Abstract
Abstract. We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.
Introduction
In this paper we develop a renormalization theory for analytic critical circle maps with several critical points. The principal result is a renormalization hyperbolicity theorem for bi-cubic circle maps, that is, analytic homeomorphisms of the circle with two critical points, both of which are of cubic type. We construct periodic orbits for renormalization for such maps (Theorem 2.8), and show (Theorem 8.3) that these orbits are hyperbolic, with two unstable directions.
The proof of Theorem 2.8 uses a new version of our complex a priori bounds [Yam99] . The proof of the bounds is of an independent importance, and we give it for arbitrary multicritical circle maps, not just bi-cubic ones (Theorem 3.2).
Preliminaries
2.1. Multicritical circle maps and commuting pairs. A multicritical circle map f : T → T is a C 3 -smooth homeomorphism of the circle T which has finitely many critical points c 0 , . . . , c l so that:
where k j ∈ 2Z + 1 and φ j is a local C 3 -diffeomorphism. We will say that a critical point described by the equation (2.1) has order k j .
The same definition becomes much simpler in the analytic cathegory (C ω ): an analytic multicritical circle map is an analytic homeomorphism of the circle with at least one critical point.
A discussion of multicritical circle maps in the real setting has recently been carried out in [GdF] .
In what follows, we will always place one of the critical points of a multicritical circle map at 0.
As discussed in some detail in [Yam02] , the space of critical circle maps is ill-suited to define renormalization. The pioneering works on the subject ( [ÖRSS83] and [FKS82] ) circumvented this difficulty by replacing critical circle maps with different objects: Definition 2.1. A C r -smooth (or C ω ) critical commuting pair ζ = (η, ξ) consists of two C r -smooth (or C ω ) orientation preserving interval homeomorphisms η : I η → η(I η ), ξ : I ξ → ξ(I ξ ), where (I) I η = [0, ξ(0)], I ξ = [η(0), 0]; (II) Both η and ξ have homeomorphic extensions to interval neighborhoods V η , V ξ of their respective domains with the same degree of smoothness, that is C r (or C ω ), which commute, η • ξ = ξ • η; (III) ξ • η(0) ∈ I η ; (IV) η (x) = 0 = ξ (y), for all x ∈ I η \ {0}, and all y ∈ I ξ \ {0}; (V) each of the maps η and ξ has finitely many critical points in V η , V ξ respectively, each of them of an odd integer order; (VI) the point 0 is a critical point for both η and ξ.
We note that the commutation condition (II) ensures that η and ξ have the same order of the critical point at 0. The mapping ξ extends to a C r -(or C ω -) diffeomorphism of open neighborhoods of η(0) and ξ • η(0). Using it as a local chart we turn the interval I into a closed one-dimensional manifold M . Condition (II) above implies that the mapping F ζ projects to a well-defined C 3 -smooth homeomorphism F ζ : M → M . Identifying M with the circle by a diffeomorphism φ : M → T we recover a multicritical circle mapping (2.3)
The critical circle mappings corresponding to two different choices of φ are conjugated by a diffeomorphism, and thus we recovered a C r -(or C ω ) smooth conjugacy class of circle mappings from a critical commuting pair.
2.2.
Commuting pairs and renormalization of critical circle maps. The height χ(ζ) of a critical commuting pair ζ = (η, ξ) is equal to r, if 0 ∈ [η r (ξ(0)), η r+1 (ξ(0))].
If no such r exists, we set χ(ζ) = ∞, in this case the map η|I η has a fixed point. For a pair ζ with χ(ζ) = r < ∞ one verifies directly that the mappings η|[0, η r (ξ(0))] and η r • ξ|I ξ again form a commuting pair.
Given a commuting pair ζ = (η, ξ) we will denote by ζ the pair ( η| I η , ξ| I ξ ) where tilde means rescaling by a linear factor λ = 1 |Iη| . Definition 2.2. We say that a real commuting pair ζ = (η, ξ) is renormalizable if χ(ζ) < ∞. The renormalization of a renormalizable commuting pair ζ = (η, ξ) is the commuting pair
The non-rescaled pair (η r •ξ|I ξ , η|[0, η r (ξ(0))]) will be referred to as the pre-renormalization pRζ of the commuting pair ζ = (η, ξ).
is a sequence of renormalizable pairs such that ζ 0 = ζ and ζ i = pRζ i−1 . We call ζ k = pRζ k−1 the k-th pre-renormalization of ζ; and ζ k the k-th renormalization of ζ and write
For a pair ζ we define its rotation number ρ(ζ) ∈ [0, 1] to be equal to the continued fraction [r 0 , r 1 , . . .] where r i = χ(R i ζ). In this definition 1/∞ is understood as 0, hence a rotation number is rational if and only if only finitely many renormalizations of ζ are defined; if χ(ζ) = ∞, ρ(ζ) = 0. Thus defined, the rotation number of a commuting pair can be viewed as a rotation number in the usual sense:
Proposition 2.1. The rotation number of the mapping F ζ is equal to ρ(ζ).
There is an advantage in defining ρ(ζ) using a sequence of heights, since it removes the ambiguity in prescribing a continued fraction expansion to rational rotation numbers in a dynamically natural way.
Let f be a multicritical circle map with a critical point at 0. Suppose that the continued fraction of ρ(f ) contains at least n + 1 terms; as usual, let p k /q k be the k-th convergent of ρ(f ) for k ≤ n + 1, and let
One obtains a critical commuting pair from the pair of maps (f q m+1 |I m , f qm |I m+1 ) as follows. Letf be the lift of f to the real line satisfyingf (0) = 0, and 0 <f (0) < 1. For each m > 0 letĪ m ⊂ R denote the closed interval adjacent to zero which projects down to the interval I m . Let τ : R → R denote the translation x → x+1.
Then the pair of maps (η|Ī m , ξ|Ī m+1 ) forms a critical commuting pair corresponding to (f q m+1 |I m , f qm |I m+1 ). Henceforth, we shall abuse notation and simply denote this commuting pair by
The m-th renormalization of f at the critical point 0 is the rescaled pair (2.4):
Denote G(x) the Gauss map
It is easy to verify that for a renormalizable pair ζ
2.3. Dynamical partitions and real a priori bounds. Following the convention introduced by Sullivan [Sul92] , we say that for an infinitely renormalizable mapping f (or a pair ζ) a quantity is "beau" (which translates as "bounded and eventually universally (bounded)") if it is bounded for all renormalizations R n f (R n ζ), and the bound becomes universal (that is, independent of the map) for n ≥ n 0 for some n 0 ∈ N. Definition 2.3. Let f be a multicritical circle map which is at least n-times renormalizable. The collection of intervals
is called the n-th dynamical partition of f .
The following standard fact is easy to verify:
Proposition 2.2. The intervals of P n (f ) are disjoint except at the endpoints. Moreover, they cover the whole circle T.
This implies, in particular:
Theorem 2.4. Suppose f is a muticritical circle map with ρ(f ) / ∈ Q. Then f is topologically conjugate to the rigid rotation R ρ(f ) .
A stronger statement is also true [EdFG16] :
Theorem 2.5. Suppose f has an irrational rotation number. Then the iterate f qn | I n+1 decomposes as
where:
As before, denote c 0 = 0, c 1 , . . . , c l the critical points of f , listed in the counter-clockwise order. Let k j denote the order of the critical point c j , and set K(f ) = {k 0 , . . . , k l }. The points φ(c j ) divide the circle into l + 1 arcs, let a 0 , a 1 , . . . , a l be their lengths listed in the counter-clockwise order, starting from the point φ(0) and set
We say that C(f ) = (K(f ), A(f )) is the marked combinatorial type of f . We say that f and g have the same (unmarked) combinatorial type if C(f ) = σC(g), where σ is a cyclical permutation, and σC(g) ≡ (σK(g), σA(g)).
For a commuting pair ζ with ρ(ζ) / ∈ Q we will defined the marked combinatorial type
, where f φ is given by (2.3), and similarly for the unmarked combinatorial type.
To define the n-th dynamical partition P n (ζ) for an n-times renormalizable commuting pair ζ = (η, ξ), we simply take the n-th dynamical partition of a map f φ as in (2.3), and pull it back to I η ∪ I ξ .
Both for multicritical circle maps and for commuting pairs we will denote by P n the set of boundary points of the n-th dynamical partition.
Theorem 2.5 implies:
Theorem 2.6. Suppose f has an irrational rotation number. Let J 1 , J 2 be two adjacent intervals of the n-th dynamical partition P n (f ). Then J 1 , J 2 are K-commensurable where K is beau.
2.4. Periodic renormalization types for bi-critical circle maps. Let us now specialize to the case when the map f has only two critical points: 0 and c with orders k 0 and k 1 respectively. Suppose ρ(f ) is irrational. We say that ρ(f ) is of periodic type if it is a periodic point of the Gauss map G. Let p be the period of ρ(f ), and assume that the orbit of c 1 never falls into the critical point 0. That ensures that for each j ∈ N the pair of maps pR j f has exactly two critical points: 0 and c j = 0 with orders k 0 , k 1 .
Definition 2.4. We say that f is of a periodic marked combinatorial type with period m, where p|m,
An equivalent way of defining a periodic marked combinatorial type uses dynamical partitions as follows; it will be useful in the applications.
Definition 2.5. Consider the 2-nd dynamical partition of the renormalization R j+m (f ) for 0 ≤ m ≤ p − 1. It has v m intervals, let us enumerate them P 1 , . . . , P vm from left to right. Let i = w j,m denote the number of the interval P i which contains c j+m . We say that f is of a periodic marked combinatorial type if:
(1) the numbers (2.7) w m ≡ w j,m are independent of j; (2) there exists N ≤ m such that for any j the two critical points of R j f are separated by an interval of P N (R j f ).
The equivalence of the two definitions is a straightforward consequence of Theorem 2.3, and will be left to the reader.
Another straightforward consequence of Theorem 2.3 is the following:
Proposition 2.7. Suppose f and g both are of a periodic marked combinatorial type. Suppose ρ(f ) = ρ(g) and f and g have the same values of w m (2.7). Then
2.5. Renormalization theorems for bi-cubic critical circle maps. In the following two statements we assume that the critical circle map has two critical points, both of which are of order 3. We call such maps bi-cubic.
Theorem 2.8. Let C be a periodic marked combinatorial type with period m. Then there exists a commuting pair ζ * ∈ C ω such that the following properties hold:
The next theorem talks about the hyperbolicity properties of the renormalization operator at the periodic point ζ * . We will see in section 6 how to make a canonical choice of the analytic diffeomorphism φ = φ ζ in (2.3), so that the correspondence
becomes a well-defined renormalization operator on the space of bi-cubic circle maps, rather than commuting pairs. We will then show:
Theorem 2.9. There exists a Banach manifold structure C U on the space of analytic bicubic critical circle maps, such that the following holds. The Banach structure is compatible with the topology of uniform convergence. The renormalization operator defined by (2.8) is an analytic operator in this Banach manifold. The point
ζ * is a hyperbolic periodic point of this operator, with two unstable eigendirections.
Holomorphic commuting pairs
Below we define holomorphic extensions of analytic commuting pairs which generalize de Faria's definition from [dF99] for the case of multicritical circle maps. We say that a critical commuting pair ζ = (η| Iη , ξ| I ξ ) extends to a holomorphic commuting pair H if there exist three simply-connected R-symmetric domains D, U, V ⊂ C whose intersections with the real line are denoted by I U = U ∩ R, I V = V ∩ R, I D = D ∩ R, and a simply connected R-symmetric Jordan domain ∆, such that
• the endpoints of I U , I V are critical points of η, ξ respectively;
and D \ V are nonempty, connected, and simply-connected;
• the maps η and ξ have analytic extensions to U and V respectively, so that η is a branched covering map of U onto (∆ \ R) ∪ η(I U ), and ξ is a branched covering map of V onto (∆ \ R) ∪ ξ(I V ), with all the critical points of both maps contained in the real line; • the maps η : U → ∆ and ξ : V → ∆ can be further extended to analytic mapŝ
with only real branched points. We shall identify a holomorphic pair H with a triple of maps H = (η, ξ, ν), where η : U → ∆, ξ : V → ∆ and ν : D → ∆. We shall also 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 .
The sets Ω H = D ∪ U ∪ V and ∆ ≡ ∆ H will be called the domain and the range of a holomorphic pair H. We will sometimes write Ω instead of Ω H , when this does not cause any confusion.
We can associate to a holomorphic pair H a piecewise defined map S H : Ω → ∆:
De Faria [dF99] calls S H the shadow of the holomorphic pair H. 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.
Definition 3.1. For µ ∈ (0, 1) let H(µ) ⊂ H denote the space of holomorphic commuting pairs H : Ω H → ∆ H , with the following properties:
Let the degree of a holomorphic pair H denote the maximal topological degree of the covering maps constituting the pair. Denote 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] :
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.
Definition 3.2. For a set S ⊂ C, and r > 0, we let N r (S) stand for the r-neighborhood of S in C. For each r > 0 we introduce a class A r consisting of pairs (η, ξ) such that the following holds:
• η, ξ are real-symmetric analytic maps defined in the domains
respectively, and continuous up to the boundary of the corresponding domains;
is a critical commuting pair.
For simplicity, if ζ is as above, we will write ζ ∈ A r . But it is important to note that viewing our critical commuting pair ζ as an element of A r imposes restrictions on where we are allowed to iterate it. Specifically, we view such ζ as undefined at any point z / ∈ N r ([0, ξ(0)]) ∪ N r ([0, η(0)]) (even if ζ can be analytically continued to z). Similarly, when we talk about iterates of ζ ∈ A r we iterate the restrictions η| Nr([0,ξ(0)]) and ξ| Nr([0,η(0)] . In particular, we say that the first and second elements of pRζ = (η r • ξ, η) are defined in the maximal domains, where the corresponding iterates are defined in the above sense.
For a domain Ω ⊂ C, we denote by D(Ω) the complex Banach space of analytic functions in Ω, continuous up to the boundary of Ω and equipped with the sup norm. Consider the mapping i :
The map i is injective, since η(0) completely determins the rescaling h. Hence, this map induces a metric on A r from the direct product of sup-norms on
This metric on A r will be denoted by dist r (·, ·). We will denote A K r the subset of A r consisting of pairs whose degree is bounded by L. Theorem 3.2 (Complex bounds for periodic type). Let C be a periodic combinatorial type and L ≥ 3. There exists a constant µ > 0 such that the following holds. For every positive real number r > 0 and every pre-compact family S ⊂ A L r of critical commuting pairs, there exists N = N (r, S) ∈ N such that if ζ ∈ S is a commuting pair with C(ζ) = C, then pR n ζ restricts to a holomorphic commuting pair H n : Ω n → ∆ n with ∆ n ⊂ N r (I η ) ∪ N r (I ξ ). Furthermore, the range ∆ n is a Euclidean disk, and the appropriate affine rescaling of H n is in H(µ).
In the next section we will give a proof of this theorem which generalizes our proof in [Yam99] . For simplicity of notation, we will assume that the pair ζ is a pre-renormalization of a multicritical circle map f , that is
This will allow us to write explicit formulas for long compositions of terms η and ξ, which will greatly streamline the exposition. Theorem 3.2 follows from Theorem 4.1 which we state and prove in the following section.
Complex a priori bounds
4.1. Power law estimate. Let f be an analytic critical circle map. Let U be a Tsymmetric annulus which is contained in the domain of analyticity of f . Suppose 0 is a critical point of f of an odd integer order k = 2m + 1 and denote R n f the n-th renormalization of f at 0 as above. Note that in what follows, rather than working with the map of the circle f , we will work with the appropriately translated lifts of f (2.4), so all of our estimates will be for maps defined in a neighborhood of the real line rather than T.
Given an interval J ⊂ R, let C J ≡ C\(R\J) denote the plane slit along two rays. Let C J denote the completion of this domain in the path metric in C J (which means that we add to C J the banks of the slits).
The following theorem is a generalization of the main estimate of [Yam99] :
Theorem 4.1. There exist universal positive constants R 0 , b, c such that the following holds. Let f be as above, and R ≥ R 0 . Then there exists n 0 ∈ N which depends on f and R such that for all n ≥ n 0 we have the following. Denote
There exist subdomains V η ⊃ •I η , V ξ ⊃ •I ξ which contain the origin, such that the map η is a branched covering
and similarly for ξ. Furthermore, for any z ∈ V η ∪ V ξ we have
where the left-hand side stands for η on V η and ξ on V ξ . Finally, the constant n 0 can be chosen to depend only on R in any family of maps which is pre-compact in the compact-open topology on U .
Fix an integer M > 0 and n ≥ M . Let
and let D m denote the Euclidean disc D(H m ). Assume that D M ⊂ U . Consider the inverse orbit:
For a point z ∈ D M we say that
is a corresponding inverse orbit if each z −(k+1) is obtained by applying to z −k a univalent inverse branch of f | U where U is a sub-interval of J −(k+1) . We derive Theorem 4.1 from the following assertion:
Lemma 4.2. There exists M > 0 which depends only on f , and can be chosen uniformly in a compact family, such that for all n > M and any z ∈ C f q n+1 (In) ∩ D M the following estimate holds:
where z n is as in (4.3) and C is a universal constant.
Poincaré neighborhoods.
By symmetry, J is a hyperbolic geodesic in C J . Consider the hyperbolic neighborhood of J of radius r that is the set of all points in C J whose hyperbolic distance to J is less than r. One verifies directly that a hyperbolic neighborhood is the union of two Euclidean discs symmetric to each other with respect to the real axis, with a common chord J. We will denote such a neighborhood D θ (J), where θ = θ(r) is the outer angle the boundaries of the two discs form with the real axis. An elementary computation yields:
(4.5) r = log cot(θ/4).
Note, that in particular the Euclidean disc D(J) ≡ D π/2 (J) can be interpreted as a hyperbolic neighborhood of J. These hyperbolic neighborhoods were introduced into the subject by Sullivan [Sul92] . They are a key tool for proving complex bounds due to the following version of the Schwarz Lemma: 
The above invariance statement becomes quasi-invariance if the domain of φ is not the whole double-slit plane, yet is large enough relative to the size of J. Let us state this as a Lemma below. Denote D R = {|z| < R}. 
where δ ∈ (0, 1) is a fixed constant.
While superficially similar to [dFdM00, Lemma 2.4], it is a much stronger statement: we do not assume that φ is conformal in all of D R (since this will clearly not be the case in our applications). Before proving this statement, let us formulate a lemma (see [War50, Lemma3] There exists C > 0 such that the following holds. Let h : Ω → D be the Riemann mapping normalized by h(0) = 0 and h (0) > 0. Then
Combining this with the Cauchy Integral Formula, we obtain (4.6)
where D is a universal constant.
Proof of Lemma 4.4. We start with an exercise in conformal mapping. Let G(z) denote the branch of √ z which is defined in C \ R <0 and preserves R >0 ; and let (this estimate could be much improved for z near ±1, but we will not require it). Using (4.6), we see that K-neighborhood of [−1, 1] in the hyperbolic metric of Ω is contained in the K -neighborhood of [−1, 1] in the hyperbolic metric of D with
Together with 4.5 and the Schwarz Lemma, this implies the claim.
Easy induction combined with real a priori bounds implies (cf. [dFdM00, Lemma 2.5]):
Lemma 4.6. For each n ≥ 1 there exist K n ≥ 1 and θ n > 0 with K n → 1 and θ n → 0 as n → ∞ such that the following holds. Let θ ≥ θ n , and let 0 ≤ i < j ≤ q n+1 be such that the restriction
is a diffeomorphism on the interior. Then the inverse branch f
) and maps it univalently into the Poincaré neighborhood D θ/Kn (f i (I n )). In what follows, a beau bound > 0 on the angle will appear; we will say that a point z −i of (4.3) -jumps if (z −i , J −i ) > . We will also say that such points have a good angle. The next statement shows that for points with a good angle the normalized distance to the intervals of (4.2) cannot increase by more than a fixed multiple. It is taken from [Yam99] . However, its proof is somewhat modified, since for a multi-critical circle map the orbit (4.2) may pass through critical values.
Lemma 4.7. We work with the notations of Lemma 4.2. Fix 1 > 0, B > 0. Let J = J −i , J = J −k be two intervals of the inverse orbit (4.2) for k > i, and let z, z be the corresponding points of (4.3). Assume that (z, J) ≥ 1 and dist(z, J) ≥ B|J|.
Proof. Suppose, f has l critical points, not counting 0. Since the orbit (4.2) forms a part of the dynamical partition of the circle of level n, there are at most l critical points for the iterate f k−i | J . Hence, there exists a sub-interval K ⊂ J which is commensurable with J such that f k−i : K → K is a diffeomorphism. By the same reasoning as in [Yam99] , the smallest closed hyperbolic neighborhood cl Proof. The proof is a very slight modification of the proof of Lemma 4.2 of [Yam99] , the only difference being that the iterate f q m+1 restricted to H m may have at most l critical points in the interior. We refere reader to figure 3 for an illustration. LetH
and letD m be the connected component of f −(qm−1) which intersects the real line over the interior ofH m . Note that the iterate f qm−1 has at most l critical points inH m and they divide it into commensurable parts. Hence, by Lemma 4.6, the domainD m ⊂ D θ 1 (H m ) where θ 1 is beau.
To fix the ideas, suppose that f −qm+1 (0) < f (0). LetĤ m denote the interval [f q m+1 (0), 0] and letD m be the domain which intersects the real line over the interior ofĤ m and such that
is a branched covering (this is a precise way of saying thatD m is a pull-back ofD m by the "branch" of f −1 defined in C [f q m+1 −qm+1 (0),f (0)] ). Since f has a critical point of order k ≥ 3 at 0, the boundary ofD m forms angles π/k < π/3 with the negative real axis. In particular, there existã 1 ,ã 2 inĤ m which divide it in commensurable parts, so that where θ 2 is beau.
Again using Lemma 4.6 for the pull-back D m ofD m by the iterate f q m+1 −qm along the real line, we see that there exist a 1 , a 2 which divide [0, f qm−q m+1 (0)] into commensurable sub-intervals such that
where θ 3 is beau, and the proof is completed.
The next statement is a direct analogue of Lemma 4.4 of [Yam99] . The proof from [Yam99] is again adapted mutatis mutandis to our situation, in the same way as the preceding proof. We will leave the details to the reader.
Lemma 4.9. Let J be the last return of the backward orbit (4.2) to the interval I m before the first return to I m+1 , and let J and J be the first two returns of (4.2) to I m+1 . Let ζ, ζ , ζ be the corresponding moments in the backward orbit (4.3), ζ = f qm (ζ ), ζ = f q m+2 (ζ ).
Suppose ζ ∈ D m . Then either (ζ , I m+1 ) > and dist(ζ , J ) < C|I m+1 |, or ζ ∈ D m+1 ; where the constants C and are beau.
Proof of Lemma 4.2. We start with a point z ∈ D M . Consider the largest m such that D m contains z. We will carry out induction in m. Let P 0 , . . . , P −k be the consecutive returns of the backward orbit (4.2) to the interval I m until the first return to I m+1 , and denote by z = ζ 0 , . . . , ζ −k = ζ the corresponding points of the orbit (4.3). Note that the intervals P −i are commensurable with J 0 . By Lemma 4.8 ζ −i either -jumps and dist(ζ −i ,
In the former case we are done by Lemma 4.7. In the latter case consider the point ζ which corresponds to the second return of the orbit (4.2) to I m+1 . By Lemma 4.9, either (ζ , I m+1 ) > , and dist(ζ , I m+1 ) ≤ C|I m+1 |, or ζ ∈ D m+1 .
In the first case we are done again by Lemma 4.7. In the second case, the argument is completed by induction in m.
Proof of Theorem 2.8
5.1. Quasiconformal conjugacies between holomorphic pairs. Let us first quote the following (see [Her86, EdFG16] ):
Theorem 5.1. Suppose ζ 1 , ζ 2 are two commuting pairs with
Then for n ≥ 1, the renormalizations R n ζ i are K n -quasi-symmetrically conjugate, and the constant K n is beau.
Combining this with Theorem 3.2 and using the standard pull-back argument (see e.g. [dF99, Yam99]), we obtain: Theorem 5.2. Let C be a periodic combinatorial type and L ≥ 3. There exists a constant K > 0 such that the following holds. For every positive real number r > 0 and every pre-compact family S ⊂ A L r of critical commuting pairs, there exists N = N (r, S) ∈ N and K > 1 such that the following holds. Let ζ 1 , ζ 2 ∈ S be two commuting pairs with
and H i n is the holomoprhic commuting pair extension of pR n ζ i constructed in Theorem 3.2, then H 1 n is K-quasiconformally conjugate to H 2 n . For bi-cubic critical circle maps we can refine this statement further by showing that the quasiconformal conjugacy is actually conformal on the filled Julia set:
Theorem 5.3. Suppose f is a bi-cubic critical circle map, ρ(f ) / ∈ Q, and let H be a holomorphic pair extension of R n f . Then any Beltrami differential µ which is R-symmetric, H-invariant, and with support supp(µ) ⊂ K(H) is necessarily trivial:
0.
Theorem 2.8 follows from the two above statements and McMullen's Tower Rigidity argument (see [dFdM00, Yam03] ). The proof of Theorem 5.3 will occupy the rest of this section.
5.2.
Zakeri's models for bi-cubic circle maps. Note that for a bi-cubic critical circle map f the combinatorial type C(f ) is expressed by a single number. Consider the family of degree 5 Blaschke products
The following is shown in [Zak99]:
Theorem 5.4. Any Blaschke product which is topologically conjugate to an element of the family (5.1) by a conjugacy which preserves 0, ∞, and S 1 has the form B • R where B is an element of the family (5.1) and R is a rotation R(z) = e ir z, r ∈ R.
Theorem 5.5. Let ρ / ∈ Q and c ∈ (0, 1). Then there exists a unique Blaschke product B(z) of type (5.1) such that the following is true:
• ρ(B| S 1 ) = ρ;
• there exists a rotation R(z) = e ir z such that f = B • R is a bi-cubic critical map and the unmarked combinatorial type of f is c.
We now prove:
Lemma 5.6. Theorem 5.3 holds under the assumption that f = B •R is as in Theorem 5.5.
Proof. Extend µ by the dynamics of f to an f -invariant, S 1 -symmetric Beltrami differential supported on
(the equality is implied by the standard properties of Julia sets, cf. e.g. [Mil06] ). Complete it to an f -invariant Beltrami diffential λ 1 inĈ by setting it equal to σ 0 on the complement of the Julia set of f . For t ∈ [0, 1], set
so that λ 0 = σ 0 . By Ahlfors-Bers-Boyarski Theorem, there exists a unique solution ψ t : C →Ĉ of the Beltrami equation ψ * t σ 0 = λ t which fixes the points 0, 1, and ∞ and continuously depends on t. By S 1 -symmetry, the rational map
is a Blaschke product. By Theorem 5.4 and Theorem 5.5,
Hence, for every t and m, the map ψ t permutes the discrete set f −m (1). Since ψ 0 = Id and ψ t continuously depends on t, ψ t fixes the points in each f −m (1). By the standard properties of Julia sets,
Proof of Theorem 5.3. By Theorem 5.5, and complex a priori bounds, there exists a critical circle map g = B • R such that the renormalization R k (g) extends to a holomorphic commuting pair G which is K-quasiconformally conjugate to H. Denote ψ such a conjugacy. The claim will follow from the above considerations if we can show that∂ψ = 0 a.e. on K(H). Indeed, assume the contrary. Then (ψ −1 ) * σ 0 is a non-trivial invariant Beltrami differential on K(G), which contradicts Lemma 5.6.
6. Cylinder renormalization of bi-cubic maps 6.1. 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 real-symmetric 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 byφ
to define the atlas on A U . The coordinate change transformations are given byφ(z) → φ(z +n)+m for n, m ∈ Z, therefore with this atlas A U is a real-symmetric complex Banach manifold. Let us denote f the critical circle map
Definition 6.1. Let U = (U 1 , U 2 , U 3 ) be an ordered triple of R/Z-symmetric equatorial annuli. We denote C U the space of triples φ i ∈ A U i , i = 1, 2, 3 such that:
• φ 1 (0) = 0;
• and each φ i is univalent in some neighborhood of the circle.
, and thus possesses a natural product Banach manifold structure. Note that the composition
is a bi-cubic circle map in A U 1 . We note:
Proposition 6.1. The correspondence Γ : C U → A U 1 , given by
is analytic.
Definition 6.2. Given an element v ∈ C U we say that it is cylinder renormalizable with period k if the following holds. Denote f = Γ(v). Then, • there exists m ∈ N such that the rotation number ρ(f ) has at least m + 1 digits in its continued fraction expansion, and k = q m ; • there exist repelling periodic points p 1 , p 2 of f in U 1 with periods k and a simple arc l connecting them such that f k (l) is a simple arc, and
k is defined and univalent in the domain C f bounded by l and f k (l), the corresponding inverse branch f −k | f k (C f ) univalently extends to C f ; and the quotient
} by the action of f k 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 );
where n(z) ∈ N is the smallest value for which f n(z) (z) ∈C f . We further require that there exists a point c in the domain of R C f such that f m (c) = 0 for some m < n(c).
Denotef the projection of R C f to C/Z with c → 0. Evidently, it is a bi-cubic circle map. We will say thatf is a cylinder renormalization of f with period k.
The relation of the cylinder renormalization procedure to critical circle maps is easy to see: Yam02] ). Suppose f is a critical 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 critical 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: Yam02] ). Let f = Γ(v) be 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 of f . Then there is an open neighborhood G ofv such that everyū ∈ G the map g ≡ Γ(ū) is renormalizable, with a fundamental crescent C g ⊂ U which depends continuously onū in the Hausdorff sense. Moreover, there exists a holomorphic motion χū :
). And finally, the renormalizationĝ is contained in A W .
To define cylinder renormalization as an operator acting on triples in C U , let us first note the following simple statement:
Proposition 6.4. SetÛ = (Û 1 ,Û 2 ,Û 3 ), and suppose, that there exists a triplev ∈ CÛ such that Γ(v) =f as above. Then such an elementv is unique.
The next proposition shows thatū →ĝ is well-defined:
Proposition 6.5. In the notations of of Proposition 6.3, let C g be a different family of fundamental crescents, which also depends continuously onū ∈ G in the Hausdorff sense. Then there exists an open neighborhood G ⊂ G of f , such that the cylinder renormalization ofū ∈ G corresponding to C g is alsoĝ.
A key property of the cylinder renormalization is the following: Proposition 6.6. In the notations of Proposition 6.3, the dependenceū →ĝ is an analytic map G → A W .
Proof. By the Theorem of Bers and Royden [BR] , the holomorphic motion χ g extends to a holomorphic motion χ g :
smaller open neighborhood G with the same equivariance property. As shown in [MSS83] , the holomorphic motion induces an analytic family of quasiconformal maps
Applying the theorem of Ahlfors and Bers, we see that the projection π g :
n =ĝ ∈ C W for all g sufficiently close to f , and the claim follows.
The following is an immediate corollary:
Proposition 6.7. In the above notation, the correspondence v → v is a locally analytic operator from a neighborhood of v ∈ C U to CÛ .
Proposition 6.8. Let f and g be two bi-critical circle maps which are analytically conjugate
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 C f ∪ f (C f ) U . 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 havef ≡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.
6.2. Cylinder renormalization operator. The cylinder renormalization of a bi-cubic commuting pair is defined in the same way as that of an analytic bi-cubic map.:
Definition 6.3. 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
We say that ζ is cylinder renormalizable with period k = q n if:
• 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
• 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 ζ ∪ η(C ζ ) \ {p + n , p − n })/ζ n C/Z; we call f the cylinder renormalization of ζ.
Theorem 6.9. Fix a periodic marked combinatorial type C, and let f ∈ A U be a bi-cubic map of this type. Then, there exist N = N (C, U ) and an equatorial annulus U 1 = U 1 (C) such that the following holds. For every n ≥ N , setting k = q n , f is cylinder renormalizable with period k. Moreover, there exists a fundamental crescent C f U with period k such that f k (C f ) ⊂ U , and, denotingf the corresponding cylinder renormalization of f , we havê
Similarly, let ζ be any bi-cubic holomorphic commuting pair of type C. Then, there exists k = k(ζ) = q n such that ζ is cylinder renormalizable with period k, and the cylinder renormalization is a bi-cubic map of type C.
The proof is a direct consequence of complex a priori bounds, and follows mutatis mutandis the proof of Lemma 4.10 of [GY18] , so we will not reproduce it here.
Proposition 6.10. Let C be a periodic marked type with period p and let ζ * be a periodic point of R with this period. Then, there exist n ∈ N, an equatorial annulus U * , and a triple U = (U * , U 2 , U 3 ) such that the following holds:
(1) there exists m such that ζ * is cylinder renormalizable with period m and the renormalization f * ∈ A U * has combinatorial type C; (2) the bi-cubic map f * is cylinder renormalizable with period k; its renormalization is equal to f * and is defined in U 1 U * ; (3) the map f * has a decompositionv * ∈ C U .
The last property implies that there exists a neighborhood V (v * ) ⊂ C U such that cylinder renormalization with period k is an analytic operator to C U . We call this operator cylinder renormalization operator corresponding to the type C and denote it
7. Constructing the stable manifold ofv *
We proceed with the above notation. Let ρ * be the rotation number of the combinatorial class C and define Below we shall demonstrate that the local stable set ofv * is a submanifold of M:
There is an open neighborhood W ⊂ C U ofv * such that S * ∩W is a smooth submanifold of M of codimension 2. Moreover, forv ∈ S * denote f = Γ(v), and let φ f be the unique smooth conjugacy between f and f * which fixes 0. Then the dependencev → φ f is smooth.
Our first step will be to show that Theorem 7.3. The set D * is a local submanifold of C U of codimension 1.
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 ofv ∈ C 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 U . Let us define a cone field C in the tangent bundle of C U given by the condition:
Lemma 7.4. Letv ∈ D k , and denote TvD k the tangent space to D k at this point. Then
Proof. Let ν ∈ C and suppose {v t } ⊂ C U is a one-parameter family such that v t =v + tν + o(t), and set f = Γ(v) and f t = Γ(v t ), so that
Then for sufficiently small values of t, π
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, then θ k → 0. Set
For k large enough,v k ∈ C U ∩ D k . Let T k = Tv k D k . Fix v ∈ C. By Lemma 7.4 and the Hahn-Banach Theorem there exists > 0 such that for every k there exists a linear functional h k ∈ (T C 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 U ) * . Necessarily v / ∈ Ker h, so h ≡ 0. Set T = Ker h. Proof of Theorem 7.3. By the above, we may select a splitting T C U = T ⊕ v · R. Denote p : T C U → T the corresponding projection, and let ψ : C U → T C U be a local chart at v * . Lemma 7.4 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 k j 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 ofv * is in G, and thus G is an open neighborhood in D * . 2
We now proceed to describing the manifold structure of S * ⊂ D * . Define a cone field C 1 in T C U consisting of tangent vectors ν in TvC U such that setting υ = DΓν, we have
Furthermore, for k > n + 2, let D k,n ⊂ D k consist of maps f such that f has the same combinatorics as C 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 [Yam03] to conclude: Theorem 7.5. 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 7.6. The set D k,n is a local submanifold of D k of codimension 1, and the cone field C 1 lies outside of its tangent bundle.
Proof. The first statement is a straightforward consequence of Implicit Function Theorem considerations. By Theorem 7.5, a vector field υ ∈ DΓTvD k,n satisfies the cohomological equation υ = ω • f − f ω where f = Γ(v) and ω is a smooth vector field on the circle. In particular, υ (c f ) = 0.
We also note:
Proposition 7.7. The cone field C 1 intersects TvC U at every pointv.
Proof. Evidently, C 1 has non-zero intersection with C and −C at every pointv. The claim follows.
Applying the proof of Theorem 7.3 mutatis mutandis (replacing D k → D * with D k,n(k) → S * ), we obtain the statement of Theorem 7.2. The following was shown in [GY18] (see also [Yam02] ):
Proposition 8.4. There exist constants a 1 > 0 and λ 1 > 1 such that for every ν ∈ C∩TvC U and n ∈ N we have ||L n ν|| > a 1 λ n 1 inf x∈R DΓν(x).
We also have:

{'paper_id': '202537010', 'title': 'Renormalization of bi-cubic circle maps', 's2_url': 'https://api.semanticscholar.org/CorpusID:202537010', 'source': 'S2ORC'}