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Dynamics of quadratic polynomials II: rigidity

This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with a-priori bounds. As a corollary, such maps are combinatorially and topologically rigid, and as a consequence, the Mandelbrot set is locally connected at the correspoinding parameter values.


Introduction
This is a continuation of the series of notes on the dynamics of quadratic polynomials. The first part of this series [L3] will be systematically used for the reference. In particular we will assume that the reader is familiar with the background outlined in §2 of Part I: quadratic-like maps, straightening, combinatorial classes, external rays, the Mandelbrot set M, secondary limbs, puzzle, etc.
Let f be a quadratic-like map which does not have non-repelling periodic points. Let us say that f satisfies the secondary limbs condition if there is a finite family of truncated secondary limbs L i of the Mandelbrot set such that the hybrid classes of all renormalizations R m f belong to ∪L i . Let SL stand for the class of quadratic-like maps satisfying the secondary limbs condition.
Here are some examples of maps of class SL: • Maps which are at most finitely renormalizable and don't have non-repelling periodic points (Yoccoz class); • Infinitely renormalizable maps of bounded type ("bounded type" means that the relative periods of all renormalizations are bounded); • Real maps which don't have non-repelling periodic points.
Recall that a quadratic-like map f has a priori bounds if there is an ǫ > 0 such that mod (R m f ) ≥ ǫ > 0 for all renormalizations.
The goal of this paper is to prove the following result: Rigidity Theorem. Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with a priori bounds.
We believe that the second assumption actually follows from the first one: Conjecture. The secondary limbs condition implies a priori bounds.
This conjecture is supported by a few partial results (see below). Note, however, that a priori bounds don't hold for all quadratics: see examples of non-locally connected Julia sets [M1].
Let QC (c) ⊂ Top(c) ⊂ Com(c) ⊂ C stand respectively for the quasi-conformal, combinatorial and topological classes of the quadratic map P c . A map P c is called combinatorially (respectively topologically or quasi-conformally) rigid if Com(c) = {c} (respectively Top(c) = {c} or QC (c) = {c}).
Corollary I. Assume that all maps of Com(c) (respectively Top(c)) satisfy the secondary limbs condition with a priori bounds. Then P c is combinatorially (respectively topologically) rigid.
The corresponding quasi-conformal rigidity problem is settled by McMullen's Rigidity Theorem [McM2] which asserts that any quadratic polynomial with a priori bounds is quasiconformally rigid.
The strongest, combinatorial, rigidity of a map P c turns out to be equivalent to the local connectivity of the Mandelbrot set M at c (see [DH1,Sch]). This property of M was conjectured by Douady and Hubbard under the name "MLC". Prior to this work it was established in the following cases: • Parabolic points (Douady and Hubbard [DH1]); • Boundaries of the hyperbolic components of M (Yoccoz, see Hubbard [H]); • At most finitely renormalizable maps (Yoccoz, see Hubbard [H], Kahn [K]).
The following Corollary adds a pool of infinitely renormalizable maps to this list. In Part I of this paper a priori bounds have been proven for all maps of class SL with sufficiently big type (in the sense of Theorems IV and IV ′ of Part I). Thus we have: Corollary II [L3]. A quadratic polynomial P c ∈ SL of a sufficiently big type is rigid, so that the Mandelbrot set is locally connected at c.
In particular, this gives first examples of infinitely renormalizable parameter values c ∈ M of bounded type where MLC holds (though one does not need the full capacity of Corollary II to produce some examples of such kind).
Remark. It is easy to construct some infinitely renormalizable parameter values of unbounded type where MLC holds (oral communication by A. Douady). First find arbitrary small copies M n of the Mandelbrot set near c = −2. Then for an appropriate subsequence n(k), the tuned Mandelbrot copies M n(1) * M n(2) * · · · * M n(l) ) shrink to a single point.
One might wonder of how big is the set of infinitely renormalizable parameter values satisfying the assumptions of Corollary II. We can show that this set has Lebesgue measure zero and Hausdorff dimension at least 1 (in preparation). Note that 1=(1/2)2 where 2 = HD(∂M) by Shishikura's Theorem [Sh].
Let us now dwell on the case of real parameter values. In this case, "sufficiently big type" means sufficiently big essential period (see [LY] for the precise definition). For maps with "small" essential period, the MLC problem is still open. However, a priori bounds have been established for all infinitely renormalizable real quadratics (see [S,MvS,L2,GS,LS,LY]). Let us say that a parameter value c ∈ R (or the corresponding quadratic polynomial P c ) is rigid on the real line if Com(c) ∩ R = {c}. Thus we have: Corollary III. Any quadratic polynomial P c without attracting cycles is rigid on the real line.
By the Milnor-Thurston kneading theory [MT], Corollary III implies: Corollary IV. Hyperbolic quadratics are dense on the real line.
The last two Corollaries were first announced by Swiatek [Sw] who approached them by methods of real dynamics. The methods of holomorphic dynamics presented in this paper were developed in [L2].
Another application of the above Rigidity Theorem is a construction of the unstable manifolds for the renormalization operator at infinitely renormalizable points of bounded type (in preparation).
Let us now outline the structure of this paper. In §2 we show that the secondary limbs condition and a priori bounds yields a definite space between the bouquets of little Julia sets. This provides us with special disjoint neighborhoods of little Julia bouquets with bounded geometry (called "standard"). Also, together with the work of Hu & Jiang [HJ, J] and McMullen [McM3] this yields local connectivity of the corresponding Julia set (Theorem I).
We start §3 with a discussion of reductions which boil the Rigidity Theorem down to the following problem: Two topologically equivalent maps (satisfying the assumptions of the theorem) are Thurston equivalent. Then we set up an inductive construction of a sequence of approximations to the Thurston conjugacy. In particular, we adjust an approximate conjugacy in such a way that it respects the standard neighborhoods of little Julia bouquets.
The last section, §4, which follows §4 of [L2], presents the proof of the Main Lemma. This lemma gives a uniform bound on the Techmüller distance between the generalized renormalizations of two combinatorially equivalent quadratic-like maps (the bound depends only on the selected secondary limbs and a priori bounds). The main geometric ingredient which makes this work is the linear growth of the principal moduli proved in Part I of this paper.
In the Appendix we collect necessary background material in the theory of quasi-conformal maps.
In conclusion let us make a couple of remarks on history and some related results and methods. The origin of our approach to the rigidity problem can be tracked back to the proof of Mostow Rigidity: from topological to quasi-conformal equivalence, and then (by means of ergodic theory) from quasi-conformal to conformal equivalence. This set of ideas were brought to the iteration theory by Sullivan and Thurston. The passage from quasi-conformal to conformal equivalence in our setting is settled by McMullen's Rigidity Theorem [McM3]. Our main task is to pass from topological to quasiconformal equivalence. A way to do this called "pull-back argument" is to start with a quasi-conformal map respecting some dynamical data, and to pull it back so that it will respect more and more data on every step. In the end it will become (with some luck) a quasi-conformal conjugacy. This method was introduced by Thurston (see [DH3] and also [McM1]) for postcritically finite maps, and exploited by Sullivan [S, MvS] for real infinitely renormalizable maps of bounded type. These first applications dealt with maps with rather simple combinatorics.
For more complicated combinatorics, a certain real version of this method based on the so called "inducing" was suggested by Jacobson & Swiatek [JS, Sw]. (Roughly speaking, "inducing" means building out of f an expanding map with a definite range.) On the other hand, by means of a purely complex pull-back argument in the puzzle framework, Jeremy Kahn [K] proved removability of non-renormalizable Julia sets (which yields the Yoccoz Rigidity Theorem) .
Our way is different from all the above, though it has some common features with them. We believe that holomorphic dynamics is the right framework for the rigidity problem, and our method is purely complex. Rather than building an induced expanding map, we pass consecutively from bigger to smaller scales by means of the generalized renormalization [L1], and carry out the pull-back using growth of moduli and complex a priori bounds [L2, L3].
Let us note that there is a different approach to rigidity problems, by comparing the dynamical and parameter planes. This method was used by Branner & Hubbard [BH] to prove rigidity of cubic maps with one escaping critical point and "non-periodic tableaux" (which corresponds to non-renormalizable quadratics). It was also used by Yoccoz to prove rigidity of at most finitely renormalizable quadratics. In the forthcoming notes we will discuss this approach in our setting.
Let us also note that the MLC problem is closely related to the problem of landing of parameter rays at points c ∈ ∂M. MLC certainly yields landing of all rays, but, on the other hand, landing of some special rays has been a basis for progress in the MLC problem. The first results in this direction (landing at parabolic and Misiurewicz points) were obtained by Douady & Hubbard (see [DH1,M2,Sch]). Recently Anthony Manning [Ma] has estimated the Hausdorff dimension of the set of rays landing at infinitely renormalizable points.
Notations and terminology. Throughout the paper f will stand for a quadratic-like map with critical point at 0.
Saying that a modulus of some annulus A is definite means that mod A ≥ ǫ > 0, where ǫ depends only on the selected truncated secondary limbs and a priori bounds. Saying that some quantity is bounded has an analogous meaning.
Given a family of compact subsets X i ⊂ U, we say that there is a definite space (at least µ > 0) in between them (in a domain U) if for any i, there exists an annulus A i ⊂ U \ ∪X i with a definite modulus (at least µ) which goes around X i but does not go around other sets X j , j = i. If U is not specified, then U = C.
We will use the following notations: D r = {z : |z| < r} is the standard disk of radius r, D ≡ D 1 is the unit disk; T r = ∂D r is the standard circle of radius r, T ≡ T 1 is the unit circle; A(r, R) = {z : r < |z| < R} is a standard annulus; similar notation is used for a closed annulus A[r, R] (or a semi-closed one). Let P c : z → z 2 + c. As usual, ω(z) ≡ ω(f, z) stands for the limit set of the forward orbit {f n z} ∞ n=0 . The set ω(0) is called postcritical. R m f is the m-fold renormalization of f . Acknowledgement. I would like to thank Curt McMullen and Yair Minsky for useful discussions. I also thank MSRI for their hospitality: Part of this work was done during the Complex Dynamics and Hyperbolic Geometry spring program 1995. This work has been partially supported by NSF grants DMS-8920768 and DMS-9022140, and the Sloan Research Fellowship.
2.1. Space and unbranching. Let J m i denote the little Julia sets of level m, that is, Finally let K m i be little filled Julia sets. We will use the notation F m for the quadratic-like map f pm near any little Julia set J m i (it should be clear from the context which one is considered). In particular, F m = R m f near the critical Julia set J m ∋ 0.
Recall that Q(µ) stands for the space of quadratic-like maps f with mod (f ) ≥ µ > 0 supplied with the Caratheodory topology (see [McM2] and §5.6 of Part I). Take a little copy M ′ ⊂ M of the Mandelbrot set with root at b. Let Q(µ, M ′ ) denote the subspace of Q(µ) consisting of renormalizable quadratic-like maps f such that the hybrid class of Rf belongs to M ′ \ {b}.
Let us have a family F of sets X a ⊂ C depending on some parameter a ranging over a topological space T . This dependence is said to be (sequencially) upper semi-continuous if for any a(i) → a, the Hausdorff limit of X a(i) is contained in X a . For example it is easy to see that the filled Julia set K(f ) of a quadratic-like map f depends upper semi-continuously on f . Let us say that a family F of sets X f ⊂ C is (upper) semi-compact if any sequence X n of these sets contains a subsequence X n(i) converging in Hausdorff topology to a subset of some X ∈ F .
Lemma 2.1. The little filled Julia sets K 1 i (f ) form a semi-compact family of sets as f ranges over the space Q(µ, M ′ ).
Proof. By the Compactness Lemma (see §5.6 of Part I), the space Q(µ, M ′ ) is compact. Moreover the quadratic-like map F 1 depends continuously on f ∈ Q(µ, M ′ ) near any K 1 i . In turn, the little filled Julia sets K 1 i depend upper semi-continuously on F 1 .
Lemma 2.2. Let f be a quadratic-like map of class SL with complex a priori bounds. Then there is a definite space in between its bouquets B m j .
Proof. Let us take a bouquet B m . Let I m stand for the set of indices j such that B m+1 j ⊂ B m . We will show first that there is a definite annulus which goes around B m+1 but does not go around other bouquets B m+1 j , j ∈ I m .
If R m f is not immediately renormalizable, then this follows from point (ii) of Theorem II (Part I). So assume that R m f is immediately renormalizable.
If B m = J m , then it is nothing to prove as there is only one bouquet B m+1 inside B m . Otherwise there are only finitely many renormalization types producing the bouquet B m (which correspond to the little Mandelbrot sets attached to the main cardioid and belonging to the selected secondary limbs). By Lemma 2.1, the bouquets B m+1 j contained in B m belong to a compact family of sets. As they don't touch each other, there is a definite space in between them.
Let N(L, ǫ) denote an ǫ · diam L-neighborhood of a set L (that is, the set of points on distance at most ǫ diam L from L). We have shown that there is an ǫ > 0 such that the neighborhood N(B m+1 , ǫ) does not intersect other bouquets B m+1 j contained in the same B m . In particular, N(B 1 , ǫ) does not intersect any other B 1 j (as all of them are contained in B 0 ≡ J(f )).

Let us show by induction that
Assuming this for m, we should show that As we already know (2.2) for j ∈ I m , let j ∈ I m . Then B m+1 j ⊂ B m k for some k = 0, while N(B m+1 , ǫ) ⊂ N (B m , ǫ), and (2.2) follows from (2.1).
What is left, is to show that there a definite space around any bouquet B m+1 j (not only around the critical one). But there is an iterate f l which univalently maps B m+1 j onto B m+1 . Pulling back the space around B m+1 we obtain the desired space about B m+1 j .

⊔ ⊓
An infinitely renormalizable map f is said to satisfy an unbranched a priori bounds condition (see [McM3]) if for infinitely many levels m, there is a definite space in between J m and the rest of the postcritical set, ω(0) \ J m . Lemma 2.3. A map f ∈ SL with a priori bounds satisfies an unbranched a priori bounds condition.
Proof. We will show that the unbranched condition can fail only if the level m is not immediately renormalizable, while m − 1 is immediately renormalizable. As the complimentary sequence of levels is infinite, the lemma will follow.
If R m−1 f is not immediately renormalizable then the bouquet B m coincides with the little Julia set J m . By Lemma 2.2, there is a definite space in between J m and J m \ J m . As ω(0) \ J m ⊂ J m \ J m , the unbranched condition holds on level m.
Assume now that both levels m − 1 and m are immediately renormalizable. Then we will show that there is a definite space in between J m and B m+1 ≡ ∪ j =0 B m+1 j . By Lemma 2.2, there is a definite space in between B m ⊃ J m and B m+1 \ B m . So we should check that there is a definite space in between J m and B m+1 ∩ B m (that is, the union of non-critical bouquets B m+1 j contained in B m ). But J m does not touch any such B m+1 j . Indeed, the only point where they can touch could be the β-fixed point β m of J m . But one can easily see that the little Julia sets of level m + 1 never contain β m . By Lemma 2.1 there is a desired space.

Remark.
If R m f is not immediately renormalizable, while R m−1 f is immediately renormalizable, then the unbranched condition can fail. Indeed in this case there are several Julia sets J m i which touch at the common fixed point β m ∈ J m . But the postcritical set ω(0) ∩ J m i can come arbitrarily close to β m (when R m f is a small perturbation of a map whose critical orbit eventually lands at β m ).
2.2. Local connectivity of Julia sets. Hu and Jiang [HJ] proved that the Feigenbaum quadratic polynomial has locally connected Julia set. The proof makes use of Sullivan's a priori bounds (see [MvS, S]). Then a more general result of this kind was worked out: Any infinitely renormalizable quadratic map with unbranched a priori bounds has locally connected Julia set (see [J,McM3]). Together with Lemma 2.3 this yields (compare Theorem V of Part I): Theorem I. Let f ∈ SL be an infinitely renormalizable quadratic polynomial with a priori bounds. Then the Julia set J(f ) is locally connected. In particular, all maps from Theorems IV and IV ′ of Part I have locally connected Julia sets.
For the sake of completeness, we will give a proof of this result. Proof. A priori bounds imply that the "little" Julia sets J m shrink down to the critical point.
with an ǫ independent of m. Clearly U m does not cover the whole Julia set.
Let Γ m ⊂ U m \U ′ m be a horizontal curve in the annulus U m \U ′ m which divides it into two sub-annuli of modulus at least ǫ/2, and Γ ′ m ⊂ U ′ m be its pull-back by f m . By the Koebe Theorem, these curves have a bounded eccentricity about 0 (with a bound depending on ǫ). Since the inner radius of curve Γ ′ m about 0 tends to 0 as m → ∞ (it follows from the fact that the sufficiently high iterates of any disk intersecting J(f ) cover the whole J(f )), the diam Γ ′ m → 0 as well. All the more, diam(J m ) → 0 as m → ∞. Let us take a δ > 0, and find an m such that J m is contained in the D δ . Let us now inscribe into D δ a domain bounded by equipotentials and external rays of the original map f . Let α m denote the dividing fixed point of the Julia set J m , and α ′ m = −α m be the symmetric point. Let us consider a puzzle piece P m,0 ∋ 0 bounded by any equipotential and four external rays of the original map f landing at α m and α ′ m . This is a "degenerate" domain of the renormalized map F m (see §2.5 of Part I). By definition of the renormalized Julia set, the preimages P m,k ≡ F −k m P m,0 shrink down to J m . Hence there is a puzzle piece P m,l contained in the D δ . As J(f ) ∩ P m,l is clearly connected, the Julia set J(f ) is locally connected at the critical point.
Let us now prove local connectivity at any other point z ∈ J(f ). This is done by a standard spreading of the local information near the critical point around the whole dynamical plane. Let us consider two cases.
Case (i). Let the orbit of z accumulates on all Julia sets J m . Let m be an unbranched level. Then there is an l = l(m) such that the puzzle piece P m,l is well inside C \ (ω(0) \ J m ).
Take now the first moment k = k(m) ≥ 0 such that f k z ∈ P m,l . Let us consider the pull-backs Q m,l ∋ z of P m,l along the orbit orb k (z) = {z, ..., f k z}. By Lemma 3.3 of Part I, this pull-back is univalent. Moreover, it allows a univalent extension to a definitely bigger domain.
By the Koebe Theorem, Q m,l has a bounded eccentricity about z. Since the inner radius of this domain about z tends to 0 as m → ∞, the diam Q m,l → 0 as well. As Q m,l ∩ J(f ) are connected, the Julia set is locally connected at z.
Case (ii). Assume now that the orbit of z does not accumulate on some J m . Hence it accumulates on some point a ∈ ω(0). Let us consider the puzzle associated with the periodic point α m (so that the initial configuration consists of a certain equipotential and the external rays landing at α m ). Since the critical puzzle pieces shrink to J m , the puzzle pieces Y l i of sufficiently big depth l containing a are disjoint from ω(0) (there are several such pieces if a is a preimage of α m ). Take such an l, and let X be the union of these puzzle pieces. It is a closed topological disk disjoint from ω(0) whose interior contains a.
Consider now the moments k i → ∞ when the orbit of z lands at int X, and pull X back to z. By the same Koebe argument as in case (i) we conclude that these pull-backs shrink to z. It follows that J(f ) is locally connected at z. ⊔ ⊓ 2.3. Standard neighborhoods. In this section we will construct some special fundamental domains near little Julia bouquets. Let us consider first the non-immediately renormalizable case when the construction can be done in a particularly nice geometric way.
Lemma 2.4. Let f be m times renormalizable quadratic map. Assume that the space in between the little Julia sets J m i is at least µ > 0. Then there are disjoint fundamental annuli A m i around little Julia sets J m i , with mod A m i ≥ ν(µ) > 0. Proof. Let us consider the Riemann surfaces S = C \ J n and S ′ = C \ f −1 J n ⊂ S. Then f : S ′ → S is a double branched covering. Let us uniformize S, that is represent it as the quotient H 2 /Γ of the hyperbolic plane modulo the action of a Fuchsian group. In this conformal representation S admits a compactification S ∪∂S to a bordered Riemann surface, with the components ∂S m i of the "ideal boundary" ∂S corresponding to the little Julia sets J m i . LetŜ = S ∪ ∂S ∪S be the double of S, that is (C\Λ(Γ))/Γ, where Λ(Γ) ⊂ S 1 is the limit set of Γ. The boundary components ∂S m i are geodesics inŜ. Moreover, these geodesics have hyperbolic length bounded by a constant L = L(µ) independent of m.
Let σ : S → S be the natural anti-holomorphic involution of S. LetS ′ = σS ′ and S ′ = S ∪ ∂S ∪S ′ ⊂Ŝ be the double of S ′ inside S. Then f allows an extension to a holomorphic double branched coveringf :Ŝ ′ →Ŝ commuting with the involution σ. Its restrictionsf |∂S m i are the double branched coverings of the topological circles ∂S m i . Let C m i (r) ⊃ ∂S m i stand for the hyperbolic r-neighborhood of the geodesic ∂S m i . By the Collar Lemma (see [Ab]), there is an r = r(L) (independent of the particular Riemann surface and geodesics) such that the collars C m i ≡ C m i (r) are pairwise disjoint. Moreover, mod (C m i ) ≥ µ(L) > 0. Let us now take such a collar C = C m i , and let Γ = ∂S m i . Let C ′ ⊂ S ′ ∩ C be the component off −1 S containing Γ. Thenf : C ′ → C is a double covering preserving Γ. As we have in the hyperbolic metric of S: there is a point z ∈ Γ such that Df (z) ≥ 2. This easily implies that Df −1 (ζ) ≤ q(a) < 1 if the hyperbolic distance between z and ζ does not exceed a. In particular, Df −1 (ζ) ≤ q = q(L, r) < 1 for all ζ ∈ C.
It follows that C ′ is contained in the hyperbolic r/q-neighborhood of Γ, and hence the annulus mod (C\C ′ ) ≥ ρ(r, q) = ρ(µ). Let now A m i = (C\C ′ ) ∩ S. ⊔ ⊓ Note that in the above lemma we don't assume a priori bounds but just a definite space between the Julia sets (which thus implies a priori bounds). Assuming a priori bounds, let us now give a different construction which works in the immediately renormalizable case as well.
Let us consider a bouquet B m ). Let us consider the domain Υ m j bounded by the pairs of rays landing at these points (defined via a straightening of F m−1 ), and p m arcs of equipotentials. Let us then thicken this domain near the points b m i as described in §2.5 of Part I (that is, replace the rays landing at b m i by nearby rays and little circle arcs around b m i ). Denote the thickened domains by U m j (see Figure 1). We also require that these domains are naturally related by dynamics so that Lemma 2.5. Let f be an m times renormalizable quadratic map of class SL with a priori bounds. Then there exist disjoint standard neighborhoods U m j of B m j with bounded geometry, and such that the annulus mod (U m i \B m i ) have a definite modulus.
Proof. By the Straightening Theorem, the renormalization R m−1 f : is K-qc conjugate to a quadratic polynomial P c : z → z 2 + c, with K dependent only on a priori bounds.
Let B ⊂ J(P c ) be the critical bouquet of little Julia sets of RP c . Let Ω(ǫ) be its neighborhood bounded by arcs of equipotentials of level 1 − ǫ, circle arcs of radius ǫ, and rays with arguments θ + t(ǫ) (see §2.4 of Part I). Here θ are the arguments of the rays landing at the co-fixed points, and t(ǫ) ∈ (−ǫ, ǫ) is selected in such a way that Ω(ǫ) is a renormalization domain for any P c from selected truncated secondary limbs.
The geometry of these domains depends only on the selected limbs and ǫ. Also, the Hausdorff distance d c (ǫ) of ∂Ω(ǫ) to B tends to 0 as ǫ → 0 uniformly over c belonging to the selected truncated limbs. Indeed, this is clearly true for a given parameter value c. Take a little δ > 0, and find an ǫ = ǫ c such that d c (ǫ c ) < δ. Then for all b sufficiently close to c, d b (ǫ c ) < 2δ. Compactness of the truncated limbs completes the argument.
It follows that for all sufficiently small ǫ (depending only on the selected limbs and a priori bounds), Ω(ǫ) belongs to the range of the straightening map. Hence these neighborhoods can be transferred to the dynamical f -plane. We obtain neighborhoods U(ǫ) of the corresponding bouquet B with bounded geometry (depending on parameter ǫ).
Moreover, as quasi-conformal maps are quasi-symmetric (see Appendix), the Hausdorff

Let us now pull this neighborhood back by dynamics to obtain standard neighborhoods
Since by Lemma 2.2 there is a definite space between the bouquets, there is also a definite space between the neighborhoods U m j (ǫ), for for all ǫ ∈ (0, ǫ * ] (with ǫ * depending only on the selected limbs and a priori bounds). Also, the moduli of U m j (ǫ) \ B m j depend only on the limbs, a priori bounds and ǫ. So they are definite, for instance in the range ǫ ∈ (0.01ǫ * , ǫ * ].

⊔ ⊓
We keep using the notations B m j , Υ m j etc. introduced before Lemma 2.5, and we also assume that the standard neighborhoods U m j satisfy the conclusions of Lemma 2.5. We will define a special qc map with bounded dilatation. This map will be called a standard straightening, or a standard local chart near the bouquet B m j . It follows from a priori bounds assumption that for any Julia set J l i there exist Jordan disks Ω l i ⊃ Π l i ⊃ J l i such that F l : Π l i → Ω l i is a quadratic-like map, and there exists a qc map with bounded dilatation conjugating F l : Π l i → Ω l i and P 0 : i does not touch other Julia sets of level m (that is, F m−1 is not immediately renormalizable) then one can select the standard neighborhood U m i as Ω m i . In this case let us define the standard straightening (2.4) as Ψ m,i .
If F m−1 is immediately renormalizable, then let us consider a family of little Julia sets and bouquets: (2.5) Let us cut Υ m j by the rays landing at the fixed point α m−1 into components Ξ m i ⊃ J m i . Since the hybrid class of F m−1 may belong to a bounded number of little Mandelbrot sets (attached to the main cardioid and intersecting the selected secondary limbs), the domains Ξ m i have a bounded geometry. Hence the maps Ψ m,i can be selected in such a way that they have bounded dilatation and Thus they glue together into a single qc map (2.3). By the rays and equipotentials near the bouquet we will mean the S m -preimages of the vertical intervals and horizontal circles in the cylinder A(1, 4). This will be also referred to as the standard coordinate system near B m j . Let us show in conclusion that the little Julia bouquets and corresponding standard neighborhoods exponentially decay. Let diam(X) stand for the Euclidean diameter of a set X.
Lemma 2.6. Let f ∈ SL be a quadratic-like map with a priori bounds. Then there exist constants λ < 1 and l 0 > 0 depending on the choice of limbs and a priori bounds such that for any two Julia bouquets B m+l Let us straighten the renormalization R m f near J m i to a quadratic polynomial P c . The dilatation K of the straightening depends only on the a priori bounds, and K-qc maps are Hölder continuous with exponent 1/K (see [A]). Hence it is enough to show that for the quadratic map P c , there exist constants λ < 1 and l 0 > 0 depending on the choice of limbs and a priori bounds such that Let us pick Γ m i as the hyperbolic geodesic in A m i . Then the hyperbolic length of this geodesic in A m i is at most π/ν. All the more, the hyperbolic length of γ m i in D 3 \ J l is bounded by the same constant. By the Collar Lemma (see [Ab]), there are exist disjoint annuli Λ m i ⊂ D 3 \ J l in the homotopy class of γ m i with mod (Λ m i ) ≥ η = η(ν) > 0. By the Grötzsch inequality, mod (D 3 Hence there is an absolute constant C such that diam B l j ≤ Ce −lη (see Appendix A1 in Part I), and (2.6) follows.
Corollary 2.7. Under the assumptions of Lemma 2.6, there exist constants λ < 1 and l 0 > 0 such that for the standard neighborhoods U m j ⊂ U m i the following estimates holds: Indeed, the standard neighborhoods U m i are commensurable with the corresponding Julia sets J m i . 2.4. Removable sets. The reader is referred to the Appendix for the definition and a discussion of removability.
Lemma 2.8 (McM2). Under the assumptions of Lemma 2.6, the post-critical set ω(0) is a removable Cantor set coinciding with ∩J m .
Proof. It was shown in the proof of Lemma 2.6 that for any z ∈ ω(0) ⊂ ∩J m , there is a nest of disjoint annuli around z with a definite modulus. Thus the first statement follows from the Removability Condition (see Appendix).
Clearly, ω(0) ⊂ ∩J m ⊂ ∩U m . Vice versa, by Lemma 2.6, ∩J m is covered by the uniformly shrinking bouquets B m i . As every B m i contains a postcritical point, ω(0) is dense in ∩J m . Let us finish this section with stating a standard fact on removability of expanding Cantor sets. Let {U i } be a finite family of closed topological disks with disjoint closures. Let us consider a Markov map g : ∪U i → C satisfying the following property: stand for the filled Julia set of g.
Lemma 2.9. For a Markov map as above, the filled Julia set K(g) is removable.
Proof. Let us select a family of annuli A j ⊂ gU j \ ∪U i homotopic to ∂(gU i ) in gU j \ ∪U i . Let consider cylinder sets U m i(0),i(1),...,i(m−1) defined by the following property: has the same modulus as A i(m−1) . This provides us with a nest of disjoint annuli with definite modulus about any z ∈ K(g). The Removability Condition concludes the proof.
3. Rigidity: beginning of the proof 3.1. Reductions. In this section we begin to prove the Rigidity Theorem stated in the Introduction. Since quadratic polynomials label hybrid classes of quadratic-like maps, this theorem can be stated in the following way: Rigidity Theorem (equivalent statement). Let f,f ∈ SL be two quadratic-like maps with a priori bounds. If f andf are combinatorially equivalent then they are hybrid equivalent.
The proof is split into three steps: Step 1. f andf are topologically equivalent; Step 2. f andf are qc equivalent; Step 3. f andf are hybrid equivalent.
The first step (passage from combinatorial to topological equivalence) follows from the local connectivity of the Julia sets (Theorem I). Indeed, a locally connected Julia set is homeomorphic to its combinatorial model (see [D]). Since the combinatorial model is the same over the combinatorial class, the conclusion follows.
The last step (passage from qc to hybrid equivalence) is taken care of McMullen's Rigidity Theorem [McM2]. Indeed, it asserts that an infinitely renormalizable quadratic-like map with a priori bounds does not have invariant line fields on the Julia set. It follows that if h is a qc conjugacy between f andf then ∂h = 0 almost everywhere on the Julia set. Thus h is a hybrid conjugacy between f andf .
So, our task is to take care of Step 2: Theorem II. Let f,f ∈ SL be two quadratic-like maps with a priori bounds. If f andf are topologically equivalent then they are qc equivalent.
In what follows we will mark with tilde the objects forf corresponding to those for f (unless another meaning is explicitly assumed). When we introduce some objects for f , we assume that the corresponding tilde-objects are automatically introduced as well.
A qc map h as above will be called a Thurston conjugacy.
Remark. It is enough to assume that h is homotopic to ψ rel postcritical sets. Then one can extend it to a qc map U →Ũ which is homotopic to ψ rel the bigger set as required above.
The following result comes from the work of Thurston (see [DH2,McM1]) and Sullivan (see [MvS, S]). It originates the "pull-back method" in holomorphic dynamics.
Lemma 3.1. If two quadratic-like maps are Thurston equivalent then they are qc equivalent.
Proof. We will use the notations for the domains and maps preceding the statement of the lemma. Let U n be the preimages of U under the iterates of f , and let c = f (0). Let h has dilatation K.
Since h(c) =c, we can lift h to a K-qc map h 1 : U 1 =Ũ 1 homotopic to ψ rel (∂U 1 , ∂U 2 , ω(0)). (Note that the dilatation of h 1 is the same as the dilatation of h, since the lift is analytic). Hence h 1 = h on these sets, and we can extend h 1 to U \ U 1 as h (keeping the same notation h 1 ). By the Gluing Lemma from the Appendix this extension has the same dilatation K. Moreover, this map is homotopic to ψ rel (ω(0), ∪ 1≤k≤2 ∂U k ). Also, it conjugates f : to the corresponding tilde-map (notice that h 1 is a conjugacy on a bigger set than h).
Let us now replace h with h 1 and repeat the procedure. We will construct a K-qc map h 2 : U →Ũ homotopic to ψ rel (ω(0), ∪ 1≤k≤3 ∂U k ) and conjugating f : to the corresponding tilde-map.
Proceeding in this way we construct a sequence of K-qc maps h n homotopic to ψ rel (ω(0), ∪ 1≤k≤n+1 ∂U k ) and conjugating f : ω(0) ∪ (U 1 \U n+1 ) → ω(0) ∪ (U\U n ) to the corresponding tilde-map. By the Compactness Lemma from the Appendix, we can select a converging subsequence h n(l) → h. The limit map h is a desired qc conjugacy.
The method used in the above proof is called "the pull-back argument". The idea is to start with a qc map respecting some dynamical data, and then pull it back so that it will respect some new data on each step. In the end it becomes (with some luck) a qc conjugacy.
Remark. For infinitely renormalizable maps of bounded type with a priori bounds, Mc-Mullen proved that the postcritical set ω(0) has bounded geometry [McM3]. It easily follows that there is a qc map h : (C, ω(f, 0)) → (C, ω(f, 0)) conjugating f tof on their postcritical sets. This is close to being a Thurston conjugacy but not quite the same, as h may be in a wrong homotopy class.
3.3. Approximating sequence of homeomorphisms. So we need to construct a Thurston conjugacy. We will construct it as a limit of an appropriate sequence of maps. Take a sufficiently small ǫ > 0, and consider the corresponding sequence of standard neighborhoods 3). By Corollary 2.7 there is an l such that U m is well inside U m−l (that is, the annulus U m−l \ U m has a definite modulus). Moreover, by Lemma 2.8 ∩J m = ω(0).
We will consecutively construct a sequence of homeomorphisms The h m are K * -qc on U m−1 \ J m , with dilatation K * depending only on the choice of limbs and a priori bounds; (iv) Dil(h m |U m−l ) ≤ 4K 4 * Dil(h m−1 |U m−l ). Such a sequence will do the job: Lemma 3.2. A sequence h m satisfying the above three properties uniformly converges to a Thurston conjugacy.
Proof. By the second property, this sequence eventually stabilizes outside ∩J m and thus it pointwise converges to a homeomorphism h : (C, ∩J m ) → (C, ∩J m ). By the last two properties, the dilatation of h m on U m−l ∩ J m is at most 4 l K * . Hence h is quasi-conformal on C \ ∩J m . But by Lemma 2.8 ∩J m = ω(0) is a removable Cantor sets. Hence h admits a qc extension across ω(0).
Further, h is homotopic to h 0 rel ω(0). Indeed, let h t , 1 − 2 −m ≤ t ≤ 1 − 2 −(m+1) , be a homotopy between h m and h m+1 given by (iii). Let ǫ m = max i diam U m i . As the U m shrink to a Cantor set, ǫ , the uniform distance between h and h t is at most ǫ m−l . It follows that the h t uniformly converge to h as t → 1. Hence h is homotopic to h 0 rel ω(0).
Since h 0 is a topological conjugacy by (i), h is a Thurston conjugacy.
3.4. Construction of h 0 . Let us supply the exterior C \ cl D of the unit disk, with the hyperbolic metric ρ. The hyperbolic length of a curve γ will be denoted by l ρ (γ), while it Euclidean length will be denoted by |γ|.
Lemma 3.3. Let A andÃ be two (open) annuli whose inner boundary is the circle T. Let ω : A →Ã be a homeomorphism commuting with P 0 : z → z 2 near T. Then ω admits a continuous extension to a map A ∪ T →Ã ∪ T identical on the circle.
Proof. Given a set X ⊂ A, letX denote its image image by ω. Let us take a configuration consisting of a round annulus L 0 = A[r, r 2 ] contained in A, and an interval I 0 = [r, r 2 ]. Let L n = P −n 0 L 0 , and I n k denote the components of P −n 0 I 0 , k = 0, 1, . . . , 2 n − 1. The intervals I n k subdivide the annulus L n into 2 n "Carleson boxes" Q n k . Since the (multi-valued) square root map P −1 0 is infinitesimally contracting in the hyperbolic metric, the hyperbolic diameters of the boxesQ n k are uniformly bounded by a constant C.
Let us now show that ω is a hyperbolic quasi-isometry near the circle, that is, there exist ǫ > 0 and A, B > 0 such that Let γ be the arc of the hyperbolic geodesic joining z and ζ. Clearly it is contained in the annulus A(1, r), provided ǫ is sufficiently small. Let t > 1 be the radius of the circle T t centered at 0 and tangent to γ. Let us replace γ with a combinatorial geodesic Γ going radially up from z to the intersection with T t , then going along this circle, and then radially down to ζ. Let N be the number of the Carleson boxes intersected by Γ. Then one can easily see that ρ(z, ζ) = l ρ (γ) ≍ l ρ (Γ) ≍ N, provided ρ(z, ζ) ≥ 10 log(1/r) (here log(1/r) is the hyperbolic size of the boxes Q n k ). On the other hand ρ(z,ζ) ≤ l ρ (Γ) ≤ CN, so that ρ(z,ζ) ≤ Cρ(z, ζ), and (3.2) follows.
But quasi-isometries of the hyperbolic plane admit continuous extensions to T (see, e.g., [Th]). Finally, it is an easy exercise to show that the only homeomorphism of the circle commuting with P 0 if identical. Proof. By the Straightening Theorem, we can assume without loss of generality that f = P c : z → z 2 + c is a quadratic polynomial. Let R : C \ K(f ) → C \ cl D be the Riemann mapping normalized by R(z) ∼ z near infinity. It conjugates P c to P 0 : z → z 2 . Let Then ω # commutes with with P 0 in an open annulus attached to the circle T. By Lemma 3.3, ω # continuously extends to T as id. Hence for any ǫ > 0 there is an r > 1 such that |ω # (z) − z| < ǫ for z ∈ A(1, r).
As the square root map P −1 0 is infinitesimally contracting in the hyperbolic metric, Take now any point z in the annulus A(r 1 4 , r 1 2 }. Using the same argument we conclude that ρ(z, ω # (z)) ≤ C(r) (with the same C(r). By induction, the same bound holds for all z.
Now we can complete the proof. Since the Riemann mapping R is a hyperbolic isometry, the hyperbolic distance between ω(z) and z in C \ J(P c ) is also bounded near J(P c ). Hence the Euclidean distance |z − ω(z)| goes to 0 as z → J(f ). It follows that the extension of ω as the identity on the Julia set is continuous.  Proof. Given an annulus A, let ∂ o A and ∂ i A stand for its outer and inner boundary components. Let us select a closed fundamental annulus A for f with smooth boundary, and let A n = f −n A. LetÃ andÃ n be similar objects forf . Then there is a diffeomorphism This diffeomorphism can be lifted to a diffeomorphism φ 1 : A 1 →Ã 1 with the same qc dilatation and such that In turn, A 1 can be lifted to a diffeomorphism φ 2 : A 2 → A 2 with the same dilatation, which matches with A 1 on ∂ o A 2 and respects dynamics on ∂ i A 2 , etc.
By the Gluing Lemma from the Appendix, these diffeomorphisms glue together into a single quasi-conformal map h 0 : On the other hand, let ψ be a topological conjugacy between f andf near the Julia sets. Then by Corollary 3.5, h 0 matches with ψ on J(f ).
3.5. Adjustment of h m . Recall that p m is the period of the little Julia sets J m j , and F m = f pm is the corresponding quadratic-like map near J m j . Let U m = ∪U m j be a standard neighborhood of the little Julia orbit J m = ∪B m j = ∪J m i , with a definite space in between the U m j and definite annuli U m j \ B m j , and let S m : U m j \ B m j → A(1, 4) be the standard straightenings (2.3). Its dilatation is bounded by a constant K * depending only on the choice of secondary limbs and a priori bounds. Let U m j (t) = S −1 m A(1, t) (note that U m j ≡ U m j (4)). The notation U m (t) is self-evident.
We say that a homeomorphism φ : U m j (2)\B m j →Ũ m j (2)\B m j is standard near the bouquet B m j if it is identical in the standard coordinates on U m j (2), that is, The dilatation of such a map is bounded by K 2 * . Note also that by Corollary 3.5, the standard map admits a homeomorphic extension across the Julia bouquet.
We will now adjust the map h m so that it will become standard near J m .
Lemma 3.7. Take an l as in §3.3. Let a homeomorphism h m : (C, J m ) → (C,J m ) be a conjugacy on J m and be K m -qc on U m−l \ J m . Then there is a homeomorphism Proof. Let us consider a retraction ψ t j : U j (4) \ B j → U j (4) \ U j (2) which is the affine vertical contraction in the standard coordinates. Its dilation is bounded by 2K 2 * . Let us extend the ψ t j to a homeomorphism ψ : C \ J → C \ U(t) by identity on C \ U(4). By the Gluing Lemma from the Appendix, ψ is also 2K 2 * -qc. Let us now define a homeomorphism h t : (C, U t , J) → (C,Ũ t ,J) as follows: In what follows we will assume that h m is adjusted as in Lemma 3.7, and will skip the "hat" in the notation for the adjusted map. Note however that unlike h m , the map Ψ m does not respect dynamics on the little Julia sets. We need to pay temporarily this price in order to make Ψ m globally quasi-conformal.

Through the principal nest
In what follows we will assume that R m f ≡ F m is not immediately renormalizable.

4.1.
Teichmüller distance between the configurations of puzzle pieces. Let us make a choice of a standard neighborhood U m of the Julia bouquet B m and the corresponding standard straightening S m , see (2.3). When F m−1 is not immediately renormalizable, this provides us with a family Y of puzzle pieces Y (k) i , see §2.6 of Part I. In the immediately renormalizable case let us start the puzzle in a slightly different way. Namely, let us consider a degenerate domain of F m (see §2.5 of Part I) bounded by external rays landing at fixed and co-fixed points α m−1 ≡ β m , −β m , and two pieces of standard equipotentials of F m−1 . Then play the puzzle by cutting this domain with external rays of F m−1 landing at α m , and pulling them back. One can easily see that this beginning is equally suitable for the puzzle game as the usual one.
As the puzzle pieces Y (k) i are bounded by equipotentials and rays, they bear the standard boundary marking, i.e. the parametrization S −1 m by the corresponding straight intervals or circle arcs.
Since h m : U m →Ũ m is the standard conjugacy (see (3.3)), it maps the pieces Y (k) i to the corresponding tilde-piecesỸ (k) i respecting the boundary marking. Given some family of puzzle pieces P i ∈ Y contained in some Y ∈ Y, let us say that a homeomorphism is a pseudo-conjugacy if it is homotopic to h m rel the boundary (∂Y, ∪∂P i ). Note that if f l : P i → Y (or f l : P i → P j ) for some iterate of f and some puzzle pieces of our family, then the pseudo-conjugacy φ is a true conjugacy between the boundary maps f l : ∂P i → ∂Y andf l |∂P i → ∂Ỹ (correspondingly ∂P j instead of ∂Y ).
In particular, the above terminology will be applied to the principal nest of puzzle pieces (see §3 of Part I): The rest of this section, except the final subsection, §4.10), will be occupied with the proof of this lemma which follows [L2], §4. As the level m is fixed, in what follows we will skip the label m in the notations of V m,n i ≡ V n i , g m,n ≡ g n etc. (unless it may lead to a confusion). In what follows referring to a qc-map, we will mean that it has a definite dilatation (depending only on the selected limbs and a priori bounds).

A point set topology lemma.
In the statement below, the objects involved need not have any dynamical meaning.
Lemma 4.1. Let P i be a family of closed Jordan disks with disjoint interiors contained in a domain Y , such that diam P i → 0. LetP i ,Ỹ be another family of disks with the same properties.
• Let h : (Y, ∪P i ) → (Ỹ , ∪P i ) be a one-to-one map, which is a homeomorphism on ∪P i and on X ≡ Y \ (∪ int P i ). Then h is a homeomorphism.
Proof. Given an ǫ > 0, there exists an N such that diam(P n ) < ǫ for n > N.
Given a point z ∈ Y , let us show that h is continuous at it. This is certainly true if z ∈ ∪ int P i , so let z ∈ X. We will show that for any nearby point ζ ∈ Y . Indeed, if ζ ∈ X ∪T it follows from the above remark. Otherwise ζ ∈ P j for some j > N, and there is point u ∈ [z, ζ] ∩ ∂P j . Then If ζ is sufficiently close to z then the first term is at most ǫ by continuity of h|X. As the second term is bounded by diam(P j ) < ǫ, and (4.2) follows. Let us now prove the second statement. As each P i is simply connected, h 0 |P i is homotopic to h 1 |P i rel ∂P i . Let h t : ∪P i →P i be a corresponding homotopy. Extend it to the whole domain Y as h 0 . We should check that this extension h t (z) : (Y, ∪P i ) → (Ỹ , ∪P i ) is continuous in two variables.
Note first that for z ∈ ∪ 1≤i≤N P i ≡ T , Given a pair (z, t), we will show that |h t (z) − h τ (ζ)| < 3ǫ as (ζ, τ ) is sufficiently close to (z, t). To this end let us consider a few cases: • If z ∈ ∂T but ζ ∈ T , then for ζ sufficiently close to z, by continuity of h and (4.3).
• Let z ∈ T . Then sufficiently close points ζ don't belong to T either. Hence by (4.3) and continuity of h, , k = 0, 1 . . . } Recall that an invariant set K is called expanding if there exist constants C > 0 and ρ ∈ (0, 1) such that |DF k (z)| ≥ Cρ k , z ∈ K, k = 0, 1, . . . Proof. Let us consider thickened puzzle piecesŶ (N ) i as in Milnor [M1] or §2.5 of Part I. Then int(FŶ is contracting by a factor λ < 1 in the hyperbolic metrics of the pieces under consideration. Let is at most λ l , and the statement follows. 4.4. First landing maps. Let us have a family of puzzle pieces P i with disjoint interiors contained in a puzzle piece X, where as usual P 0 ∋ 0 stands for the critical puzzle piece. Let us also have a Markov map G : ∪P i → X which is univalent on all non-critical pieces P i , i = 0, and the double branched covering on the critical one, P 0 . The Markov property means that if int(GP i ∩ P j ) = ∅ then GP i ⊃ P j . Let A be the corresponding Markov matrix: A ij = 1 if int(GP i ∩ P j ) = ∅, and A ij = 0 otherwise.
Let P ≡ P 0 . A string of labelsī = (i(0), . . . , i(l − 1)) is called admissible if A i(k),i(k+1) = 1 for k = 0, . . . , l − 2), and i(k) = 0 for k < l − 1. The length l of the string will be denoted by |ī|. To any admissible string corresponds a cylinder of rank l defined by the following property: Note that G l−1 univalently maps P l i onto P i(l−1) . Let us denote by Ωī ≡ P l i the cylinders mapped onto the critical puzzle piece (so that i(l − 1) = 0). The first landing map is defined as follows: T z = G l−1 z for z ∈ Ωī, |ī| = l. This map is univalent on all pieces Ω i (identical on the critical piece Ω 0 ).
Proof. Let us pull H back to the pieces P i , i = 0, that is, let us consider the map Hence these maps glue together into a single map K-qc map equal to H 1 on ∪∂P i , and equal to H outside of it. We will keep notation H 1 for this map. Let us do the same pull-back with H 1 . We will obtain a K-qc pseudo-conjugacy . Repeating this procedure over again, we obtain a sequence of K-qc pseudo-conjugacies By the Compactness Lemma from the Appendix we can pass to a limit K-qc map φ : By Lemma 4.1 this map is homotopic to h rel (∂X ∪ ∂Ω j ), and hence is a desired pseudoconjugacy.
Let us now do a bit more (assuming a bit more). Let us consider the generalized renormalization of G on P 0 , that is, the first return map g : ∪V j → P 0 . Let b = g(0) = G t 0 be its critical value.
Proof. As H and H ′ match on ∂P 0 , they glue together into a singe K-qc pseudo-conjugacy H : (X, ∪P i , b) → (X, ∪P i ,b) coinciding with H 1 on P 0 and coinciding with H 0 on X \P 0 (see the Gluing Lemma in the Appendix). By Lemma 4.3, there is a K-qc map φ : (X, ∪Ω j ) → (X, ∪Ω j ) homotopic to h rel (∂X ∪ ∂Ω j ), and conjugating the first landing maps. As H : b →b, we have: φ : G k 0 →G k 0, k = 1, . . . , t. In particular, φ respects the G-critical values: Recall that the domains V i are the pull-backs of the Ω j by G : P 0 → X, that is, the components of (G|P 0 ) −1 Ω j . It follows that φ can be lifted to a K-qc map ψ : (P 0 , ∪V i ) → (P 0 , ∪Ṽ i ) homotopic to h rel (∂P 0 ∪ ∂V i ). (This lift is uniquely determined by the diagram G • ψ|P 0 = φ • G|P 0 and the homotopy condition.) This map ψ is the desired pseudo-conjugacy.
4.5. Initial constructions. Now the reader should consult §3.2 of Part I of this paper [L3], where the initial Markov partition (3-3) of the Yoccoz puzzle piece Y (0) is constructed. We will apply it to the renormalized map F . Let us recall some notations. The first piece of the partition, Y ≡ Y (0) , is bounded by the external rays landing at the fixed point α, and the equipotential E. The central piece of this partition, V 0 , is the first piece of the principal nest. It is obtained by pulling back a puzzle piece Z (1) ν attached to the co-fixed point α ′ (that is, F (α ′ ) = F (α)). There is a double branched covering F s : V 0 → Z (1) ν . All the puzzle pieces of the initial partition intersecting the Julia set J(F ) are univalent pull-backs of either Y or V 0 . Let us denote the pieces of this partition by P i , in such a way that where p is the number of external rays of F landing and α. With these notations, where K is the residual Cantor set (of the points whose orbits never land at ∪ 0≤i≤p−1 P i ).
Lemma 4.5. In the decomposition (4.6), diam P i → 0 and the set K is a removable Cantor set.
Proof. The first statement follows from Lemma 4.2. To prove removability of K, let us consider the domains Q 1 and Q 2 introduced in §3.2 of Part I. Then F p Q i covers Q 1 ∪ Q 2 , and K is the set of points which never escape Q 1 ∪ Q 2 . By a little thickening of these domains, we obtain a Bernoulli map F p :Q 1 ∪Q 2 → C (so that int(F pQ i ) containsQ i ). By Lemma 2.9, the Julia setK of this map is removable. All the more, K ⊂K is removable (one can actually see that K =K).
Let us now go back to §4.2 of Part I where the fundamental domain Q near the fixed point α is constructed. Recall that γ ∈ Y (1) is the periodic point of period p, γ ′ = −γ is the "coperiodic" point, and R(γ ′ ) is the family of rays landing at γ ′ . Also, let X = Y (0) ∪ 1≤i≤p−1 P i . This domain is bounded by the rays landing at α and equipotential F −1 E. Furthermore D is the connected component of Y (1) \ R(γ ′ ) attached to α, and F −p : D → F −p D is the branch of the inverse map fixing α.

x°α°γ
Proof. By the secondary limbs and a priori bounds assumptions, geometry of the configuration (Y, Y (1) , Y (1+p) , R(γ), R(γ ′ )) is bounded (see §4.1 of Part I). Hence Q * −2 and Q * 1 have a bounded hyperbolic diameter in Y . For the same reason, Q * has a bounded hyperbolic diameter in F p D * . As F −p : F p D * → F p D * is a hyperbolic contraction, the diameters of Q * −k in F p D are bounded by the same constant. All the more, they are bounded in a bigger domain Y .
To prove the second statement, note that by bounded geometry of the initial ray-equipotential configurations, there is a definite space in between Q * −1 and the Q * −k , k = 0, 1 . . . For the same reason, there is a definite annulus T 0 ⊂ F p D * about Q * 0 which does not intersect Q * −k , k = 2, 3 . . . . Then T −i = F −ip T 0 ⊂ F p D * is the annulus with the same modulus going around Q * −i and disjoint from Q * −k with |k − i| > 1.
Our first essential step towards the Main Lemma is the following: Lemma 4.7. The Teichmüller distance between the configurations (Y, Proof. Recall that F s (V 0 ) = P ν , and F (P i ) univalently covers Y . Let us consider a point a = F s+1 0 ∈ X. We will construct a qc map (Y, a) → (Ỹ ,ã) respecting the boundary marking.
In particular, this map conjugates F p : Q * → F p Q * to the corresponding tilde map.
As F p univalently maps Q * −k−1 onto Q * −k , φ can be re-defined on the Q * −k , k ≥ 0, in such a way that it becomes the pseudo-conjugacy between the configurations with the same dilatation. (Just let φ(z) =F −kp • φ • F kp (z) for z ∈ Q * −k ). It follows that φ(a) andã belong to the same piece of the family {Q * −k } ∞ k=−2 By Lemma 4.6 the hyperbolic distance between φ(a) andã inỸ is bounded.
Consider now the double branched covering F s+1 : (V 0 , 0) → (Y, a) with the critical point at 0, and the corresponding tilde map. As φ 1 : (Y, a) → (Ỹ ,ã) respects the critical values for these maps, it can be lifted to a map φ 2 : V 0 →Ṽ 0 with the same dilatation respecting the boundary marking.
Let us now construct a qc pseudo-conjugacy φ 3 between corresponding non-critical puzzle pieces P i andP i . It is easy as any non-central puzzle piece P i under some iterate F l i is univalently mapped onto either Y or V 0 . In the first case let φ 3 be the pullback of φ : Y →Ỹ ; in the second let it be the pull-back of φ 2 . This pull-back preserves the dilatation and respects the boundary marking. This provides us with a qc map φ 3 : ∪P i → ∪P i respecting the boundary marking of the puzzle pieces.
The latter property means that φ 3 matches with h on ∪∂P i . By the first part of Lemma 4.5 and Lemma 4.1, these maps glue together into a single homeomorphism coinciding with φ 3 on ∪P i and with h outside, homotopic to h rel ∂Y ∪ ∂P i (we will still denote it φ 3 ).
By the Gluing Lemma from the Appendix, this homeomorphism is qc on Y \ K. By the second part of Lemma 4.5, the residual set K is removable, and thus φ 3 is automatically quasi-conformal across it (with the same dilatation). ⊔ ⊓ The next step towards the Main Lemma is the following: Lemma 4.8. The configurations (V 0 , ∪V 1 i ) and (Ṽ 0 , ∪Ṽ 1 i ) stay bounded Teichmüller distance away.
Proof. Let us consider the first return b = g 1 0 of the critical point back to V 0 . We will construct a qc map respecting the boundary marking. Let u = F s+1 b ∈ X (where F s maps V 0 onto P ν ). Let φ be a pseudo-conjugacy given by Lemma 4.7.Then φ(u) andũ belong to the same piece of the tiling X∩J(F ) = ∪ −∞<k≤2 (Q * −k ∩ J(F )). By Lemma 4.6, the hyperbolic diameters of these pieces in Y (and the corresponding tilde-pieces) are bounded by a constant ρ dependent only on the selected limbs and a priori bounds. Hence ρỸ (φ(u),ũ) ≤ ρ.
So, we have constructed a qc map (4.8) which carries the critical value b = g 1 (0) to the critical valueb =g 1 . Lemma 4.4 completes the proof. 4.6. Inductive step (non-central case). Let us now inductively estimate the Teichmüller distance between the configurations (V n−1 , ∪V n i ) and (Ṽ n−1 , ∪Ṽ n i ). Let τ n stand for the maximum of this Teichmüller distance and log Dil(h), where as above, h stands for the conjugacy between F andF ). Recall that µ n = mod (V n−1 \ V n ) denote the principal moduli.
The following lemma is the main step of our construction.
Remark. We don't assume that the non-critical puzzle-pieces V n i , i = 0, are well inside V n−1 , since this is not the case on the levels which immediately follow long central cascades (see Theorem II of Part I). to be degenerate which actually happens in the beginning.
Proof. Let us skip n in the notations of the objects of level n, so that V n i ≡ V i , g n ≡ g, µ n ≡ µ, etc. Also, let V n−1 ≡ ∆ and g(0) ≡ c 1 . As above, the corresponding objects for f are marked with tilde. Thus we have two generalized polynomial-like maps g : ∪V i → ∆ andg : ∪Ṽ i →∆, which are pseudo-conjugate by a K = e τ -qc map φ : (∆, ∪V i ) → (∆,Ṽ i ). (4.9) The objects on the next level, n + 1, will be marked with prime: Let λ(ν) be the maximal hyperbolic distance between the points in the hyperbolic plane enclosed by an annulus of modulus ν. Note that λ(ν) = O(e −ν ) as ν → ∞ (see Appendix A1 in Part I). Set λ = λ(µ).
Our goal is to lift φ to a K(1 + O(λ))-qc pseudo-conjugacy (4.10) The problem is that φ need not respect the positions of the critical values: φ(c) =c. Let us consider the first landing map T : ∪Ω j → V 0 . By Lemma 4.3, the pseudo-conjugacy φ admits the pull-back to a K-qc pseudo-conjugacy φ 1 : (∆, ∪Ω j ) → (∆, ∪Ω j ). (4.11) This localizes the positions of the critical values in the sense that φ 1 (c 1 ) andc 1 belong to the same domain Ω s ⊂ V k (see Figure 3) and hence the hyperbolic distance between these points inṼ k is at most λ. Figure 3. Localization of the critical values.
Case (iii). Let us finally assume that V j = V 0 is critical. Then a belongs to a pre-critical puzzle-piece (4.13)). By the Moving Lemma, there is a (1 + O( √ λ))-qc map ψ 3 : (∆ ′ , φ(a)) → (∆ ′ ,ã). Let us now define a map (4.14) as follows: φ ′ 0 = ψ 3 • H. So in all cases we have constructed a (1 + O(λ 1/4 ))-qc map (4.14). It is still not the desired map (4.10), though. Now Lemma 4.4 completes the proof. 4.7. Through a central cascade. Let V m ⊃ V m+1 ⊃ ... ⊃ V m+N be a cascade of central returns, so that the critical value g m+1 0 belongs to V k , k = m + 1, ..., m + N − 1, but escapes V m+N . Proof. We will adjust the proof of Lemma 4.9 to this situation. Let g = g m+1 , µ = mod (V m \ V m+1 ), etc. By definition, there is a K = e τ -qc pseudo-conjugacy: Let us consider the first landing map T : ∪Ω j → V m+1 corresponding to g, Ω 0 = V m+1 . By Lemma 4.3, T andT are pseudo-conjugate by a K-qc map Let us take a family of puzzle pieces V m+1 i ⊂ A m+1 = V m \V m+1 and pull them back to the annuli A m+2 , . . . , A m+N . We obtain a family of puzzle pieces W m+k i , together with a Bernoulli map Let us now lift φ 1 to the annuli A m+k →Ã m+k , k = 2, . . . , N. We obtain a K-qc map )-qc, respects the boundary marking and positions of the critical values. Consider now the topological disks Q 1 and Q 2 in V m+N univalently mapped by g onto V m+N . The Bernoulli map g : Q 1 ∪ Q 2 → V m+n produces a family of cylinders Q t i ,ī = (i(0), i(1), . . . , i(t − 1)), such that Moreover, by Lemma 2.9, the residual set K = ∩Q t is removable.
The map φ 3 can be consecutively lifted to the maps with the same dilatation respecting the boundary marking. By the Gluing Lemma, they are organized in a single qc map with the same dilatation. As K is removable, this map automatically extends across K: The maps (4.18) and (4.17) glue together into a single K(1 + O(e −mu ))-qc map Take now a family of cylinders W m+k i of the Bernoulli map (4.15) (whereī are finite strings of symbols). The map φ 4 is naturally lifted to a qc pseudo-conjugacy Φ with the same dilatation which respects this family of cylinders. Moreover, every W m+k i contains a piece and all puzzle pieces V m+k j are obtained in such a way. As φ 4 respects the V m+k−1 -pieces, k ≤ N, the new map Φ respects the V m+k j -pieces. Thus Φ is a K(1 + O(e −µ ))-qc pseudoconjugacy between g m+k andg m+k , so that τ m+k ≤ log K + O(e −µ ), k = 1, . . . m + N.
Let us proceed with the estimate of τ m+N +1 . Take the first return a of the critical point back to V m+N , and construct a K(1 + O(e − µ 4 ))-qc map To this end let us go through Cases (i), (ii), (iii) of the proof of Lemma 4.9 using the {W m+N In the first two cases the argument is the same as above. However, the last case is different since the pre-critical puzzle-pieces Q 1 and Q 2 are not necessarily well inside of V m+N . To take care of this problem let us consider the first "escaping moment" t when b ≡ g t a ∈ Q 1 ∪ Q 2 . Then b ∈ U m+N +1 i for some U-domain from (4.18). Then there is a domain Λ ⊂ Q 1 ∪ Q 2 containing a which is univalently mapped onto U m+N +1 i by g t . Moreover By means of g : Q 1 ∪ Q 2 → V m+N , the map (4.18) can be turned into a qc map (with the same dilatation) (4.20) But by Theorem III from Part I [L3], the principal moduli µ i(k)+1 grow at linear rate: µ i(k)+1 ≥ Bk, where the constant B depends only on µ 1 . Hence the sum (4.20) is bounded by log K * + C(µ 1 ). In turn, by Theorem I of Part I the modulus µ 1 is bounded by a constant depending only on the selected limbs and a priori bounds. Hence τ n ≤ log K * + A, where A depends only on the choice of limbs and a priori bounds. The Main Lemma is proved. ⊔ ⊓ 4.9. Last cascade. If the map F ≡ F m = R m f is not renormalizable then the principal nest consists of infinitely many central cascades, and the Main Lemma gives a uniform bound on the Teichmüller distance between the corresponding generalized renormalizations. Otherwise the principal nest ends up with an infinite central cascade V n−1 ⊃ V n ⊃ . . . shrinking to the little Julia set J m+1 of the next renormalization g n = F m+1 ≡ R m+1 f . All levels n, n + 1, . . . of this final cascade are called the renormalization levels.
Proof. Recall that k n = V k−1 \ V k . The map H : A n →Ã n admits a lift to qc maps (with the same dilatation) H k : A n+k →Ã n+k homotopic to h rel the annuli boundary. These maps match to a single qc map φ : V n−1 \ J m+1 →Ṽ n−1 \J m+1 with the same dilatation conjugating F m+1 toF m+1 . By Corollary 3.5, this map (and the whole homotopy between it and h) matches with h on J(F m+1 ). 4.10. Spreading around. Let us consider the pieces P j ⊂ Y ≡ Y (0) of the initial partition (4.6), and the Markov map G : ∪P i → Y (see §3.2 of Part I). Let us consider the first landing map to V 0 , T 0 : ∪Ω 0 i → P 0 . By Lemma 4.7 and Lemma 4.3, there is a qc pseudo-conjugacy φ 0 : (Y, ∪Ω 0 i ) → (Ỹ , ∪Ω 0 i ). Let us also consider the following maps: • The first landing maps to V n corresponding to the generalized renormalization g n : ∪V n i → V n−1 : T n : ∪Ω n i → V n , Ω n i ⊂ V n−1 ; • The first landing maps to V n corresponding to G: (4.21) By the Main Lemma and Lemma 4.3, there is a sequence of qc pseudo-conjugacies φ n : (V n−1 , ∪Ω n i ) → (Ṽ n−1 , ∪Ω n i ), n < N + 1, where N is the first DH renormalizable level (if F is non-renormalizable then N = ∞). Let us turn it inductively into a sequence of pseudo-conjugacies between S n andS n (with the same dilatation). Indeed, using (4.21), we can define it as follows: As these maps match with H n−1 on the boundaries ∂O n−1 i , the glue together into single qc conjugacy (4.22) with the same dilatation.
If F is non-renormalizable, we obtain an infinite sequence of qc pseudo-conjugacies H n (with uniformly bounded dilatation). As the pieces V n i shrink as n → ∞, there is the limit qc map Assume F is renormalizable. Let I be the family of little Julia sets J m+1 i contained in Y , J m+1 ≡ J m+1 0 . Let us consider the last pseudo-conjugacy (4.22) on the renormalization level N. Let us replace it on V n−1 by the pseudo-conjugacy constructed in Lemma 4.11. Spread it around by the landing map S N , that is, set As these maps match on the ∂O N with H N , they glue together into a homeomorphism (4.24) quasi-conformal on Y \ i∈I J m+1 i (with dilatation depending only on the choice of limbs and a priori bounds), and homotopic to h rel ∂Y i∈I J m+1 i . Let us consider the backward orbit Y ≡ Y 0 , Y −1 , . . . , Y −r+1 of Y under f such that Y −k ∋ f r−k 0, where r is the first return time of the critical orbit to Y . The disks Y −k have disjoint interiors. Let us pull the map H back to these disks, that is, set As this map respects the boundary marking of the Y −k , it extends to to the whole plane as h m , which provides the desired next approximation to the Thurston conjugacy (see §3.3).
The Rigidity Theorem is proved.

5.
Appendix: Quasi-conformal maps 5.1. There are a few Russian and English sources on the basic theory of quasi-conformal maps: [A, B, Kr, LV, V]. A homeomorphism h : U → V , where U, V ⊂ C, is called quasi-conformal (qc) if it has locally integrable distributional derivatives ∂h,∂h, and |∂h/∂h| ≤ k < 1 almost everywhere. As this local definition is conformally invariant, one can define qc homeomorphisms between Riemann surfaces.
One can associate to a qc map an analytic object called Beltrami differential, namely µ =∂ h ∂h dz dz , with µ ∞ < 1. The corresponding geometric object is a measurable family of infinitesimal ellipses (defined up to dilation), pull-backs by h * of the field of infinitesimal circles. The eccentricities of these ellipses are ruled by |µ|, and are uniformly bounded almost everywhere, while the orientation of the ellipses is ruled by the arg µ. The dilatation Dil(h) ≡ K h = (1 + µ ∞ )/(1 − µ ∞ ) of h is the essential supremum of the eccentricities of these ellipses. A qc map h is called K-qc if Dil(h) ≤ K.
One of the most remarkable facts of analysis is that any Beltrami differential with µ ∞ < 1 (or rather a measurable field of ellipses with essentially bounded eccentricities) is locally generated by a qc map, unique up to post-composition with an analytic map. Thus such a Beltrami differential on a Riemann surface S induces a conformal structure quasi-conformally equivalent to the original structure of S, Together with the Riemann mapping theorem this leads to the following statement: Measurable Riemann Mapping Theorem. Let µ be a Beltrami differential onC with µ ∞ < 1, Then there is a quasi-conformal map h :C →C which solves the Beltrami equation: |∂h/∂h| = µ.
In what follows by a conformal structure we will mean a structure associated to measurable Beltrami differentials µ with µ ∞ < 1. We will denote by σ the standard structure corresponding to zero Beltrami differential.
Another fundamental property of the space of qc maps is compactness: Compactness Lemma. The space of K-qc maps h : C → C normalized by h(0) = 0 and h(1) = 1 is compact in the uniform topology on the Riemann sphere.
The following gluing property is also important: Gluing Lemma. Let us have two disjoint domains D 1 and D 2 with a piecewise smooth arc γ of their common boundary. Let D = D 1 ∪ D 2 ∪ γ. If h : D → C is a homeomorphism such that h|D i is K-qc, then h is K-qc.
One of Sullivan's leading ideas was the idea of the Teichmüller metric on the space of deformations of a conformal dynamical systems. The prototype for this metric is the classical Teichmüller metric on the space of marked Riemann surfaces. A Riemann surface (perhaps with boundary) is said to be marked if it is endowed with a preferred basis of the fundamental group and a preferred parametrization of the boundary components. The Teichmüller distance dist(S 1 , S 2 ) between two marked Riemann surfaces is defined as the infimum of the dilatations K h , where h : S 1 → S 2 runs over qc homeomorphisms in the homotopy class respecting the marking.
Let D be a simply connected domain conformally equivalent to the hyperbolic plane H 2 . Given a family of subsets {S k } n k=1 in D, let us say that a family of disjoint annuli A k ⊂ D\∪S k is separating if A k surrounds S k but does not surround the S i , i = k. The following lemma is used in the present paper uncountably many times: Moving Lemma. • Let a, b ∈ D be two points on hyperbolic distance ρ ≤ρ. Then there is a diffeomorphism φ : (D, a) → (D, b), identical near ∂D, with dilatation Dil(φ) = 1 + O(ρ), where the constant depends only onρ.
Proof. As the statement is conformally equivalent, we can work with the unit disk model of the hyperbolic plane, and can also assume that a = 0. Also, it is enough to prove the statement for sufficiently small ρ.
Hence for sufficiently small ρ > 0, f is a local orientation preserving diffeomorphism. As f : ∂D → ∂D, f is a proper map. Hence it is a diffeomorphism. Finally, (5.1) yields that the Beltrami coefficient µ f = O(ρ), so that the dilatation Dil(f ) = 1 + O(ρ).
Let Q ⊂ C, h : Q → C be a homeomorphism onto its image. It is called quasi-symmetric (qs) if for any three points a, b, c ∈ Q such that q −1 ≤ |a − b| |b − c| ≤ q, we have: It is called κ-quasi-symmetric if κ(1) ≤ κ. It follows from the Compactness Lemma that any K-qc map is κ-quasi-symmetric, with a κ depending only on K.
Let us note that in the upper half-plane model, the Ahlfors-Börling extension of a qs map R → R is affinely equivariant (that is, commutes with the action of the complex affine group z → az + b).
A direct calculation shows that the dilatation of the covering map θ on the fundamental intervals I λ and −I λ is comparable with (log r) −1 . Henceh is C(κ, r)-qs on this interval. By equivariance it is C(κ, r)-qc on the rays R + and R − .
5.2. Removability. A compact set X ⊂ C is called removable if for any neighborhood U ⊃ X, any conformal map h : U \ X → C admits a conformal extension across X. Let us show that removability is quasi-conformally invariant.
Lemma 5.1. Let φ : (C, X) → (C,X) be a qc map. If the set X is removable thenX is removable as well.
Proof. Let σ be the standard conformal structure on C. LetŨ ⊃X be a neighborhood of X, and leth :Ũ \X → C be a conformal map. Let as consider a conformal structureμ on C which is equal to (h • φ) * (σ) on h(Ũ \X), and is equal to σ outside. By the Measurable Riemann Mapping Theorem, there is a qc map ψ : C → C such thatμ = ψ * (σ).
Let U = φ −1Ũ . Then the function h = ψ −1 •h • φ : U \ X → C is conformal. As X is removable, it admits a conformal extension across X. We will use the same notation h for the extended function. Then the formulah = ψ • h • φ −1 provides us with a conformal extension ofh acrossX.
Let us now show that removable sets are also qc-removable.