Universality of Critical Circle Covers - Springer Link

Commun. Math. Phys. 228, 371 – 399 (2002)

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Mathematical Physics

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Universality of Critical Circle Covers ´ atek2, G. Levin1, , G. Swi¸ 1 Department of Mathematics, Hebrew University, Jerusalem 91904, Israel. E-mail: [email protected] 2 Department of Mathematics, Penn State University, University Park, PA 16802, USA.

E-mail: [email protected] Received: 25 April 2001 / Accepted: 20 March 2002

Abstract: For a class of critical circle covers we show that properly re-scaled first return maps to a neighborhood of the critical point converge to universal limits. For that to occur, the critical point has to be sufficiently flat.

1. The Setting 1.1. Introduction. A brief history of phase-space universality. The universality in one-dimensional dynamics was discovered numerically by Feigenbaum [7, 8], and Coullet–Tresser [2] in the family x → bx(1−x), b ∈ [1, 4], of quadratic maps of the unit interval, and then proved rigorously by computer assisted and analytic methods, see [13, 14, 6]. Main feature of the quadratic dynamical system is that it is essentially non-linear: it has a critical point (which is folding in that case). Later, similar observations were made for some important high-dimensional non-linear dynamical systems such as the Lorenz system [1, 32]. Universality of critical circle automorphisms was also studied, see [26]. A more general approach which did not rely on computer-assisted estimates appeared first in [27]. This led to a proof of the phase-space universality, see also [24], and to the parameter universality [20], for unimodal mappings. Similar work was done for critical circle automorphisms with irrational rotation number, see [3, 4, 31]. For more discussion and a complete list, see survey [25] and references therein. Both authors were supported by Grant No. 98-00080 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. A part of this work was done while they both stayed at the Mathematical Institute of the Polish Academy of Sciences in Warsaw. The first author acknowledges KBN Grant 2 P03A 00917 of the Polish Academy of Sciences for its support. Partially supported by NSF grant DMS-0072312.

´ atek G. Levin, G. Swi¸

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There are two types of universality: in the phase space, and (related to it) in the parameter space. We focus on the universality in the phase space. Informally speaking, it can be stated as follows. Given a class of dynamical system, we start with one of them f , and construct a sequence of induced systems fi inductively: f0 = f while fi+1 is the first return map of fi restricted to an appropriate portion of the phase space of fi (which catches the nonlinearity of fi ) and then re-scaled to a unit size. The universality means that the sequence fi converges in an appropriate functional space to a system f∗ of our class, and f∗ is universal, i.e. f∗ does not depend on the map f started with. We prove that the universality exists also in the smooth interval maps which are increasing on each branch and have single (inflection) critical point. Such maps have been studied in [15, 16]. Typical example provides the family of generalized Arnold maps x → dx + θ −

d sin(2π x)(mod1), 2π

θ ∈R

(1)

with d an integer greater than 1. Note that each such map is semi-conjugate to a map x → dx(mod1). For the setting and precise statement of our results, see Theorem 1. Inducing. From the standpoint of one-dimensional dynamics, phase-space universality can be described as the study of first-return maps of the initial map to suitably chosen neighborhoods of the critical point. The goal is to prove that as the sizes of such neighborhoods shrink, the first return maps tend to limits which are moreover independent of the original large-scale dynamics. Whenever a result of this type can be established, it then becomes an important tool in the study of the large-scale dynamics. As the study of Feigenbaum universality began in the late 1970s along these lines, a different approach which also relied on first returns to shrinking neighborhoods of the critical point appeared in [11]. This method can be characterized as induced hyperbolicity. The goal was to show that under appropriate conditions first return maps to shrinking neighborhoods of the critical point become more and more expanding. In the simplest sense, the critical branch of the first return map, that is, the branch whose domain contains the critical point and which, therefore, is definitely not expanding, occupies less and less room in smaller and smaller scales. One can observe here that although the focus on first return maps is the same as in Feigenbaum type universality, the goals of both approaches are contradictory: if induced hyperbolicity occurs, no non-trivial limit exists. With time, the quest for induced hyperbolicity led to the development of “inducing algorithms” which were the recipes for taking the “right” sequences of first return maps for any kind of unimodal map, see [22, 12]. The next step was to study sequences of renormalized induced maps obtained by these algorithms with the goal of finding universality, whether or not induced hyperbolicity might occur. Here one has to realize that the renormalization group analysis is not appropriate to apply to the first return maps directly. The simplest reason is that these maps will have many branches, some of which will always exhibit increasing expansion thus negating the possibility of taking limits in the traditional sense. However, by simply ignoring most branches except for a few special ones, we can obtain a sequence of induced maps which are all of the same type at least topologically and then look for renormalization limits. Such is the nature of the “unimodal Fibonacci map”. In this case we take a sequence of first return maps dictated by the inducing algorithm and then for each of them ignore all branches but two, one of

Universality of Critical Circle Covers

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which contains the critical point and the other the critical value of the first return map. A renormalization group approach in this case was first proposed in [19], although in the particular case considered in that paper of Fibonacci unimodal maps with quadratic singularity no non-trivial renormalization limits exist. In an unpublished study of Fibonacci unimodal maps occurring in families of the type z2 + c, universality was discovered and proved for > 1, see [30]. In this paper we carry these ideas to the case of critical circle covers. They are in many ways similar to unimodal maps. The fact that the maps are of degree greater than 1 implies that expansion is taking place “on average” and that justifies a quest for induced hyperbolicity. It turns out that the inducing algorithms developed for unimodal maps extend naturally to this case, as does the example of the Fibonacci map. In this sense, the closest analogue to our work is the study in [30] of unimodal Fibonacci maps for maps with higher order singularities. Hidden universality. One has to be aware that the link between universality for induced maps and the large-scale dynamics in the general case including the Fibonacci map is more tenuous than in the classical examples of renormalizable unimodal maps, i.e. of Feigenbaum-type universality, or critical circle automorphisms. In any of those cases there exists a global attractor, both metric and topological, which is nothing but the closure of the critical orbit and whose geometry is automatically universal as a consequence of phase-space universality. In the case of general induced maps, we not only restrict ourselves to first return maps into small neighborhoods of the critical point, but also ignore almost all branches of such maps. As the result, it is not clear how the existence of renormalization limits will be visible on the level of the large-scale dynamics. Certainly, orbits from a residual set for a Fibonacci map are dense in the phase space and so fail to detect the postcritical set in whose geometry the universality would be visible. It is an open question whether almost all orbits in the sense of Lebesgue measure might also be dense. It is even conceivable that in some examples to which our results apply an SRB measure equivalent to Lebesgue measure could exist. In a case like this, universality would really be hidden from the point of view of a typical trajectory, although it certainly always remains visible from the critical trajectory, or in the structure of the corresponding Julia set in the complex plane, see [30]. Technical comments. The basic technique for establishing universality in a onedimensional map has not changed since it was described in [27]. For a smooth map the steps are as follows: 1. Establish certain real estimates, known as “bounded geometry” using the cross-ratio technique. 2. These real bounds together with Köbe type distortion estimates imply the normality of appropriately re-scaled (i.e. renormalized) induced maps, hence limits can be taken. 3. These limits are automatically analytic, in the so-called Epstein class (a key observation which we like to call Sullivan’s principle). 4. Polynomial-like extensions (see below) with complex bounds analogous to real bounds but involving estimates in the complex plane are now established for maps in the Epstein class. 5. Universality can now be proved for maps in the Epstein class using techniques of complex dynamics, on which we will comment more extensively in a moment. 6. Universality is finally inferred for the renormalized maps induced by the original smooth dynamics.


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In the key step of proving universality for complex maps we follow the “tower method” [24] when certain limit objects called towers are built from renormalized maps and universality becomes a rigidity statement about the towers. Thus the main idea can be described as follows: Consider the set of limit maps {G} of the sequence {fi }. Show that any map G extends from the real line to a polynomial-like map, i.e. to a complex analytic covering map from a finitely many disjoint topological discs onto a bigger disc [5, 19]. Prove that any two limit maps G1 , G2 coincide. To this end, construct two towers with the maps G1 , G2 in the bottom, and prove that these towers are linearly conjugate. In turn, this method is conditional: it cannot be developed without two a priori properties (important by themselves) of the limit maps G: so-called bounded geometry, and polynomial-like extensions with uniform complex bound. Consequently, most work of the paper is concentrated on the proof of the bounded geometry and the complex bound for the considered maps. Then we show that the tower construction works in our setting as well, following [24]. The main difference in our situation is that the domain of the mapping is disconnected. In this sense, the setting in analogous to that for the unimodal Fibonacci map, where universality was investigated in [30].

1.2. Class of mappings. Definition 1.1. Consider a class G m , m ≥ 1 and > 1, of mappings of the following form: • Every g ∈ Gm is a proper C m map from the union of two disjoint open intervals I0 and I1 onto an interval I . The restriction of g to either connected component of the domain (such restrictions will be called branches) is an increasing C 1 map onto I . • I0 ∩ I1 = ∅ and I0 ∪ I1 ⊂ I . • The derivative of g is positive except for a precisely one critical point ζ ∈ I0 , • g restricted to I0 can be represented as g(x) = g(ζ ) + H sgn(x − ζ )|x − ζ | , where H is C m onto the interval {x : x + g(ζ ) ∈ I } with H positive everywhere and H (0) = 0. We will write G m for >1 Gm . Combinatorial type. Definition 1.2. Let g ∈ G 1 . We say that a non-negative integer k is the combinatorial type of g if g 1 (ζ ), · · · , g k (ζ ) ∈ I1 while g k+1 (ζ ) ∈ I0 . Obviously, not every g ∈ G 1 has a combinatorial type. If it does and k = 0, then we say that g makes a close return. Otherwise we may say that it shows a non-close return with depth k.


375

1.3. Inducing. Inducing step defined. Let g ∈ G 1 have combinatorial type k. A new mapping g1 ∈ G 1 induced by g is obtained as follows. In the case of the close return, g1 is the first return map into I0 , however with the domain restricted only to those points, which return no later than in the second iteration. In the case of a non-close return it is the first return map, but with the domain restricted only to those points which return after 1 or k + 1 iterations. In other words, we take the first return map into I0 , but from among infinitely many branches only keep the central one, and the monotone branch with the lowest return time. It is clear that the new map g1 still belongs to G 1 . Extendibility. Each branch of g1 has a dynamical extension, from which the same iterate of g maps onto I . The domains of these extensions are still contained in I0 though not necessarily disjoint and have the form h ◦ g, where h is an iterate of g and a diffeomorphism onto I . Real bound. n n n Fact 1.1. Suppose that (gn )∞ n=1 , gn : I0 ∪ I1 → I , is a sequence of mappings from 3 G . Assume that every gn has a combinatorial type bounded by k, no more than k close returns may occur in a row and gn+1 is derived from gn by an inducing step. Assume also that for x in a neighborhood of ζ which is the critical point of g1 the representation

g1 (x) = g1 (ζ ) + sgn(H (x))|H (x)| is valid, where H is a C 3 local diffeomorphism with H (ζ ) = 0 and > 1. For every > 1 and k there is K and for every sequence (gn ) which satisfies these hypotheses there is i0 so that for all i ≥ i0 , |I i+1 | ≤ K. dist(I i+1 , ∂I i ) For a proof see [15] for maps with negative Schwarzian derivative, and [16] for C 3 maps. Here is a notion of bounded real geometry. Definition 1.3. Let g ∈ G 1 with I 0 ∩ I 1 = ∅. After subtracting from the range I the endpoints of I0 and I1 , one gets an open set with five connected components. Let µ(g) denote the maximum of the five ratios which feature the length of I in the numerator and the length of one of these five components in the denominator. 1.4. Limits of smooth maps under renormalization. Let us begin by defining a distance in G 1 . Definition 1.4. Let g, gˆ ∈ G 1 , set up so that g : I0 ∪ I1 → I and gˆ : Iˆ0 ∪ Iˆ1 → Iˆ The distance between g and gˆ is going to the maximum of the following: • the distance between the critical points, • the Hausdorff metric distances between the closures I and Iˆ, I0 and Iˆ0 , I1 and Iˆ1 , • the supremum of |g(x) − g(x)| ˆ on the intersection of both domains: I0 ∪ I1 , Iˆ0 ∪ Iˆ1 .


376

The Epstein class. If I is an open segment of the real line, we will write CI for the doubly-slit plane, CI := (C \ R) ∪ I . Recall that a real-analytic diffeomorphism h onto its image J belong to the Epstein class provided that h−1 continues analytically to a univalent map defined on CJ . Definition 1.5. For > 1 which is an odd integer, we define the Epstein class GE as the subset of G of maps for which the off-critical branch g|I1 as well as the map H from the decomposition of the critical branch specified in Definition 1.1 are diffeomorphisms which belong to the Epstein class. See [27] and [16] for an overview of the Epstein class. We write GE =

E G2n+1 .

n≥1

There is a natural concept of convergence in GE . Definition 1.6. Let (gi ) be a sequence of maps which all belong to some GE . We say that gi converge to some g ∈ GE if and only if: • the critical points ζi of gi converge to the critical point ζ of g, • the closures of ranges I i converge to I in the Hausdorff metric, • the inverses of the off-critical branches gi|I i and of critical decompositions Hi (see 1 Definition 1.1) for maps gi converge to the inverses of corresponding maps for g, uniformly on compact subsets of CI . Extendibility and normality. If g : I0 ∪ I1 → I belongs to GE for some , then g : I0 ∪ I1 → I also a member of GE is called an extension of g if I ⊃ I , I0 ⊃ I0 , I1 ⊃ I1 and g is an analytic continuation of g. Definition 1.7. A map g ∈ G E with range I is called !-extendible if it has an extension with range I and dist(I, ∂I ) ≥ !|I |. Lemma 1.1. Let gn : I0n ∪ I1n → I n be a sequence of maps from GE for some fixed . Suppose all In are contained in some bounded set; for some fixed δ > 0 we have |In | > δ for every n and all gn are !-extendible with a fixed positive !. Let us also suppose that µ(gn ) ≤ K for some fixed K and every n, see Definition 1.3. Then the family (gn ) is normal in GE , every limit G remains !-extendible and µ(G) ≤ K. Proof. Without loss of generality ζn → ζ , In → I and In → I , where In are extension ranges. We also have dist(I, ∂I ) ≥ !|I |. If we pick any interval J positioned so that I ⊂ J ⊂ J ⊂ I , then the Epstein class inverses of the off-critical branches and critical decompositions form normal families in CJ . Moreover, the limits cannot be constant since images of the intervals In have length bounded away from 0 by δ/K.


377

Limits in the Epstein class. The following fact establishes the importance of the Epstein class. Proposition 1. Let g ∈ G3 , where is an odd integer greater than 1. Consider the sequence (gn )∞ i=1 in which g1 = g, for every n ≥ 1, the map gn has a combinatorial type bounded by some fixed κ, no more than κ close returns may occur in a row, and gn+1 is derived from gn by an inducing step. Let g˜ n mean gn conjugated by an affine map in such a way that the critical point of g˜ n is at 0 and the length of its range is 1. Suppose that for a fixed K and every n, the real bound µ(gn ) ≤ K holds (see Definition 1.3). For every g like that there is ! > 0 and the family (g) ˜ n is normal with the convergence specified by Definition 1.4; moreover the limit G of every convergent subsequence belongs to GE , is !-extendible and µ(G) ≤ K. Proposition 1 is proven in Sect. 2.2. The hypothesis about µ(gn ) ≤ K can be verified using Proposition 2 in Sect. 2.1. Convergence of renormalization. Definition 1.8. If g ∈ G 1 , then define the normalizing map Lg to be the unique affine transformation characterized by the conditions Lg (ζ ) = 0, Lg (I1 ) is to the right of 0, |Lg (I )| = 1. Theorem 1. For every positive integer κ there exists 0 as follows. Let g, gˆ ∈ G3 for some odd integer > 0 , with critical points ζ and ζˆ , respectively. Consider sequences n n ∞ n ˆn ˆn ˆn (gn )∞ n=1 , (gˆ n )n=1 , gn : I0 ∪I1 → I , gˆ n : I0 ∪ I1 → I , and assume that the following conditions are satisfied: g1 = g, gˆ 1 = g, ˆ for every n ≥ 1 maps gn , gˆ n have the same combinatorial type which is bounded by κ, no more κ close returns can occur in a row, and gn+1 , gˆ n+1 are derived by an inducing step, see Sect. 1.3, from gn , gˆ n , respectively. Then, for every such sequence of maps g, g, ˆ the sequence of distances, in the sense −1 of Definition 1.4, between Lgn ◦ gn ◦ Lgn and Lgˆn ◦ gˆ n ◦ L−1 tends to 0. gˆ n Moreover, Lemma 2.5 allows one to determine 0 specifically in some combinatorial cases (comp. Definition 1.3).

2. Bounded Geometry 2.1. Real bounds. Our main goal is the following proposition: Proposition 2. Fix a positive integer k. Consider a sequence of mappings gi : I0i ∪I1i → I i from G3 . Assume that, for every i ≥ 0, gi has a combinatorial type which is bounded by k, gi+1 is derived from gi by an inducing step and that at most k close returns occur in a row. Then, for every k there is 0 > 1 and for every pair k, there is K0 , independent of g, such that if > 0 , then for every sequence gi as above lim sup µ(gi ) ≤ K0 . Moreover, Lemma 2.5 allows one to determine 0 specifically in some combinatorial cases (comp. Definition 1.3).


378

Initial conventions and remarks. For g ∈ G 1 the complement of the domain of g in I consists of three intervals. Dividing I by the lengths of these intervals we get three ratios. |I | |I | Let µ(g) ˜ denote the maximum of these ratios. We have µ(g) = max µ(g), ˜ , |I0 | |I1 | . The following is quite similar to Proposition 2, but the claim is weaker since only ratios µ˜ are considered and the hypothesis is weaker as well since > 1 suffices. Also, we will use notations of the type (K, L), K, L ⊂ R to denote the convex hull of K and L with both endpoints removed, (K, L] for the convex hull with only the right endpoint included, etc. Proposition 3. Consider a sequence of mappings gi : I0i ∪ I1i → I i from G3 . For every i ≥ 0 assume that gi has a combinatorial type which does not exceed some fixed k and gi+1 is derived from gi by an inducing step. In particular, this assumption implies that for every i ≥ 0, gi (ζ ) ∈ I0i ∪ I1i . Assume also that no more than k consecutive close returns can occur. ˜ i ) ≤ K1 . For every k, > 1 there is K1 , independent of g, so that lim sup µ(g Recall Fact 1.1 which introduces an a priori bound K which depends on k. We will use the following tool for bounding the distortion. Fact 2.1. Let f ∈ G 2 . For every M > 0 there is K ≥ 1 with the following property. Let J ⊂ I in the domain of f satisfy the following conditions for some n > 0: • f n has a diffeomorphic inverse branch φ defined on I , • intervals φ(J ), f (φ(J )), · · · , f n−1 (φ(J )) are pairwise disjoint, dist(J, ∂I ) • < M. |J | Then for any x, y ∈ J ,

|φ (x)| ≤ K. |φ (y)|

This is Lemma 2 in [10]. An immediate consequence of Fact 1.1 is that |Iqi |

dist(Iqi , ∂I i )

≤ K

for i ≥ 1 and q = 0, 1 where K depends on K, k and , which by Fact 2.1 implies bounded distortion. Here is another simple fact: Fact 2.2. Consider an interval (a, b), a < 0 < b and > 1. Let Q denote the map Q(x) := sgn(x)|x| . Then for every x the estimate Q (x)

|(a, b)| ≤ K2 |Q(a, b)|

holds, where K2 only depends on . Fact 2.2 follows by a direct elementary calculation.


379

An a priori bound on eccentricity. If an interval I is compactly contained in (a, b), let us consider the eccentricity |(a, I )| |(I, b)| , . e(I, (a, b)) := max |(I, b)| |(a, I )| Lemma 2.1. lim sup e(Iqi , I i ) ≤ K3 i

for q = 0, 1 and K3 which depends on K and . Proof. Suppose that gi shows a close return. Then e(I0i+1 , I i+1 ) ≤ Q e(I0i , I i ),

(2)

where Q depends on K, . Likewise, e(I1i+1 , I i+1 ) ≤ Qe(I1i , I i ).

(3)

To see the second estimate, we observe that by Fact 1.1 e(I1i+1 , I i+1 ) differs only by a constant from e(I˜, I i+1 ), where I˜ is the extension domain for the mapping onto I i . Then the estimate follows from Fact 2.2 and real bounds. If gi makes a non-close return of depth k, then e(I1i+1 , I i+1 ) ≤ Qe(I0i , I i ) by the same reasoning. Also,

e(I0i+1 , I i+1 ) ≤ Q1 e(I , I i ),

where I is the preimage of I0i by the k th iterate of gi|I i and Q1 depends on K and . 1

But e(I , I i ) ≤ Q2 e(I1i , I i ), where Q2 depends on K by the real bound, so e(I0i+1 , I i+1 ) ≤ Q e(I1i , I i ). Changing Q if needed, estimates (2) and (3) can be replaced with e(Iqi+1 , I i+1 ) ≤ Qe(Iqi , I i ).

For a sequence of 0 ≤ p ≤ k close returns ended with a non-close one, we get i+p+1 i+p+1 e(I0 ,I ) ≤ Q2 e(I1i , I i ) and i+p+1

e(I1

, I i+p+1 ) ≤ Q2 e(I0i , I i ).

Passing to logarithms of the eccentricities we get a linear recursion with matrix −1 0 1 0 with maximal eigenvalue

√1

< 1. Lemma 2.1 follows.

Lemma 2.1 and the real bound mean that each of the connected components of I i \[I0i , I1i ] is at least a definite fraction of I i , in the sense that lim inf of the pertinent ratios are bounded below by positive constants which depend on k, explicitly or through K.


380

Estimates on the middle gap.. Assume for definiteness that I0 is to the left of I1 . Our goal is to prove that lim sup

|(I0i , I1i )| Ii

is positive. First, consider r1 (gi ) =

|I0i |

|(I0i , I1i )|

.

Lemma 2.2. lim sup r1 (gi ) ≤ K4 , i

where K4 depends on K, k and . Proof. Suppose first that gi shows a close return. If r1 (gi ) < 1, clearly r1 (gi+1 ) ≤ Q1 , where Q1 depends on K. Otherwise, we can write r1 (gi+1 ) ≤ Q2 r1 (gi ) since the central branch can be factored into diffeomorphisms with bounded distortion on a singular part for which the needed distortion property is evident. Thus, if gi begins a sequence of p ≤ k close returns, for each 1 ≤ j ≤ p, we get r1 (gj ) ≤ max(Qk1 , Qk2 r1 (gi )). Now suppose that gi−1 makes a non-close return. Then I0i is contained in the preimage −1 by the central branch of I := gi−1 (I0 ) ∩ I1i−1 . In other words, I is the preimage of I0 by the diffeomorphic branch. With the usual convention that I0 < I1 , we see that between I0i−1 and I we can still find the preimage J by the diffeomorphic branch of (∂l I i−1 , I0−1 ). For large i, that last interval has length comparable to |I i−1 |, in a way made precise by Lemma 2.1. By the usual factorization and extendibility of the diffeomorphic branch, we see that |I |/|J | ≤ Q3 , where Q3 depends on K and . But now applying the critical branch to the picture and estimating the distortion in a way analogous to the close return case, we see that r1 (gi ) ≤ max(Q1 , Q2 Q3 ) which only depends on K and . The lemma follows.

Again assume I0 < I1 (I1 to the right of I0 ) and introduce the notation r2 (gi ) = |I1i | . |(I0i , I1i )| Lemma 2.3. Let > 1. Then lim sup r2 (gi ) ≤ K5 , i

where K5 depends on K, k and .


381

Proof. Estimating the distortion ass in the proof of Lemma 2.2, we get in the case when gi makes a close return: r2 (gi+1 ) ≤ Q1 r2 (gi ). For a non-close return, r2 (gi+1 ) ≤ Q1 r1 (gi ). The claim follows from Lemma 2.2.

Finally, we introduce ratios m0 (gi ) =

|(∂l I i , I0i )| |[I0i , I1i )|

and

m1 (g1 ) =

|(I1i , ∂r I i )|)| |(I0i , I1i ]|

.

Lemma 2.4. Let > 1 and assume for definiteness that I0 < I1 . Then

lim sup i

|I i | |(I0i , I1i )|

≤ K6 ,

where K6 depends on K, k, . Proof. In the light of previously proven lemmas, it suffices to prove that the upper limits of mq (gi ), q = 0, 1 are similarly bounded. Note that both ratios are asymptotically bounded below by positive constants, by Lemma 2.1. Let i be large enough so that the asymptotic relations of the previous lemmas hold. Suppose that gi shows a close return. Then m0 (gi+1 ) ≤ Q m0 (gi ) m1 (gi+1 ) ≤ Q m1 (gi ).

and

In the case of a non-close return, we similarly get m0 (gi+1 ) ≤ Q m1 (gi ) m1 (gi+1 ) ≤ Q m0 (gi ).

and

A very simple linear recursion develops for log mq (gi ) which proves the needed bounds. Lemmas 2.1 and 2.4 imply Proposition 3.


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The sizes of domains. For g ∈ G 1 define a pair of ratios ρq (g) := |(I|I0 ,Iq |1 )| with q = 0, 1. In writing this definition, we adopted the usual convention that I0 < I1 (I0 to the left of I1 ). Lemma 2.5. Consider a sequence of mappings gi : I0i ∪ I1i → I i from G3 . For every i ≥ 0 assume that gi+1 is derived from gi by an inducing step. In particular, this assumption implies that for every i ≥ 0, gi (ζ ) ∈ I0i ∪ I1i . Suppose that for every i the combinatorial type of gi is defined and bounded by k. Assume also that no more than k close returns can occur in a row. For every k there is 0 > 1 and if > 0 , then lim sup ρq (gi ) ≤ K6 for q = 0, 1, with K6 which depends only on k and , but not the sequence (gi ). For any rotation-like combinatorics, i.e. one which consists of blocks of a fixed number p ≥ 0 of close returns interspersed with single non-close returns of depth 1, we get 0 = 2. Proof. Note first of all that the hypothesis of Proposition 3 is fulfilled in the current setting. We fix some > 1 and assume that i is chosen large enough so that the asymptotic relations developed in Proposition 3 hold. Also, constants Qi are meant to depend only on k, . We observe based on Proposition 3 that ρq (gi ) are asymptotically bounded below by K1−1 . From Facts 1.1 and 2.1 we know that diffeomorphic branches have bounded distortion. Let us first consider the case when gi makes a close return. Then ρ0 (gi+1 ) ≤ Q1 ρ0 (gi ). By the lower bound on ρ1 (gi ), using bounded distortion and Fact 2.2 we get ρ1 (gi+1 ) ≤ Q2

|(I0i , I )| , |I |

where I is the preimage of I0i by the diffeomorphic branch. Using bounded distortion and Proposition 3, we obtain |(I0i , I )| ≤ Q3 ρ0 (gi )ρ1 (gi ) |I | which leads to ρ1 (gi+1 ) ≤ Q4 ρ0 (gi )ρ1 (gi ). Similarly approaching the case when gi makes a non-close return with depth ki , we obtain ρ1 (gi+1 ) ≤ Q5 ρ0 (gi ), ρ0 (gi+1 ) ≤ Q6 ρ0 (gi )ρ1ki (gi ). The ki th power in the second relation comes from the fact that we need to consider the ki th preimage of I0i by the diffeomorphic branch gi|I i . 1

Introducing vector Ri = (log ρ0 (gi ), log ρ1 (gi ))T , we get two linear recursion relations: −1 0 Ri + Q Ri+1 ≤ 1 1


if gi makes a close return and Ri+1

383

−1 ki −1 Ri + Q ≤ 1 0

in the case of a non-close return. The inequality between vectors is meant to be component-wise and Q = (Q5 , Q6 )T . Define a subsequence gij by the condition that gij −1 runs over the set of exactly those maps showing a non-close return. Then Rij +1 ≤ Mj Rij with

−1 pj −1 0 qj −1 , Mj := 1 1 1 0

where qj := kij +1 −1 ≥ 1, while pj := ij +1 − ij − 1 ≥ 0 is the number of close returns which occur. By the hypothesis of Lemma 2.5 both parameters are bounded above by k. Calculating further, −1 pj −pj

−1 −1 0 0 qj −1 qj −1 −pj Mj = = 1− 1 1 1 0 1 0 1 1−−1

−pj −pj −1 qj −(qj −1) − qj −1 . −1 = −p j 0 In the case of rotation-like combinatorics characterized by qj = 1 and pj = p for all j , we get −p−1 1− −1 . −1 Mj = −p 0 Matrix Mj has spectral radius less than 1 provided that Tr Mj − det Mj < 1. For every p that is satisfied, provided that > 2. In the general case, we have to look for products of perhaps different matrices Mj . To handle them, let us estimate the entries of Mj from above: k k−1 −1 . Mj ≤ N := 1 0 We replaced qj by its upper estimate k and for pj we set 0 or ∞ whichever was going to make the entry of Mj bigger. Matrix N has spectral radius less than 1 provided that 1 1 + < k −1 . −1 If 0 is chosen as the root of the corresponding equation and > 0 , then the coefficients of Mj · · · Mj +m decay exponentially with m, and thus the sequence Rij , hence the entire Ri is bounded. Proof of Proposition 2. Proposition 2 follows directly from Proposition 3 and Lemma 2.5.


384

2.2. Limits in the Epstein class. In this section, we give a proof of Proposition 1. Let us adopt the notations and hypotheses of that proposition. The proof we give is of the same type as the argument in [19]. Observe first that it will be sufficient to prove the normality of the sequence and the fact that every limit G is in the Epstein class. What remains is then showing that µ(G) ≤ K and the extendibility. The bound on µ(G) follows directly from the hypothesis of Proposition 1, since µ is a continuous function on G 1 with the distance given |I0 | by Definition 1.4. In addition, we observe that the real bound ≤ K1 holds, dist(I0 , ∂I ) with G : I0 ∪ I1 → I , where K1 is independent of the limit G. This follows from Fact 1.1. To prove the uniform extendibility of G claimed in Proposition 2, suppose it is the limit of a subsequence g˜ nj . Consider G which is the limit for a further subsequence of g˜ nj −k . All the same assertions hold for G and k inducing steps applied to G yield an affine re-scaling of G. But now the extendibility of that re-scaling follows from the real bound for G . Existence of C 0 limits. It is quite easy the see that every subsequence of g˜ n has a C 0 limit in the following sense. We first choose the subsequence so that the closures of intervals I n , I0n , I1n converge in the Hausdorff metric to their limits I ∞ , I0∞ , I1∞ , respectively. By the real bound from Fact 1.1 and the distortion estimate of Fact 2.1, on every compact subinterval of I0∞ ∪ I1∞ the family is equicontinuous and hence one can take a subsequence converging to C 0 . By a diagonal argument we obtain a subsequence which converges on I0∞ ∪ I1∞ almost uniformly. We also observe that these limits are strictly monotone, again by the bounded distortion and the bound on µ(gn ). We may also assume that the limit restricted to I0∞ factors as in Definition 1.1 with H which is only continuous. The issue is now in showing that H and the off-critical branch of the limit belong to the Epstein class. The postcritical set. Lemma 2.6. Let g ∈ G 3 . Suppose that the sequence (gn )∞ n=1 is constructed starting with g1 = g by deriving each gn+1 from gn by the inducing step. Suppose that every gn has a combinatorial type bounded by κ and that no more than κ close returns may occur in a row. Then the Lebesgue measure of the set {f i (ζ ) : i > 0} is zero. Proof. The proof follows an analogous argument in [21]. Suppose that ω(ζ ) has positive measure and let x be its point of density, other than ζ . Since ω(ζ ) is a minimal set, for every n > 1 there is an interval Un x which is the domain of the first entry map into I0n , that is some iterate φn of g maps Un onto I0n and extends as a diffeomorphism onto the larger range I n . Note that the lengths of Un go to 0, or otherwise there would be a wandering interval. By Fact 1.1 the distortion of φn is bounded on Un , uniformly with respect to n. But ω(ζ ) ∩ I0n ⊂ I0n+1 ∪ I1n+1 . By Proposition 3, |I0n+1 ∪ I1n+1 | ≤η 0 is the restriction of that fixed diffeomorphism to some interval in the form g i (Iqn ) with i ≤ Qnq . Hence, by the conclusion to Lemma 2.6, Q i=0

sup{| log |hi (y)| − log |hi (y )|| : y, y in the domain of hi } → 0

(4)

with increasing n. In the Poincaré model, hP = (hQ )P ◦ (χQ )P ◦ · · · ◦ (h1 )P = HQ ◦ · · · H1 ◦ (χQ )P ◦ · · · (χ1 )P with −1 Hi = (χQ )P ◦ · · · ◦ (χi )P ◦ (hi )P ◦ (χi )−1 P ◦ · · · ◦ (χQ )P .

Since (χi )P are contractions, sup{|Hi (x) − x| : x ∈ R} ≤ sup{|(hi )P (x) − x| : x ∈ R}. In conjunction with (4), sup{|HQ ◦ · · · ◦ H1 (x) − x| : x ∈ R} → 0 as n tends to ∞. Returning from the Poincaré model, we see that h−1 = h˜ ◦ χ˜ Q ◦ · · · ◦ χ˜ 1 , where !h˜ − id ! → 0 and each χ˜ i is equal to Ai ◦ χi ◦ Ai , where Ai , Ai are affine. In particular, the whole composition has a univalent extension to CI , where I is the image of h.


386

Conclusion. Let us take a C 0 -convergent subsequence gnj and concentrate on the offcritical branches. By the preceding discussion, the inverse of γnj of each of them can be written as h˜ j ◦ λj , where h˜ j tend to the identity and the λj is the inverse of an Epstein diffeomorphism. We have limj →∞ γnj = lim j → ∞λj on the real domain. But limj →∞ λj extends to the appropriate doubly slit plane and hence limj →∞ gnj is an Epstein diffeomorphism. The same argument works if γnj is the inverse of the diffeomorphic component of the decomposition of the critical branch of gnj given by Definition 1.1. 3. Polynomial-Like Extensions If g ∈ G E , then its critical branch gI0 has an analytic continuation as a branched cover of CI , and the off-critical branch g|I1 can be continued as an univalent map onto CI . Let us denote these continuations G0 and G1 , respectively. Definition 3.1. A polynomial-like extension of g ∈ G E consists in choosing analytic continuations G0 , G1 of the branches g0 , g1 , respectively, and bounded open topological disks >, >0 , >1 subject to the following requirements: • all three disks are symmetric with respect to the real line, > ∩ R = I and >q ∩ R = Iq for q = 0, 1, • >1 ∩ >0 = ∅ while >q ⊂ > for q = 0, 1, • Gq is a proper holomorphic map of >q onto > for q = 0, 1. Commonly, we will specify a polynomial-like extension by providing a map φ : >0 ∪ >1 → >, where φ acts by G0 on >0 and by G1 on >1 . Definition 3.2. A polynomial-like extension φ of a g ∈ G E has a geometric bound C provided that: • µ(g) ≤ C −1 (comp. Definition 1.3), • >, >0 , >1 are C −1 -quasiconformal disks, • annuli > \ >q , q = 0, 1, have modulus at least C. We say that φ has complex bound C if the third condition only is satisfied. Proposition 4. Fix a positive integer k, an odd integer > 1, and a positive number K. Then there exist i0 > 0 and C > 0 as follows. Consider a sequence of mappings gi : I0i ∪ I1i → I i from GE . For every i ≥ 0 assume that gi has a combinatorial type which is bounded by k, and that at most k close returns can occur in a row. For every i, gi+1 is derived from gi by an inducing step. For every i ≥ 0, the following a priori real bound holds: |I i+1 | ≤ K. dist(I i+1 , ∂I i ) For every sequence (gi ) which satisfies these hypotheses, for every i ≥ i0 , the map gi has a polynomial-like extension with the complex bound C (see Definitions 3.1 and 3.2.)


387

This statement and Propositions 1–2 will imply Proposition 5. Fix a positive integer k and an odd integer > 0 (k), where 0 (k) comes from Proposition 2. Then there exists C > 0 as follows. Consider a sequence of mappings gi : I0i ∪ I1i → I i from G . For every i ≥ 0 assume that gi has a combinatorial type bounded by k, at most k close returns can occur in a row, and gi+1 is derived from gi by an inducing step. Let G be the limit of a convergent subsequence of the sequence of normalized maps g˜ i of gi (see Definition 1.5 and Proposition 1). Then the map G belongs to the Epstein class and has a polynomial-like extension with geometric bound C (see Definitions 3.1 and 3.2.) The rest of the section is devoted to the proof of Propositions 4 and 5. Proof of Proposition 4. It follows basically the proof of a similar statement (Theorem A) of [16], where the existence of quasi-polynomial-like mappings is shown with complex bounds for a subsequence of the indexes i: there is a subsequence ik → ∞ such that each gik has an extension as in Proposition 4, but without the second assumption of Definition 3.1, i.e. >i0 , >i1 can overlap. Here we prove the existence of polynomial-like maps, without overlaps, with complex bounds and for the whole sequence of indexes i ≥ i0 . We split the proof into steps emphasizing the differences with the corresponding statements of [16]. There is also a new idea which will allow us to construct polynomiallike maps with complex bounds from the quasi-polynomial-like maps. Step 1. It is enough to show the existence of polynomial-like maps with complex bounds for a subsequence in , such that i0 < N and in+1 − in ≤ N , for a fixed N , such that N depends only on the combinatorics k, the degree , and the real bound K. Indeed, for intermediate indices we can form the ranges >i by inducing. The complex bounds will undergo a change by each inducing step which is bounded in terms of and k. We determine N like this in the course of the proof. Step 2. Notations ∂l and ∂r refer to the left and right endpoint of an interval, respectively. Recall that ζ stands for the critical point of g. Lemma 3.1. Assume that there exist C > 0, and !0 , such that if for some ! ∈ (0, !0 ) and i ≥ 0 the following holds: max(|∂l I i+1 − ζ |, |∂r I i+1 − ζ |) < (1 + !) · dist(∂I i , ζ ).

(5)

Then the statement of the proposition holds, where one can take >i = D∗ (I i ) (round disc with the diameter I i ). Proof. We repeat the proof of Lemma 3.1 (case I) of [16]. Consider the component of the domain of the first return map to >i = D∗ (I i ) which intersects g(I0i ) (recall that I0i = I i+1 ). By the Schwarz Lemma and because g is in Epstein class this component is inside D∗ (g(I0i )). If ! would be equal to zero, then because in this case I0i is well inside I i = I0i−1 (the real bound), g −1 (D∗ (g(I0i )) is compactly contained in D∗ (I i ). Provided ! ∈ (0, !0 ) and we choose !0 sufficiently small, the same is still true provided (5) holds. Note that the modulus of D∗ (I i ) \ g −1 (D∗ (g(I0i )) is bounded from below by a positive constant which depends on and K only. Again by Schwarz the non-central domain is mapped univalently inside D∗ (I i ). Note that the central >i0 and non-central >i1 components are automatically disjoint in this case because >i = D∗ (I i ) and, hence, the images g(>i0 ), g(>i1 ) are contained in round discs based on disjoint intervals as on their diameters.


388

Let us fix ! ∈ (0, !0 ), and denote by jn all the indexes i such that the condition (5) is satisfied. It follows from Step 1 and the lemma above that: If there exist N > 0 (which depends on k, , and K) such that j0 ≤ N and jn+1 − jn ≤ N for every n, then Proposition 5 follows. Step 3. Either |∂l I i+1 − ζ | ≥ (1 + !)|∂r I i − ζ | |∂r I

i+1

i

− ζ | ≥ (1 + !)|∂l I − ζ |

for all jn < i < jn+1 , for all jn < i < jn+1 .

or

(6) (7)

This follows from the inclusion I i+1 ⊂ I i , i ≥ 0. In the sequel, we will say that jn is a “right” (resp. “left”) index if (7) (resp. (6) holds. Step 4. Roughly speaking, we will show that if jn+1 − jn is sufficiently large, then for many i, jn < i < jn+1 the claim of Proposition 5 holds. Since the orientation can be changed by applying a linear reflection, without loss of generality jn is a right index. >From the description of the inducing step in Sect. 1.3, we see that g1 (ζ ) and g(ζ ) are on the same side of ζ precisely if a close return occurs. Indeed, if g makes a close return, then g1 (ζ ) = g(ζ ). On the other hand, if the return is not close, then g(ζ ) and the non-central component J of g1 are on the opposite sides from ζ , hence, g1 (ζ ) is on the same side from ζ as J is (otherwise the iterates of ζ by g1 escape). >From the combinatorics allowed in Proposition 5, the sign of gi (ζ ) − ζ stays the same for no more than k consecutive indices i. Let im , m = 1, · · · , m0 , be the subsequence of all i, jn < i < jn+1 , for which gi shows a non-close return and gi (ζ ) < ζ . If we throw in i0 = jn and im0 +1 = jn+1 , then im+1 − im < 2(k + 1) ≤ 4k for each m = 0, · · · , m0 . Step 5. If T = (a, d) and J = [b, c] ⊂ T , then define Poin(T , J ) = Poin(a, b, c, d) :=

(c − b)(d − a) . (b − a)(d − c)

Let us fix i = im . Then I i+1 ⊂ I i ⊂ · · · ⊂ I i−k(i) , where k(i) ≥ 0 is minimal so that gi−k(i)−1 has again a non-close return. The sequence k(i) is uniformly bounded (by k). Following [16], define V = Vim +1 = [∂l I im −k(im ) , ∂r I im +1 ]. Let also Iˆ = Iîm +2 be an interval containing ζ with ∂r Iîm +2 = ∂r I im +2 and so that an extension of the central branch of gim +1 maps Iˆ onto V homeomorphically. We have Poin(g(V ), g(Iˆ) ≤ Poin((∂l I im−1 , ∂r I im ), gim (Iˆ)).

(8)

This is precisely the inequality of Lemma 3.4 in [16] (in our notations). Step 6. Here we prove a refinement of Lemma 3.5 of [16]: Lemma 3.2. There exist δ > 0 and N1 , so that δ and N1 depend only on k, , and K, such , m = 1, 2, ..., m , that, for every pair jn , jn+1 with jn+1 − jn > 3N1 there is a subset im < N , and of the set of indexes im , such that im+1 − im < N1 , i1 − jn < N1 , jn+1 − im 1 for every i = im , Poin(g(V ), g(Iˆ)) ≤ (1 + δ)−1 , where V = Vi +1 , Iˆ = Iîm +2 . m


389

Proof. By estimate (8) it suffices to estimate from below the numbers Am := Poin−1 [∂l I im−1 , ∂r I im ]), gim (Iîm +2 ) .

(9)

Moreover, since gim (Iˆ) ⊂ l(I im \ I im +1 ), where l(T \ J ) denotes the left component of T \ J , the left hand side in the previous inequality is bounded from below by Poin−1 [∂l I im−1 , ∂r I im ]), l(I im \ I im +1 ) . (10) Normalize so that ζ = 0 and write I i = [−xi , yi ], where by our assumption that jn is a right index, we have 0 < (1 + !)xi < yi+1 < yi for jn < i < jn+1 . Then (x im−1 − xim )(yim + xim +1 ) Poin−1 [∂l I im−1 , ∂r I im ]), l(Iim \ Iim +1 ) = (yim + xim−1 )(xim − xim +1 )

xim−1 − xim (1 + !)xim + xim +1

≥ (1 + !)xim + xim−1 xim − xim +1

λim − 1 (1 + !)λim + 1

= , λim − 1 (1 + !) + λim (11) where we denote

λim = xim /xim +1 and λim = xim−1 /xim .

Therefore,

λim − 1

(1 + !)λim + 1

Am ≥ . λim − 1 (1 + !) + λim

(12)

λim = xim−1 /xim ≥ xim−1 /xim−1 +1 = λim−1 .

(13)

Observe that

Assume first that for some fixed m and some r > 0, λim ≤ (1 + r)λim . Denote t = λim . Then the expression in (12) is bounded from below by the function A(r, t) =

(t − 1) ((1 + !)(1 + r)t + 1) . ((1 + !) + t) ((1 + r)t − 1)

(14)

It is easy to see that it is increasing in t > 1. The real bound implies that λim ≥ λim −1 ≥ K := 1 + 1/K > 1 for all m. Hence, one can assume that t ≥ K and A(r, t) ≥ A(r, K ). On the other hand, A(r, K ) → 1 + 2δ as r → 0, where we denote δ = 0.5!(K − 1)/(1 + ! + K ) > 0. It follows that there exists r > 0 dependent only on the real bound and ! such that the expression in (12) is at least 1 + δ, provided λim ≤ (1 + r)λim for a given m.


390

Fix such r > 0. Assume now that the inequality λim ≥ (1 + r)λim holds for all m

≥ (1 + r)N −1 . between some M and M + N . It implies λiM+N ≥ (1 + r)N , λi M+N But then provided N is chosen large enough but fixed (dependent only on r and !) the expression in (12) would be at least 1 + !/2, so that δ = !/2 in this case. Therefore, the statement holds with N1 = 4kN . , so that the conclusion of the lemma from the Step 7. Let us fix an index i = im previous step holds. Then we can apply Lemma 3.6 of [16]: the map gi+1 extends to a quasi-polynomial-like map G : >0 ∪>1 → >, where the range > is defined as the round disc D∗ (V ) (where V = Vim +1 ) with the interval V \ I i+1 deleted. Here >0 is proper inside > so that the modulus of the annulus > \ >0 is bigger than a universal positive constant C0 (i.e. C depends on and the real bound K only) while >1 is contained inside >, but intersects >0 (in the complex plane). To obtain from this “quasi-map” a polynomial-like map with definite complex bounds, we proceed as follows. We consider the first return map G of the map G to the central component >0 of G restricted on the real line to two components I0i+2 , I1i+2 of the map gi+2 (so G on the real line is just gi+2 ). Then we claim that this new map will be a polynomial-like map with universal complex bounds. It is enough to show that two components >l and >r of the map G are disjoint, where the notations mean that the intersection of >l with the real line lies to the left from the intersection of >r with the real line. Note that as for the map G, the trace I0i+1 of the central domain >0 on the real line lies to the left from the trace I1i+1 of the non-central domain >1 on the real line. Assume the contrary: >l and >r do overlap. Note that

g nl (>l ) = g nr (>r ) = >0 , g n0 (>0 ) = g n1 (>1 ) = >, for appropriate iterates of the original map g. Observe that by the definition of the inducing step, we have: nl < nr , and n1 < n0 . Therefore, g nl (>l ) = >0 while g nl (>r ) ⊂ >1 ), hence, from the assumption, >0 intersects g nl (>r ). Now apply g n1 . Then g n1 ◦ g nl (>r ) = >0 while g n1 ◦ g nl (>l ) = g n1 (>0 ) is a preimage of > in the chain of preimages by g between > and >0 , and >0 , g n1 (>0 ) still intersect each other. Apply g one more time. Now we have that g(>0 ) is contained in a round disc with some diameter J1 and g(g n1 (>0 )) is contained in a disc with diameter J2 . On the other hand, J1 , J2 are disjoint, a contradiction with the assumption that >l and >r are not disjoint. Step 8. We conclude the proof of Proposition 4. As we have proved in the preceding steps, there is a universal N∗ , such that, for every j , one of the maps gj , gj +1 , ..., gj +N∗ has a polynomial-like extension with universal complex bound. By Step 1, it is enough to end the proof. Proof of Proposition 5. Let g˜ be a limit of a subsequence of g˜ i . Since > 0 (k), there is a universal K, such that µ(g) ˜ < K, in particular, a universal real a priori bound holds for g. ˜ Then we find i0 and C as in Proposition 4. Consider now the map G0 = G, which is the limit of a convergent subsequence g˜ i(j ) . Passing to subsequences, one can assume also that the sequences g˜ i(j )−h converge, for every h = 0, 1, 2, ...i0 . Let G−h : I0−h ∪ I1−h → I −h be corresponding limits. By the construction, G−h+1 is derived from G−h (up to re-scaling) by an inducing step, h = 1, ...i0 . Therefore, by


391

Proposition 4, the map G = G0 has a polynomial-like extension φ : >0 ∪ >1 → > with the complex bound. Note however that > is not necessarily a quasi-disk. Indeed, > is obtained by at most N number of times by inducing a polynomial-like extension φh : >h0 ∪>h1 → >h of G−h , where h ≤ N , and the range >h is either a round disk based on I −h as on diameter, or a round disk based on [∂l I −h−r , ∂r I −h ] as on diameter but with the slit [∂l I −h−r , ∂r I −h ] \ I −h . Here r is universally bounded, and the right and the left sides could be reversed. On the other hand, since the geometry is uniformly bounded, and r is bounded as well, the critical value of G−h is not too close as well as not too far, from the boundary of >h . Therefore, one can replace >h by a quasi-disk replacing the slit in >h by a fixed angle around the slit with the vertex at ∂l I −h . Since annuli >h \ >hq , q = 0, 1, has modulus at least C, if we fix the above angle narrow enough (dependent only on the bounded geometry and C), we still obtain a polynomial-like extension of G−h with geometric bound C/2. By the same bounded geometry property, after at most N inducing, we come to the desired polynomial-like extension of G with a universal geometric bound.

4. Towers The definitions and arguments in this section follow the template of [24] with modifications due mostly to the fact that the Julia set is totally disconnected.

4.1. Definition, compactness and construction. For the purposes of this paper, we define a tower as follows. E Definition 4.1. Consider a sequence (gn )M n=0 , possibly M = ∞, of mappings from G , all normalized so that ζ = 0. Suppose each of them has a combinatorial type bounded by κ and that gi−1 is derived from gi by an inducing step, see Sect. 1.3. Suppose that polynomial-like extensions φn : V0n ∪ V1n → U n are also given. If all φn , n = 0, · · · , M have geometric bound C > 0 (see Definition 3.2), then we say that this pair of sequences of maps forms a tower with combinatorial bound κ and geometric bound C.

The map g0 will be called the bottom map of the tower. Since the sequence φn : V0n ∪ V1n → U n defines a tower uniquely, we will often identify them. Lemma 4.1. Suppose that T = (φn )M n=1 is a tower with geometric bound C. For every C > 0 there is N so that for every such tower and every 0 ≤ n < n + N ≤ n ≤ M we n n have U ⊂ V0n with mod (V0n \ U ) > 1. Proof. Since the geometric bound in particular involves the bound on µ(gn ), the lengths of the real ranges U n ∩ R decline exponentially with n at a rate controlled from both sides by C. By the quasi-disk requirement of the geometric bound, for every n, D(0, C1−1 |U n ∩ R|) ⊂ Un ⊂ D(0, C1 |U n ∩ R|) for some C1 ≥ 1 depending on C.


392

Limiting towers. Definition 4.2. Suppose that gn is a sequence of maps from GE for a fixed . Let φn be a polynomial-like continuation of gn is the sense of Definition 3.1. Let g∗ ∈ GE as well, and φ∗ be its polynomial-like continuation. We will say that (φn ) converge to its limit φ∗ provided that • gn converge to g∗ in the sense of Definition 1.6, n n n ∗ ∗ • sequences of compact sets > , >0 , >1 converge in the Hausdorff metric to > , >0 ∗ and >1 , respectively. Since maps gn are determined by φn , we will often say simply that polynomial-like maps φn converge, with the understanding that the corresponding gn = φn|R are indeed in some G . j m(j )

Definition 4.3. Let Tj = {φn }n=1 , j > 0, be a sequence of (finite or infinite) towers so ∗ that, for each j , m(j ) ≤ ∞ and m∗ := lim inf m(j ). A tower T∗ = {φn∗ }m n=1 is described as a limiting tower of that sequence if, for every n > 0, there is a sequence {jk,n }k>0 , such that j

• the sequence φnk,n of maps of the level n of the corresponding towers Tjk,n converges in the sense of Definition 4.2 to a map φn∗ , • the sequence jk,n is a subsequence of the previous one jk,n−1 , j m(j )

j

n,j

n,j

Lemma 4.2. Consider a sequence of towers Tj = (φn )n=1 , φn : V0 ∪ V1 → U n,j , which have the same geometric and combinatorial bounds and such that 0 < C1 < diam U 0,j < C2 with C1 and C2 independent of j . Suppose also that for every n there j j is !n > 0 and maps gn := φn|R are !n -extendible for every j . Then Tj has limiting tower in the sense of Definition 4.3. The limiting tower has the same geometric and combinatorial bounds and the map on its nth level is !n -extendible. Proof. The construction of the limiting tower is inductive. Suppose that a subsequence j jk,n has already been chosen in such a way that the sequence φnk,n converges to φn∗ in the sense of Definition 4.2. Then jk,n+1 is chosen as a further subsequence so as to ensure jk,n+1 that φn+1 also converge in the same sense, see Lemma 1.1. The geometric bound and j

extendibility clearly remain the same. It is also true that for every n all φnk,n with k large m∗ enough, as well as the limiting map φn∗ have the same combinatorial type. So, (φn∗ )n=1 is a tower.

4.2. Expansion in towers. In this section, we work with some fixed infinite tower T = (φn )∞ n=0 with the geometric bound C and combinatorial bound κ. We will often refer to the “tower parameter” N obtained from Lemma 4.1. By our usual convention, φn : V0n ∪ V1n → U n . k Postcritical sets. Let Pn denote the closure ∞ of the set {φn (0) : k = 1, 2, · · · }. Then the postcritical set of the tower is P := n=0 Pn . We will use the following lemma, which in particular implies that the union of the postcritical sets of the elements of any tower is closed.


393

Lemma 4.3. If g ∈ G and g1 is derived from g by an inducing step, then the postcritical set of g1 is equal to the postcritical set of g intersected with the domain of g1 . By induction, for towers it means that that postcritical set of φn+j intersected with the real domain of φn is the same as the postcritical set of φn , in a sharp contrast to Julia sets. In particular, P ∩ U n = Pn , so P is closed. Hyperbolic metric. Let ρn denote the hyperbolic metric on U n \ P . As n tends to ∞, the metric elements dρn (z) decrease and tend to dρ(z), where ρ is the hyperbolic metric of C \ P . We will use notations |f (z)|ρn to mean the norm of the derivative with respect to ρn , similar for ρ instead of ρn . / P , then |φn (z)|ρn > 1 and |φn (z)|ρ ≥ 1. Lemma 4.4. If z ∈ V0n ∩ V1n \ P and φn (z) ∈ Proof. We will prove the first claim. Without loss of generality, n = 0. Suppose that p p z ∈ V0 , p = 0 or 1. Let ρ˜ denote the hyperbolic metric on V0 \ φ0−1 (P ). Then the norm of φ0 (z) acting from ρ˜ into ρ0 is 1, while the element of ρ˜ at z is greater than the element of ρn at z by the Schwarz lemma. Next, we show that |φ0 (z)|ρkN ≥ 1 for every k, where N is the parameter of the p tower T . This is true by the previous step, since φ0 restricted to V0 is an iterate of φkN . Now passing to the limit with k, we get the claim of the lemma. Let us state the following simple fact. Fact 4.1. Let X and Y be hyperbolic regions and Y ⊂ X and z ∈ Y . Let ρX and ρY be the hyperbolic metrics of X and Y , respectively. Suppose that the hyperbolic distance in X from z to X \ Y is no more than D. For every D there is λ0 > 1 so that |ι (z)|H ≤ λ10 , where ι : Y → X is the inclusion, and the derivative is taken with respect to the hyperbolic metrics in Y and X, respectively. Lemma 4.5. Suppose that z ∈ V0n ∪ V1n and φn (z) ∈ / V0n ∪ V1n . There is λ > 1, independent of n or z and depending on the parameters of T only, so that |φn (z)|ρ ≥ λ. Proof. Without loss of generality, we suppose that n = 0 and similarly to the proof of the previous lemma, and realize φ0|V p as an iterate of φm , where m = kN . We will show 0 (z)| that |φm ρm ≥ λ > 1, where λ only depends on the tower geometry. By Lemma 4.4, this implies that |φ0 (z)|ρm ≥ λ and the claim of the current lemma follows as we pass to the limit with m. Let us choose the units so that |U 0 ∩R| = 1. Since φ0 (z) ∈ / V00 ∪V10 , then dist(z, P ) ≥ C1 > 0, where C1 only depends on the geometric parameter of the tower. Moreover, one can construct a path from z to a point w ∈ φ0−1 (P ) \ P which is contained in U 0 , has bounded Euclidean length and keeps Euclidean distance C1 away from P . Then the element of the metric ρm , m = kN, k ≥ 1 is uniformly bounded along such a path, and hence the length of the path in ρm is also uniformly bounded. Using Fact 4.1 with −1 (P ), we see that |ι (z)| ≤ 1 , with λ > 1 which only X = Um \ P and Y = V0m \ φm 0 λ0 depends on the geometry of the tower. On the other hand, φm = φm ◦ ι and since φm is 1 (z)| a local isometry as a map from Y onto X, |φm ρm = |ι (z)| ≥ λ0 .


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A distortion bound. Let us start by quoting a general fact. Fact 4.2. Let X and Y be hyperbolic regions and Y ⊂ X and z1 , z2 ∈ Y . Let ρX and ρY be the hyperbolic metrics of X and Y , respectively. Suppose that the hyperbolic distance in Y from z1 to z2 is no more than D. Then for every D, there is β > 1 so that β

1 > |ι (z1 )|H ≥ |ι (z2 )|H ≥ |ι (z1 )|H , 1/β

where the subscript H indicates that the derivative is taken with respect to the hyperbolic metrics in the domain and range. See [23], Corollary 2.27 for a proof. Lemma 4.6. Let γ be a simple path in U m \ P , m ≥ 0 and suppose that the length of k , k ≥ 1, defined in a γ with respect to ρ is at most D. Let ψ be an inverse branch of φm neighborhood of γ and let x1 , x2 be the endpoints of γ . For every D there exists α > 1 so that for every γ , m, k, |ψ (x1 )|αρ ≤ |ψ (x2 )|ρ ≤ |ψ (x1 )|1/α ρ . Proof. Without loss of generality, m = 0. Choose n = pN , p > 0. Then φ0k on a j neighborhood of ψ(γ ) is φn , for some positive j . Let us consider W which is the j −j connected component of the domain of φn which contains ψ(γ ). Then Y := W \φn (P ). j Then J := φn acting from Y onto X := U n \ P is a local isometry with respect to the j pertinent hyperbolic metrics. Then J = φn ◦ ι, where ι is the inclusion of Y into X. In particular, for any x ∈ γ , j

|ψ (x)|ρn = |(φ )n (ψ(x))|−1 ρn = |ι (ψ(x))|H .

(15)

The hyperbolic distance in Y between ψ(x1 ) and ψ(x2 ) is no greater than the length of ψ(γ ), which is the same as the length of γ in the metric ρn . For n sufficiently big, this is less than 2D. Now we use Fact 4.2. What we get is that β

1 > |ι (ψ(x1 ))|H ≥ |ι (ψ(x2 ))|H ≥ |ι (ψ(x1 ))|H , 1/β

where β > 1 only depends on D. From there and estimate (15), we get |ψ (x1 )|βρn ≤ |ψ (x2 )|ρn ≤ |ψ (x1 )|1/β ρn . Allowing n to tend to ∞, we conclude the proof.

We get the following corollary. Lemma 4.7. Choose a point z in V00 ∪ V10 and suppose that for some n, j > 0, the point j j +1 z := φn (z) belongs to V0n ∪ V1n , but φn (z) ∈ / V0n ∪ V1n . Let γ be a simple path with an endpoint at z , contained in C \ P and whose length in ρ does not exceed D. Let ψ j be the inverse branch of φn which is defined in a neighborhood of γ and sends z to z. For every D there are K and λ > 1 otherwise only depending on the parameters of T so that the length of ψ(γ ) in ρ does not exceed Kλ−n .


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Proof. It is enough to prove the statement for n in the form kN with fixed N > 0 and j all k. In view of Lemma 4.6, it will be enough to show that |(φn ) (z)|ρ ≥ λn0 with λ0 > 1 depending only on the tower. We choose N to be an integer multiple of the tower parameter N , moreover, large enough so that for every n if w ∈ U n , then φn+N (w) ∈ V0n+N ∪ V1n+N . Let j0 + 1 j be the largest iterate of z by φ0 which is still well defined and denote z1 := φ00 (z). By Lemma 4.5, |φ0 (z1 )|ρ ≥ λ0 > 1. On the other hand, by our choice of N , z1 ∈ V0N 2 (z ) is well defined. Now we can repeat the argument (or use induction) with z and φN 1 replaced by φ0 (z1 ) and all indices in the tower shifted by N . Julia sets. Given a tower T and n ≥ 0, Jn is the filled-in Julia set of the polynomial-like map φn , i.e. the set of points which can be iterated by φn forever. We remark that the Julia sets are totally disconnected and Jn ⊂ Jn+1 for n ≥ 0, see [17] and [18] although those facts are not used in our proof. What we will use is the following fact: Proposition 6. The union of sets Jn is dense in C. It will be enough to prove that it is dense in V00 , since V0m exhaust C and for m > 0 the same argument applies. Let z ∈ V00 be given and without loss of generality z ∈ / Jk k k k k for any k. Observe now that for any k if w ∈ V0 ∪ V1 , but φk (w) ∈ / V0 ∪ V1 , then the distance from w to Jk in ρ is bounded by some D which only depends on the geometry j of the tower. Since z ∈ / Jk , for some j we can set w := φk (z). Then we consider a j simple path with length D which connects w to Jk its pull-back γk by φk as described in Lemma 4.7. The length of γk in ρ is bounded by Kλ−k , where K and λ depend only on D and the parameters of the tower. Since this tends to 0 as k increases, Proposition 6 follows. 4.3. Rigidity. ˆ ˆ ∞ Definition 4.4. Two towers T = (φn )∞ n=0 and T = (φn )n=0 are said to be conjugated by a homeomorphism H of the complex plane onto itself provided that for any n ≥ 0 the functional equation H (φn (z)) = φˆ n (H (z) is satisfied for all z ∈ (V n ∪ V n ) H −1 (Vˆ n ∪ Vˆ n ). 0

1

0

1

Depending on additional properties of H we will talk about the conjugacy being quasiconformal, linear, etc. ˆ ˆ ∞ Lemma 4.8. Suppose that towers T = (φn )∞ n=0 and T = (φn )n=0 are combinatorially equivalent, i.e. for every n ≥ 0 the combinatorial types of φn|R and φˆ n|R are the same, see Definition 1.2. Then T and Tˆ are quasiconformally conjugate. Proof. For any n, the postcritical sets of φn and φˆ n are mapped onto each other by a Kquasisymmetric homeomorphism of the line, with K bounded in terms of the geometric parameters of the towers. Then a quasiconformal conjugacy Hn between φn and φˆ n


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is obtained by the standard pull-back argument. This goes back to [28], see also [9] and [16]. This Hn is a conjugacy in the ordinary sense, in particular the domains and ranges correspond. Each Hn can be extended to the whole plane with the same maximal dilatation, independent of n. Note that if the range of φn is contained in the range of φn , n > n, then each branch ˆ And of φn is a restriction of some iterate of φn . The same is true with φ replaced by φ. so it follows that Hn also satisfies the functional equation Hn (φn (z)) = φˆ n (Hn (z)) on the domain of φn intersected with the preimage of the domain of φˆ n . Hence H = limn →∞ Hn is a conjugacy in the sense of Definition 4.4.

Theorem 2. Suppose that T = (φn )∞ n=1 is a tower for which all maps gn := φn|R are !extendible with a fixed positive !, see Definition 1.7. If T is quasiconformally conjugate to another tower Tˆ , then the conjugacy is necessarily linear. Invariant line field. Suppose that the towers T and Tˆ are conjugated by a quasiconformal map H . Then the measurable conformal structure H ∗ (dz) is invariant under the action of maps φn∗ acting from the sets, where H satisfies the conjugacy condition. However, we want it to be invariant for φn acting from its entire domain. To show that, we choose n = n+N (N can be chosen common for both towers). The reasoning is now similar to one used in the proof of Lemma 4.8. Let φn (z) = φnk (z). Then points z, φn (z), · · · , φnk−1 (z) belong to the domain of φn and similarly points H (z), φˆ n (H (z)), · · · , φˆ nk−1 (H (z)) belong to the domain of φˆ n . So, by the functional j ∗ equation H (dz) is invariant under all φn j = 0, · · · , k − 1 at the images of z, and so also invariant under φn which is the composition. Unless H is affine, this invariant measurable conformal structure is non-trivial, i.e. different from dz on a set of positive measure. Recall that a measurable complex structure |a(z)|b(z) a(z)dz+b(z)dz is usually identified with its invariant line field ν(z) := which a(z)|b(z)| ∗ is well-defined wherever b(z) "= 0. As the consequence of the invariance of H (dz) the form ν(z) dz dz is invariant. Invariant line fields cannot exist. We will now repeat the argument from Sect. 6.4 in [24], with changes that don’t go far beyond changing the notation. Throughout this section ν denotes the invariant line field for tower T . Definition 4.5. A line field ν(z) is called holomorphic on an open set U if every z0 ∈ U dz has an open neighborhood W ⊂ U and ν(z)|W dz = h∗ ( dz dz ) for some function h which is univalent on W . Lemma 4.9. Suppose that a line field given by ν is invariant under the action of all maps φn for some tower T . There is no open set U on which ν is holomorphic. Proof. Pick n so large that U is contained in the domain of φn and intersects Jn . This is possible by Proposition 6. If z ∈ U and φn (z) " = 0, then ν is also holomorphic in a neighborhood of φn (z). Thus, we can view U as a forward invariant set, except for the forward critical orbits which a priori may not belong to U . By the straightening theorem, see [5] and [17], the union of the images of U under φn covers U n . Thus, ν is


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holomorphic on Un \ P . But every point in P has a preimage by φn which is not in P , and so ν is holomorphic on Un . But that is not possible, since if ν is holomorphic near the critical value and invariant, then it is not holomorphic near the critical point. We will use the following fact: Fact 4.3. For every n ≥ 0, φn admits no invariant line field supported on its Julia set Jn . See [16]. Hence, ν would have to be supported on C \ ∞ n=0 Jn . Let z0 be a Lebesgue point of ν(z), in particular z0 ∈ / Jn for all n. Suppose that z0 ∈ V0n0 ∪ V1n0 , n0 ≥ 0. For every n ≥ 0 let Ln denote the linear map z → z · |Un ∩ R|. This allows dz to construct a collection of line fields νn dz := L∗n (ν dz dz ). We view νn as elements of ∞ 1 ∗ ∞ L (C) = (L ) (C). Since the L norms of νn are all 1, the compactness of the unit ball in the weak-* topology allows to pick a subsequence nj in such a way that νnj converge weak-* to an L∞ function ν∞ . Note that a priori ν∞ does not have norm 1 and does not correspond to a line field. p Next, for every j we have a point vj = φnjj (z0 ), where pj is chosen so that vj ∈ n n V0 j ∪ V1 j , but φnj (vj ) is no longer in the domain of φnj . Such pj exists since z0 ∈ / Jnj . −1 Then the sequence wj := Lnj (vj ) is bounded and by possibly replacing nj with a subsequence, we may assume that wj converges to w∞ . Now, we construct a sequence of re-scaled and shifted towers Tj . The tower Tj is the sequence of maps L−1 nj +k ◦ φnj +k ◦ Lnj +k for k = 0, 1, · · · . Note that the line field νnj is invariant under Tj for any j . Now let T∞ be a limiting tower for the sequence Tj . It exists by Lemma 4.2 and note that for any n if ψn ∈ T∞ , dz dz ψn∗ ν∞ = ν∞ . dz dz Observe that w∞ belongs to the closure of the domain of ψ0 , and if it actually belongs to the domain, then ψ0 (w∞ ) is not in the domain of ψ0 . Next, pick a Euclidean disk D(w∞ , r) of some hyperbolic diameter δ > 0 with respect to the hyperbolic metric on C \ P∞ , where P∞ is the postcritical set of the tower T∞ . Let D denote D(w∞ , r/2). Then Lnj (D(w0 , r)) for j large enough are contained in hyperbolic discs centered at vj with radius 2δ. This time, we are referring to the hyperbolic metric ρ of C \ P . Take the preimage Dj of Lnj (D) by the inverse branch ζj p of φnjj which sends vj to z0 . By Lemma 4.7, Dj is contained in the hyperbolic ball of radius Kδλ−nj , λ > 1, with all constants independent of j . Then the Euclidean diameter of Dj tends to 0 at least as fast. Since z0 was a Lebesgue point of ν and each Dj is the image of the Euclidean disk D with uniformly bounded distortion, ν|Dj − ν(z0 ) tends to 0 in the sense that its L1 norm on Dj divided by the measure of Dj converges to 0. Then (ζj ◦ Lnj )∗ (ν(z0 ) dz dz ) dz ∞ ∗ converge in the weak-* topology on L (D) to the same limit as (ζj ◦ Lnj ) (ν|Dj dz ), and that is ν∞|D by definition of ν∞ . If Lj denotes an affine map which fixes z0 and has derivative (diam Dj )−1 , then Lj ◦ ζ ◦ Lnj form a normal family of function on D;


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moreover, the limits are all univalent by the bounded distortion of ζj . Thus, ν∞

dz dz = lim (ζj ◦ Lnj )∗ ν(z0 ) dz j →∞ dz dz = lim (Lj ◦ ζj ◦ Lnj )∗ ν(z0 ) j →∞ dz dz , = F ∗ ν(z0 ) dz

where F is a limit function of the sequence Lj ◦ ζ ◦ Lnj . Hence, ν∞ is holomorphic on D in contradiction to Lemma 4.9. This contradiction ends the proof of Theorem 2. Proof of the convergence of renormalization. We will now derive Theorem 1 from Theorem 2 and Proposition 1. Adopt the hypotheses and notation of Theorem 1. In addition, write γn for Ln ◦gn ◦L−1 n and γˆn for Lˆ n ◦ gˆ n ◦ Lˆ −1 . n Rigidity of the limits. First, pick any subsequence nj in such a way that γnj converge to G ˆ We will prove that G = G. ˆ The idea is to put them on the bottom of a tower and γˆnj to G. ˆ1 = G ˆ and consider the subsequence nj − 1. and use Theorem 2. Thus, write G1 = G, G Then pick a subsequence of this in such a way that maps Lnj ◦ gnj −1 ◦ Lnj converge to ˆ 2 . This is possible by Proposition 1 and G1 , G ˆ1 G2 and Lˆ nj ◦ gˆ nj −1 ◦ Lˆ nj converge to G ˆ ˆ are derived from G2 , G2 , respectively, by an inducing step. Maps G1 and G1 have the same combinatorial type bounded by κ. In this way we construct sequences (Gn )∞ n=1 ˆ n )∞ . Since all these maps are in the Epstein class with uniform extendibility, and (G n=1 polynomial-like extensions exist by Proposition 5. This makes the sequences (Gn )∞ n=1 ˆ n )∞ into combinatorially conjugate towers, hence quasiconformally conjugate and (G n=1 by Lemma 4.8. By Theorem 2 these towers are linearly conjugate, and since the real ˆ were normalized to length 1, G = G. ˆ ranges of G, G Conclusion. Now suppose that Theorem 1 fails, which means that there is an infinite subsequence with the property that the distance from γnj to γˆnj remains at least δ > 0. By Proposition 1 without loss of generality these sequences converge, and we get a contradiction with the “rigidity of the limits” obtained in the preceding paragraph. Acknowledgement. We thank the referees for constructive comments.

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