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Hyperbolicity of Renormalization for bi-cubic circle maps with bounded combinatorics

We construct a hyperbolic attractor of renormalization of bi-cubic circle maps with bounded combinatorics, with a codimension-two stable foliation.


Introduction
Renormalization theory, which has revolutionized one-dimensional dynamics, has developed around two key examples: unimodal maps of the interval, and maps of the circle with a single non-degenerate critical point (critical circle maps). Renormalization techniques for critical circle maps first appeared in the works [ORSS83] and [FKS82], and were further developed by O.E. Lanford [Lan86], who described a conjectural renormalization picture for such maps in full generality. This set of conjectures, known as Lanford's Program, became the subject of intensive study for several decades (see the historical account in [Yam02]). A cornerstone of Lanford's Program is the existence of a horseshoe attractor for renormalization of critical circle maps with irrational rotation numbers. The symbolic dynamics on the attractor is described by (a compactification of) the natural extension of the action of the Gauss map G(ρ) = {1/ρ} on the rotation numbers of the maps. In a suitable Banach manifold, the attractor is hyperbolic with a single unstable direction, which can be informally thought of as the direction of change of the rotation number of the map. The maps with the same irrational rotation number form stable manifolds of the horseshoe, and renormalization hyperbolicity promotes the topological equivalence of such maps to a smooth one with an appropriate degree of smoothness.
Lanford's Program was completed by the second author in [Yam02,Yam03]. A natural direction of generalizing it is to consider maps with more than one critical point -in which case, adding each new critical point would create an additional combinatorial invariant for the maps, which should result in an extra unstable direction for renormalization.
The problem of extending Lanford's Program is already very interesting for maps with two critical points. In [Yam19], the second author considered renormalization of analytic maps of the circle with two cubic critical points (bi-cubic), and proved that every periodic combinatorial type corresponds to a unique hyperbolic periodic orbit of renormalization with two unstable directions, in the space of analytic maps of the circle. Then, in [ESY20], D. Smania and the two authors proved that, restricted to the set of bi-cubic maps whose rotation numbers are irrationals of a type bounded by some B ∈ N, renormalization has an attractor A B . Periodic orbits are dense in A B so it would seem natural to extend the hyperbolicity result of [Yam19] to A B , similarly to how the theory developed for maps with a single critical point [Yam02,Yam03].
However, the attractor A B contains maps with a single critical point of order 9. These appear as limits of renormalizations of bi-cubic maps, for which the two critical orbits "collide". As was shown by the second author [Yam03], in the space of unicritical analytic circle maps, renormalization has only one unstable direction at these points. Near these maps, as critical points move (combinatorially) closer, the two unstable cone fields constructed in [Yam19] "collide". In the limit, they collapse into a single unstable cone field for unicritical circle maps. The same problem would, of course, arise for any number of critical points greater than one.
In this paper, we overcome the difficulty, by working in a different function space, in which the collapse does not happen, extending the definition of renormalization operator to this space, and then proving that the attractor is uniformly hyperbolic for the new definition. As has been the second author's experience in the subject, the correct choice of a function space and an appropriate definition of renormalization play a crucial role. 1 The idea which we develop in this paper is applicable to renormalization horseshoes for maps with arbitrary criticality, constructed by I. Gorbovickis and the second author in [GY21].
Our main result can be summarized as follows: There exists a suitable function space into which the space of bi-cubic maps considered in [ESY20] naturally embeds, in which: • renormalization is a compact analytic operator; • for each B ∈ N, the attractor for renormalization of bi-cubic maps of type bounded by B is uniformly hyperbolic with stable foliation of codimension two.
1.1. Organization of the paper. In Section 2, we give basic definitions which are used in the rest of the paper. In the following two sections, 3 and 4, we review the theory of renormalization for commuting pairs (real and complex) and cylinder renormalization, respectively. In section 5, we define renormalization in a suitable space of triples. Finally, in section 6 we prove our Main Theorem in the space of triples.

Preliminaries
2.1. Bi-cubic circle maps. When we talk about the circle, we will usually mean the affine manifold T = R/Z, which is canonically identified with the unit circle S 1 via the exponential map θ → exp(2πiθ). We let R ρ (x) = x + ρ mod Z denote the rotation by an angle ρ ∈ T. Given a circle homeomorphism f , we denote by ρ(f ) its rotation number. A bi-cubic circle map is an orientation preserving circle homeomorphism of class C ω with two critical points both of which are of cubic type.
To fix the ideas, we will place one of the two critical points at 0 ∈ T. Note that a C ω map of T is a germ of a complex-analytic (and real-symmetric) map of an annular neighborhood of T in C/Z to C/Z. Suppose, f is a bi-cubic map with an irrational rotation number ρ(f ) / ∈ Q whose critical points are 0 and c. By a theorem of Yoccoz [Yoc84], there exists an orientation preserving homeomorphism φ : T → T which is a conjugacy between f and a rigid rotation: . We will say that the signature of f is the pair C(f ) = (ρ(f ), δ), where δ ∈ (0, 1) is the length of the arc of T going from 0 to φ(c) in positive (counterclockwise) direction. Clearly, f is topologically conjugate to another bi-cubic map g by an orientation preserving map fixing 0, and sending the other critical point of f to a critical point of g if and only if C(f ) = C(g).
Examples of bi-cubic maps are given by a family of Blaschke products where p, q ∈ C with |p| > 1, |q| > 1 and t ∈ [0, 1]. It was considered by Zakeri in [Zak99], who proved the following statements: and in what follows we will always assume a j ∈ N. Note that an irrational number in (0, 1) has a unique infinite continued fraction expansion; a continued fraction expansion of a rational number is necessarily non-unique. This is due to the fact that [n] = [n − 1, 1]; we will always use the shorter expansion for a rational. The Gauss map x acts as a forgetful map on the continued fraction expansions of numbers in (0, 1): Suppose that the continued fraction expansion of a number ρ = [a 0 , a 1 , . . .] has at least m terms; its k-th continued fraction convergent for k ≤ m is the rational number p k q k ≡ [a 0 , a 1 , · · · , a k−1 ].
Recall, that the denominators {q k } k∈N satisfiy the following recursive formula: q 0 = 1, q 1 = a 0 and q k+2 = a k+1 q k+1 + q k for all k ≥ 1. (2.2) If ρ = ρ(f ) ∈ (0, 1) is the rotation number of a circle homeomorphism f , then f q k (x) is a closest return of a point x ∈ T in the following sense. Suppose q k+1 is defined (that is, the continued fraction expansion of ρ has at least k+1 terms). Denote I k (x) the arc of T whose endpoints are x, f q k (x) and such that f q k+1 (x) / ∈ I k (x). Then for all j < q k , the iterate If we denote t the iterates q k + q k+1 , for 0 ≤ ≤ a k+1 (compare with (2.2) then f t (x) ∈ I k (x) and all of these iterates are closest returns in the above sense: there is no smaller iterate of x in the subinterval of I k bounded by x and f t (x). In fact, all of the closest returns of a point x ∈ T are of this form, see Figure 1.
Let us denote QI the set of quadratic irrationals in (0, 1), that is, the set of numbers whose continued fraction expansion is eventually periodic. In other words, ρ ∈ QI iff there exist k ≥ 0 and m ∈ N such that G k (ρ) = G k+m (ρ). For B ∈ N we will say that ρ ∈ (0, 1) \ Q is of a type bounded by B if the coefficients in its continued fraction expansion are bounded by B. We denote the set of such numbers by BT B . We also set BT = ∪ B BT B and call the numbers in BT irrationals of bounded type. A number ρ is of bounded type iff it satisfies the Diophantine condition with exponent 2.
A multicritical circle map can be associated to a multicritical commuting pair ζ = (η|I η , ξ|I ξ ) as follows. Let x ∈ I ξ be any regular point of ξ, and glue the endpoints of the interval I = [x, ξ(x)] via ξ. More precisely, consider the onedimensional analytic manifold S, obtained by using ξ as a local chart in a small neighborhood of x. Consider the map f : I → I defined as follows: Letf : S → S be the projection of f to a map of the analytic circle S, and let h be a diffeomorphism of the same smoothness as ζ, which maps S → T and sends 0 ∈ S to 0 ∈ T. It is straightforward to check that the mapf projects to a well-defined multicritical circle map: Of course, f ζ is not defined uniquely, since there is a priori no canonical way of identifying the manifold S with the circle T. Rather, we recover a conjugacy class of multicritical cicle maps from a pair ζ from this gluing construction.
We say that a pair ζ is bi-cubic if f ζ is a bi-cubic map. The rotation number ρ(ζ) of a commuting pair is defined as note that the latter is well-defined, since f ζ is defined up to a conjugacy. If ρ(ζ) / ∈ Q, the signature C(ζ) is defined as it is similarly well-defined.
Let f be a multicritical circle map with a rotation number ρ = ρ(f ) whose continued fraction expansion has at least m + 1 terms, and as usual, denote p n /q n the continued fraction convergents of ρ, defined for n ≤ m + 1. Let f be the lift of f such that 0 < f (0) < 1 (note that D f (0) = 0). For 1 ≤ n ≤ m, let I n be the closed interval in R, containing the origin as one of its boundary points, which is projected onto I n (0) ⊂ T = R/Z. We define ξ : I n+1 → R and η : I n → R by where T (x) = x + 1 is the unit translation. Then the pair is a multicritical commuting pair. In what follows, for the simplicity of the notation, we will denote this pair simply by (f qn+1 | In , f qn | In+1 ).
Suppose now that f is bi-cubic, then the pair ζ f,n corresponds to a multicritical circle map f ζ f,n (3.1) which has either two cubic critical points, or a single critical point of order 9. The latter happens if the orbit of one of the critical points of f hits the other one for an iterate m ≤ q n+1 . Let us say that a multicritical commuting pair ζ is bi-cubic if f ζ is a bi-cubic circle map.
Moreover, we define the renormalization of ζ as the normalization of pR(ζ): If ζ is a bi-cubic commuting pair with χ(R j ζ) < ∞ for 0 ≤ j ≤ n − 1, we say that ζ is n-times renormalizable. Moreover, if χ(R j ζ) < ∞ for all j ∈ N, we say that ζ is infinitely renormalizable. It is straightforward to check that the rotation number ρ(ζ) is equal to a finite or infinite continued fraction and that renormalization acts as the Gauss map G(ρ) on the rotation numbers: The n-th renormalization R n f of a bi-cubic circle map f is the commuting pair This definition is clearly consistent with the above definition of renormalization of commuting pairs: R(R n f ) = R n+1 f, and ρ(Rf ) = G 2 (ρ(f )). The action of renormalization on signatures is given by the expanding cocycle Section 3] where the corresponding cocycle is constructed for renormalization acting on bi-critical commuting pairs). Compactness properties of infinite orbits of a renormalization transformation are known as a priori bounds; such bounds allow us to discuss attractors of renormalization, and play a crucial role in establishing rigidity results. Real a priori bounds for an infinitely renormalizable bi-cubic map concern pre-compactness of renormalizations in the C r topology. Real a priori bounds were proved for multicritical circle maps with an arbitrary irrational rotation number in [EdF18]. These bounds were improved in [EdFG18] to be universal (independent of the map). We give a precise statement for bi-cubic circle maps: Theorem 3.4. There exists n 0 ∈ N such that for any n ≥ n 0 and any bi-cubic circle map f with an irrational rotation number, the sequence {f qn+1 | In } is universally bounded in the C 1 norm.
Complex a priori bounds concern convergence of analytic extensions of renormalizations in the complex plane, and will require a detailed discussion.

Complex a priori bounds and convergence of renormalization.
Let us recall the definition of a holomorphic commuting pair given in [Yam19], which generalizes the original definition of de Faria [dF99].
with only real critical values, and ∆ Û 0, ∆ V 0; We shall call ζ the commuting pair underlying H, and write ζ ≡ ζ H . When no confusion is possible, we will use the same letters η and ξ to denote both the maps of the commuting pair ζ H and their analytic extensions to the corresponding domains U and V . We can naturally view a holomorphic pair H as three triples We say that a sequence of holomorphic pairs converges in the sense of Carathéodory convergence, if the corresponding triples do. We denote the space of triples equipped with this notion of convergence by H.
We let the modulus of a holomorphic commuting pair H, which we denote by mod (H) to be the modulus of the largest annulus A ⊂ ∆, which separates C \ ∆ from Ω. (1) mod (H) ≥ µ; (2) |I η | = 1, |I ξ | ≥ µ and |η −1 (0)| ≥ µ; Let the degree of a holomorphic pair H denote the maximal topological degree of the covering maps constituting the pair. Denote by H K (µ) the subset of H(µ) consisting of pairs whose degree is bounded by K. The following is an easy generalization of Lemma 2.17 of [Yam03]: Lemma 3.7. For each K ≥ 3 and µ ∈ (0, 1) the space H K (µ) is sequentially compact.
We say that a real commuting pair ζ = (η, ξ) with an irrational rotation number has complex a priori bounds, if there exists µ > 0 such that all renormalizations of ζ = (η, ξ) extend to holomorphic commuting pairs in H(µ). The existense of complex a priori bounds is a key analytic issue of renormalization theory. For holomorphic commuting pairs corresponding to bi-cubic circle maps, with bounded type rotation number, complex a priori bounds were established [ESY20]: Theorem 3.8. Let ζ be a bi-cubic analytic commuting pair with ρ(ζ) ∈ BT B . Then ζ has complex a priori bounds, with µ = µ(B).
Based on this result, the following renormalization convergence statement was obtained in [ESY20]: Theorem 3.9 (Exponential convergence of the renormalizations). Let ζ 1 and ζ 2 be two analytic bi-cubic circle maps with same bounded type rotation number ρ ∈ BT B . Then there exist n 0 > 0, C > 0 and 0 < λ = λ(B) < 1 such that for all n ≥ n 0 and for all r ≥ 0, we have where dist C stands for the Carathéodory distance.
Theorems 3.8 and 3.9 imply the following: There exists a Carathéodory compact set A B of analytic commuting pairs such that the following holds: • for every bi-cubic pair ζ 1 with ρ(ζ 1 ) ∈ BT B , the renormalizations R n (ζ 1 ) converge to A B geometrically fast in the Carathéodory sense, with rate λ(B).
Note, that pairs in A B can have two cubic critical points, or only one, of order 9 -the maps latter correspond to the situations when the two cubic critical points "collide" in the limit.
Remark 3.11. Note also that every pair ζ = (η, ξ) ∈ A B consists of maps which decompose as where φ η and φ ξ are holomorphic in a neighborhood of the origin. The space of pairs with this cubic symmetry property is clearly renormalization-invariant, and limit points of renormalization belong to this space by the standard machinery of renormalization theory.
Let us denote P B ⊂ A B the set of periodic orbits of renormalization with rotation numbers in BT B . As shown in [Yam19], all periodic signatures are represented in P B . From the standard considerations of compactness, as well as the density of periodic orbits of the cocylcle (3.3), we have Proposition 3.12. The set P B is dense in A B in the Carathéodory sense.

Cylinder renormalization of bi-cubic maps
In this section, we recall the definition of cylinder renormalization that acts in the space of bi-cubic analytic circle maps. It was introduced in [Yam19], and is based on the previous definition for the unicritical case in [Yam02] (see also [GY20] for a more general exposition). 4.1. Some functional spaces. Firstly, let us denote π the natural projection C → C/Z. For an equatorial annulus U ⊂ C/Z let A U be the space of bounded analytic maps φ : U → C/Z continuous up to the boundary, such that φ(T) is homotopic to T, equipped with the uniform metric. We shall turn A U into a realsymmetric complex Banach manifold as follows. DenoteŨ the lift π −1 (U ) ⊂ C. The space of functionsφ :Ũ → C which are analytic, continuous up to the boundary, and 1-periodic,φ(z + 1) =φ(z), becomes a Banach space when endowed with the sup norm. Denote that spaceÃ U . For a function φ : U → C/Z denoteφ an arbitrarily chosen lift π(φ(π −1 (z))) = φ. Observe that φ ∈ A U if and only ifφ =φ − Id ∈Ã U . We use the local homeomorphism betweenÃ U and A U given bỹ to define the atlas on A U . The coordinate change transformations are given bỹ φ(z) →φ(z + n) + m for n, m ∈ Z, therefore with this atlas A U is a real-symmetric complex Banach manifold; we denote A R U its real slice, consisting of the realsymmetric maps φ(T) = T.
is a Riemann surface conformally isomorphic to the cylinder C/Z (we will call a domain C f with these properties a fundamental crescent of f k ); Denotef the projection of R C f to C/Z with c → 0. We will say thatf is a cylinder renormalization of f with period k.
Note that if f ∈ A R U is a multicritical circle map, thenf is also a multicritical circle map.
The relation of the cylinder renormalization procedure to renormalization of pairs is as follows:

Proposition 4.2 ([Yam02]). Suppose f is a multicritical circle map with rotation
number ρ(f ) ∈ R \ Q. Assume that it is cylinder renormalizable with period q n . Then the corresponding renormalizationf is also a multicritical circle map with rotation number G n (ρ(f )). Also, Rf (in the sense of pairs) is analytically conjugate to R n+1 f . The next property of cylinder renormalization is the following:

Proposition 4.3 ([Yam02]
). Let f be cylinder renormalizable with period k = q n , with a cylinder renormalizationf corresponding to the fundamental crescent C f , and let W be any equatorial annulus compactly contained in the domain U 1 off . Then there is an open neighborhood G of f in A U such that every map g ∈ G is cylinder renormalizable, with a fundamental crescent C g ⊂ U which depends continuously on g in the Hausdorff sense. Moreover, there exists a holomorphic motion χ g : ∂C f → ∂C g over G, such that χ g (f (z)) = g(χ g (z)). And finally, the renormalizationĝ is contained in A W .
The next proposition shows that g →ĝ is well-defined: Proposition 4.4. In the notations of Proposition 4.3, let C g be a different family of fundamental crescents, which also depends continuously on g ∈ G in the Hausdorff sense. Then there exists an open neighborhood G ⊂ G of f , such that the cylinder renormalization of g ∈ G corresponding to C g is alsoĝ.
A key property of the cylinder renormalization is the following (see Proposition 6.7 of [Yam19]): Proposition 4.5. In the above notation, the correspondence g →ĝ is a locally analytic operator from a neighborhood of f ∈ A U to A W . Proposition 4.6. Let f and g be two bi-critical circle maps which are analytically conjugate φ • f = g • φ in a neighborhood U of the circle. Suppose that f is cylinder renormalizable with period k = q n , and let C f be a corresponding fundamental crescent of f . Suppose Then, g is also cylinder renormalizable with the same period and with a fundamental crescent C g = φ(C f ); and denotingf andĝ the corresponding cylinder renormalizations of f and g respectively, we havê f ≡ĝ.
Proof. The domain C g is a fundamental crescent by definition. Denote C f , C g the closures of the two fundamental crescents minus the endpoints. The analytic conjugacy φ projects to an analytic isomorphism which conjugatesf toĝ. By Liouville's Theorem,φ is a translation; since it fixes the origin, it is the identity. 4.2. Cylinder renormalization operator. The cylinder renormalization of a bicubic commuting pair is defined in the same way as that of an analytic bi-cubic map.: Definition 4.7. Let ζ = (η, ξ) be a bi-cubic holomorphic commuting pair which is at least n times renormalizable for some n ∈ N. Denote D η the domain of the pair ζ. Denote ζ n = (η n , ξ n ) ≡ pR n (ζ) : D ηn → C. We say that ζ is cylinder renormalizable with period k = q n if: (1) there exist two compex conjugate fixed points p + n , p − n of the map η n with p ± n ∈ ±H, and an R-symmetric simple arc l connecting these points such that l ⊂ D ηn and η n (l) ∩ l = {p + n , p − n }; (2) denoting C the R-symmetric domain bounded by l and η(l) we have C ⊂ D ηn , and the quotient of C ∪ η(C) \ {p + n , p − n } by the action of ζ n is a Riemann surface homeomorphic to the cylinder C/Z (that is, C is a fundamental crescent of η). Clearly, the action of ζ induces a bi-critical circle map f with rotation number ρ(R n ζ) on the quotient (C ∪ η(C) \ {p + n , p − n })/ζ n C/Z; we call f the cylinder renormalization of ζ.
Note that the existence of the two conjugated points required in (1) is guaranteed for all holomorphic commuting pairs without fixed points in the real line (see e.g. [GY20, Section 4.3]).
Theorem 4.8. Let ζ be any bi-cubic commuting pair which extends to a holomorphic commuting pair H ζ . Then ζ is cylinder renormalizable with period q 0 = 1. Furthermore, for every µ > 0 there exists an equatorial annulus U such that if H ζ ∈ H(µ), then the cylinder renormalization lies in A R U .
The proof follows mutatis mutandis the proof of Lemma 4.10 of [GY20], so we will not reproduce it here. Let A B be the attractor of renormalization for pairs with rotation numbers in BT B (Theorem 3.10). Compactness considerations imply: Proposition 4.9. There exists an equatorial annulus U , m, k ∈ N, and a neighborhood V = V(A B ) in Carathéodory topology such that the following holds: (1) every ζ ∈ V is cylinder renormalizable with period 1 and the cylinder renormalization Let us fix B ∈ N. By complex a priori bounds, and in view of the above discussion, we have the following: Proposition 4.10. There exist n = n(B) ∈ N, µ = µ(B) > 0, an R-symmetric disk ∆ ⊂ C, and an equatorial R/Z symmetric annulus U such that for every multicritical circle map f ∈ A U with either two cubic or one degree nine critical points such that there are at least n + 1 terms in the continued fraction expansion of ρ(f ) all bounded by B the following holds: (1) R n−1 f extends to a holomorphic pair H f ∈ H(µ) with range ∆; (2) H f is cylinder renormalizable with period 1, and the corresponding cylinder renormalization g f ∈ AŨ withŨ U .
Let us denote and call it the cylinder renormalization of f . For every ζ ∈ A B , denote f ζ the cylinder renormalization of ζ with period 1 in A U , and let Clearly, the cylinder renormalization operator R cyl is an analytic operator on an open neighborhood ofÃ B in A U . As we will discuss below, it was proved in [Yam19, Theorem 2.9] that bi-cubic periodic orbits of R cyl are hyperbolic with a stable manifold of codimension 2. However, the attractor of renormalizationÃ B is not going to be uniformly hyperbolic under the action of R cyl . Indeed,Ã B contains unicritical circle maps with a critical point of order 9. For these orbits of R cyl the two unstable directions collapse into one. In the next section we will define a different Banach manifold, into which the dynamics of R cyl will conveniently embed -and in which the main result will hold.

The space of triples
Let us introduce several more Banach spaces. For an equatorial annulus A ⊂ C/Z, we will let D A be the space of all bounded holomorphic functions from A → C. IfÃ ⊂ C is the lift of A, then D A can be identified with the space of 1-periodic bounded holomorphic functions inÃ. We will equip the latter with the sup norm, which turns D A into a Banach space.
Similarly, for a domain U ⊂ C, we let G U denote the space of bounded holomorphic functions U → C/Z, again with the natural Banach structure. Finally, B U will denote the standard Banach space of bounded holomorphic functions from a domain U ⊂ C to C with the sup norm. Let q(x) = x 3 .
Definition 5.1. Let U = (U 1 , U 2 , U 3 ) be an ordered triple of an R/Z-symmetric equatorial annulus U 1 and two R-symmetric domains U 2 , U 3 in C. We denote C U the space of triples • each of the maps φ i is locally conformal in a neighborhood of every real point of its domain of definition.
Clearly, C U is an open subset of D U1 × B U2 × G U3 , and thus possesses a natural product Banach manifold structure. It is real-symmetric, and we denote C R U its real slice, consisting of triples of real-symmetric maps. Denote the composition We note: Proposition 5.2. The correspondence Γ : C U → A U1 , given by is analytic.
We will interpret cylinder renormalization as an operator acting on triples in the real slice C R U . Now, let f be in the domain of R cyl (see Proposition 4.10) and set (η, ξ) = R n f . The map η is a branched covering of a domain W η → ∆ of degree 9. Hence, it uniquely decomposes as η = ψ 3 • q • ψ 2 • q • ψ 1 , where ψ i are conformal maps onto the images; each partial composition is a branched covering; the domain of ψ 1 is W η , the range of ψ 3 is ∆. Now let where U = (U 1 , U 2 , U 3 ) with U 1 = U , and the domains U 2 , U 3 appropriately and uniformly chosen. We set , and refer to it as a lift of f to the space of triples. Note that from definition, the lift of cylinder maps to the space of triples is injective.
With the definition of the lift at hand, we can define renormalization in a appropriate subspace of the triples space C R U . By definition of R cyl (see equation (4.1)), Lemma 4.10 and Proposition 5.2, we have that there exists a neighborhood V B ofÂ B ≡ Λ( A B ) such that, the operator given bŷ is well defined and it is analytic. We will prove our Main Theorem for the renormalization operatorR defined in (5.2), and acting in the Banach manifold V B .
5.1. Periodic orbits ofR in the space of triples. We will now specialize to real-symmetric maps, and consider the real slice C R U of the space C U consisting of maps with real symmetry, with the induced real Banach manifold structure (cf. [Yam02,GY20]). As seen from Proposition 3.12, periodic orbits ofR are dense in A B ; let us denote the set of these orbits byP B . Let v * ∈P B be a periodic point of R with period p ∈ N. Let us set and denote ρ * = ρ(f * ) and let C be the signature of f * . We define such that C(Γ(v)) = C} ⊂ D * . As shown in [ESY20], all the maps in S * are C 1+α -conjugate with each other, with α = α(B). We will see that the local stable set of v * is a submanifold of C R U : Theorem 5.3. There is an open neighborhood W ⊂ C R U of v * such that S * ∩ W is an analytic submanifold of C R U of codimension 2. It is the local stable manifold ofR at v * : for every v ∈ S * ∩ W the iterateŝ R pk −→ k→∞ v * at a geometric rate.
Moreover, for v ∈ S * denote f = Γ(v), and let φ f be the unique smooth conjugacy between f and f * which fixes 0. Then the dependence v → φ f is analytic.
Let us define a cone field E in the tangent bundle of C U given by the condition: We first show that: As before denote p k /q k the reduced form of the k-th continued fraction convergent of ρ * . Furthermore, define D k as the set of v ∈ C R U for which ρ(g) = p k /q k and 0 is a periodic point of g with period q k , where g = Γ(v). As follows from the Implicit Function Theorem, this is a codimension 1 submanifold of C R U . Let us define a cone field E in the tangent bundle of C R U given by the condition: Lemma 5.5. Let v ∈ D k , and denote T v D k the tangent space to D k at this point.
Proof. Let ν ∈ E and suppose {v t } ⊂ C R U is a one-parameter family such that v t = v + tν + o(t), Then for sufficiently small values of t, Elementary considerations of the Intermediate Value Theorem imply that for every k there exists a value of θ ∈ (0, 1) such that the map f θ = R θ • Γ(v * ) ∈ D k . Moreover, if we denote θ k the angle with the smallest absolute value satisfying this property, Fix v ∈ E. By Lemma 5.5 and the Hahn-Banach Theorem there exists > 0 such that for every k there exists a linear functional h k ∈ (T C R U ) * with ||h k || = 1, such that Ker h k = T k and h k (v) > . By the Alaoglu Theorem, we may select a subsequence h n k weakly converging to h ∈ (T C R U ) * . Necessarily v / ∈ Ker h, so h ≡ 0. Set T = Ker h. Proof of Theorem 5.4. By the above, we may select a splitting T C R U = T ⊕ v · R. Denote p : T C R U → T the corresponding projection, and let ψ : C R U → T C R U be a local chart at v * . Lemma 5.5 together with the Mean Value Theorem imply that p • ψ : D k → T is an isomorphism onto the image, and there exists an open neighborhood U of the origin in T , such that p • ψ(D k ) ⊃ U. Since the graphs D k are analytic, we may select a C 1 -converging subsequence D kj whose limit is a smooth graph G over U. Necessarily, for every g ∈ G, ρ(g) = ρ * . As we have seen above, every point g ∈ D * in a sufficiently small neighborhood of v * is in G, and thus G is an open neighborhood in D * .
We now proceed to describing the manifold structure of S * ⊂ D * . Let c f = 0 denote the non-zero critical point of f . Define a cone field Furthermore, for k > n + 2, let D k,n ⊂ D k consist of maps f such that f has the same combinatorics as E up to the level n, and the n-th renormalization R n (f ) is uni-critical (that is, two critical orbits collide). Elementary considerations again imply that for every f ∈ S * there exists a sequence {f k,n(k) } converging to it. Since the n-th renormalization of f ∈ D k,n is a uni-critical circle mapping, we can apply the results of [Yam02] to conclude: Theorem 5.6. All maps f ∈ Γ(D k,n ) are smoothly conjugate. Moreover, letf be an arbitrary base point in D k,n . Then, for f ∈ D k,n the conjugacy φ f which realizes f =f and sends 0 to 0 can be chosen so that the dependence f → φ f is analytic in S n .
We note: Proposition 5.7. The set D k,n is a local submanifold of D k of codimension 1, and the cone field E 1 lies outside of its tangent bundle.
Proof. The first statement is a straightforward consequence of Implicit Function Theorem considerations. By Theorem 5.6, a vector field υ ∈ DΓT v D k,n satisfies the cohomological equation and ω is a smooth vector field on the circle. In particular, υ (c f ) = 0.
We also note: Proposition 5.8. The cone field E 1 intersects T v D * at every point v.
Proof. Evidently, E 1 has non-zero intersection with E and −E at every point v. The claim follows.
By Proposition 5.7, applying the same arguments as in the proof of Theorem 5.4 (replacing D k → D * with D k,n(k) → S * ), we obtain that S * is a submanifold of C R U of codimension 2. By the definition of S * , it satisfies the statement of Theorem 5.3. 6. Hyperbolicity ofR.
Recall thatP B ⊂ C R U denotes the set of periodic points ofR with rotation numbers of type bounded by B. We claim: Theorem 6.1. Let B ∈ N. The setP B is uniformly hyperbolic with two-dimensional unstable bundle.
Informally speaking, one unstable direction corresponds to adding a constant to φ 3 , and the other one corresponds to adding a constant to φ 2 . The key in the above theorem, and the difference from the result of [Yam19] is the uniformity of hyperbolicity. The two unstable cones constructed in [Yam19] collide when the two critical points combine into one in the limit.
Let us v * be as above and denote We have (see [GY20]): Proposition 6.2. The linear operator L is compact.
Proof. By [Yam02, Proposition 9.1], see also [GY20, Proposition 6.4], the linear operator DR cyl | f * is compact. On the other hand, the operator DΛ| R cyl (f * ) is bounded. Therefore, the operator L being the composition of a bounded operator with a compact operator, is compact.
Hence, the cone field E contains an unstable direction ofR. To see the existence of the other unstable direction, we will consider two separate cases. Let c * = c f * be the second critical point of f * = Γ(v * ), and let n be as in Proposition 4.10. We will say that the critical points of f * are well-separated, if each of the arcs bounded by these points contains a subarc of the form [x, f qn+1 (x)].
Case 1: the critical points of f * are well-separated. We then have: Proposition 6.4. There exist constants a 2 > 0 and λ 2 > 1 such that the following holds. Let ν ∈ T v * D * ∩ E 1 . Set υ = DΓν and υ ± k = DΓL k (±ν) for k ∈ N. Then, Since periodic orbits are dense inÂ B , we can use the standard considerations of uniform hyperbolicity to derive the main result. We can use, for instance, Lemma VII.1 of [Man83] which discusses uniform hyperbolicity in Banach space setting: x depend continuously on x. b 2 ) For all n ≥ 0 and x ∈ f n (K) x is injective for all n ≥ 0 and x ∈ f n (K). b 4 ) There exist C > 0, µ > 0 and 0 < ε < γµ/3 satisfying for all x ∈ K, n ≥ 0, m ≥ 0. Moreover, if n − m ≥ 0 then is the projection associated to the splitting E = H (n) x ⊕ G x , v < r}, there exists a map ϕ : ∆ r → E satisfying the following properties: x ϕ(x, v) = v and f n (ϕ(x, v)) − f n (x) ≤ δ e −nµ for all n ≥ 0. In particular, this implies ϕ(x, 0) = x. III) ϕ is continuous, for every x the map ϕ(x, ·) is C k+γ and its derivatives depend continuously on (x, v). Moreover, its first derivative at (x, 0) is the inclusion map.
We conclude: Theorem 6.7. For every B ∈ N, the attractorÂ B is uniformly hyperbolic in C R U with two unstable directions. It has a codimension two stable foliation by analytic manifolds. If v 1 and v 2 belong to the same stable leaf of the foliation then C(v 1 ) = C(v 2 ). In a neighborhood ofÂ B , the converse is also true.
Proof. Using Lemma 6.6 with K =P B we see that if v ∈ P B then there exists an open neighborhood U (v) ⊂ C R U and r > 0 such that for every w ∈ P B ∩ U (v), the stable manifold W s loc (w) is an analytic graph of codimension two over the ball ∆ of radius r around the origin in T v (W s (v)). SinceP B is dense inÂ B , passing to converging subsequences, we obtain a codimension two foliation F s B with analytic leaves throughÂ B which is the closure of the stable foliation ofP B . Trivial considerations of continuity imply that it has the desired properties.