, f = ˜ f .Now we define h : Σ Z → F such that h ( M ) = f where f is a bi-infinitelyrenormalizable map with combinatorics M . Since the existence and uniqueness of f has been proven, the map h is well-defined. Let A := h (Σ Z ), then h : Σ Z → A is surjective.To see the injectivity of h , we assume that if there exist M = M such that h ( M ) = h ( M ) = f . Then by the definition of h , f is bi-infinitely renormalizablewith combinatorics M and M . Clearly, M k = M k for all k ≥
0. However, bythe injectivity of the renormalization operator R (see [20, Proposition 2.2]), R k f is a singleton for k ∈ Z − and then M k = M k for all k ∈ Z − . Thus M = M .Finally, the continuity of h follows easily from Lemma 3.1 and we are done. (cid:3) To prove Theorem A, it remains to prove the following lemma:
Lemma 5.3. If f k → f in A , then h − ( f k ) → h − ( f ) in Σ Z . HE FULL RENORMALIZATION HORSESHOE FOR MULTIMODAL MAPS 31
Proof.
Let M = ( M l ) l ∈ Z and M k = ( M kl ) l ∈ Z be the combinatorics of f and f k ( k ∈ N ) respectively. It sufficies to show that for any l ∈ N , M kl = M l for all − l ≤ l ≤ l and k sufficiently large.Since the periodic points of an infinitely renormalizable map are all repelling,the post-critical set moves continuously in a neighborhood of f . Thus it is clearthat M kl = M l for 0 ≤ l ≤ l and k large.To prove M kl = M l for all − l ≤ l < k sufficiently large, we just need toprove R − l f k → R − l f for − l ≤ l <
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