Renormalization and 3-Manifolds which Fiber over the Circle

By Curtis T. McMullen

PRINCETON UNIVERSITY PRESS

Copyright © 1996 Princeton University Press
All rights reserved.
ISBN: 978-0-691-01153-0

Contents

1 Introduction, 1,
2 Rigidity of hyperbolic manifolds, 11,
3 Three-manifolds which fiber over the circle, 41,
4 Quadratic maps and renormalization, 75,
5 Towers, 95,
6 Rigidity of towers, 105,
7 Fixed points of renormalization, 119,
8 Asymptotic structure in the Julia set, 135,
9 Geometric limits in dynamics, 151,
10 Conclusion, 175,
Appendix A. Quasiconformal maps and flows, 183,
Appendix B. Visual extension, 205,
Bibliography, 241,
Index, 251,

CHAPTER 1

Introduction

In the late 1970s Thurston constructed hyperbolic metrics on most 3-manifolds which fiber over the circle. Around the same time, Feigenbaum discovered universal properties of period doubling, and offered an explanation in terms of renormalization. Recently Sullivan established the convergence of renormalization for real quadratic mappings.

In this work we present a parallel approach to renormalization and to the geometrization of 3-manifolds which fiber over the circle. This analogy extends the dictionary between rational maps and Kleinian groups; some of the new entries are included in Table 1.1.

Both discussions revolve around the construction of a nonlinear dynamical system which is conformally self-similar.

For 3-manifolds the dynamical system is a surface group acting conformally on the sphere via a representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given a homeomorphism ψ: S ->S, we seek a discrete faithful representation satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some α [member of] Aut([??]). Such a ρ is a fixed point for the action of ‘1/J on conjugacy classes of representations. This fixed point gives a hyperbolic structure on the 3-manifold

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which fibers over the circle with monodromy ψ. Indeed, the conformal automorphisms of the sphere prolong to isometries of hyperbolic space H3, and Tψ is homeomorphic to H3/Γ where Γ is the group generated by α and the image of ρ.

For renormalization the sought-after dynamical system is a degree two holomorphic branched covering F : U ->V between disks U [subset] V [subset] C, satisfying the Cvitanovic-Feigenbaum functional equation

Fp(z) = α-1F(αz)

for some α [member of] C*. The renormalization operator Rp replaces F by its pth iterate Fp, suitably restricted and rescaled, and F is a fixed point of this operator.

In many families of dynamical systems, such as the quadratic polynomials z2 + c, one sees cascades of bifurcations converging to a map f(z) = z2 + c∞ with the same combinatorics as a fixed point of renormalization. In Chapter 9 we will show that Rnp(f) converges exponentially fast to the fixed point F. Because of this convergence, quantitative features of F are reflected in f, and are therefore universal among all mappings with the same topology.

Harmonic analysis on hyperbolic 3-space plays a central role in demonstrating the attracting behavior of Rp and ψ at their fixed points, and more generally yields inflexibility results for hyperbolic 3-manifolds and holomorphic dynamical systems.

We now turn to a more detailed summary.

Hyperbolic manifolds. By Mostow rigidity, a closed hyperbolic 3-manifold is determined up to isometry by its homotopy type. An open manifold M = H3/Γ with injectivity radius bounded above and below in the convex core can generally be deformed. However, such an M is naturally bounded by a surface [partial derivative]M with a conformal structure, and the shape of [partial derivative]M determines M up to isometry.

In Chapter 2 we show these open manifolds with injectivity bounds, while not rigid, are inflexible: a change in the conformal structure on [partial derivative]M has an exponentially small effect on the geometry deep in the convex core (§2.4). This inflexibility is also manifest on the sphere at infinity [??]: a quasiconformal conjugacy from Γ = π1(M) to another Kleinian group Γ’ is differentiable at certain points in the limit set Λ. These deep points x [member of] Λ have the property that the limit set is nearly dense in small balls about x—more precisely, Λ comes with distance r1+ε of every point in B(x, r).

Chapter 3 presents a variant of Thurston’s construction of hyperbolic 3-manifolds that fiber over the circle. Let ψ : S ->S be a pseudo-Anosov homeomorphism of a closed surface of genus g ≥ 2. Then the mapping torus Tψ is hyperbolic. To construct the hyperbolic metric on Tψ, we use a two-step iterative process.

First, pick a pair of Riemann surfaces X and Y in the Teichmüller space of S. Construct the sequence of quasifuchsian manifolds Q(ψ-n(X), Y), ranging in a Bers slice of the representation space of π1(S). Let M = lim Q(ψ–n(X), Y). The Kleinian group representing π1 (M) is totally degenerate—its limit set is a dendrite.

For the second step, iterate the action of ψ on the space of representations of π1(S), starting with M. The manifolds 1/Jn(M) are all isometric; they differ only in the choice of isomorphism between π1(M) and π1(S). A fundamental result of Thurston’s—the double limit theorem—provides an algebraically convergent subsequence ψn(M) ->Mψ. The theory of pleated surfaces gives an upper bound on the injectivity radius of M in its convex core. Therefore any geometric limit N of ψn(M) is rigid, and so ψ is realized by an isometry α Mψ ->Mψ, completing the construction.

Mostow rigidity implies the full sequence converges to Mψ. From the the inflexibility theory of Chapter 2, we obtain the sharper statement that ψn(M) ->Mψ exponentially fast.

The case of torus orbifold bundles over the circle, previously considered by Jørgensen, is discussed in §3.7. We also give an explicit example of a totally degenerate group with no cusps (see Figure 3.4).

Renormalization. The simplest dynamical systems with critical points are the quadratic polynomials f(z) = z2 + c. In contrast to Kleinian groups, the consideration of limits quickly leads one to mappings not defined on the whole sphere.

A quadratic-like map g : U ->V is a proper degree two holomorphic map between disks in the complex plane, with [bar.U] a compact subset of V. Its filled Julia set is K(g) = [intersection]g-n(V).

If the restriction of an iterate fn: Un ->Vn to a neighborhood of the critical point z = 0 is quadratic-like with connected filled Julia set, the mapping fn is renormalizable. When infinitely many such n exist, we say f is infinitely renormalizable.

Basic results on quadratic-like maps and renormalization are presented in Chapter 4. In Chapter 5 we define towers of quadratic like maps, to capture geometric limits of renormalization. A tower

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a collection of quadratic-like maps with connected Julia sets, indexed by levels s > 0. We require that 1 [member of] S, and that for any s, t [member of] S with s<t, the ratio t/s is an integer and fs is a renormalization of ft/ss.

A tower has bounded combinatorics and definite moduli if t/s is bounded for adjacent levels and the annuli Vs – [bar.U]s are uniformly thick. In Chapter 6 we prove the Tower Rigidity Theorem: a hiinfinite tower T with bounded combinatorics and definite moduli admits no quasiconformal deformations. (This result is a dynamical analogue of the rigidity of totally degenerate groups.)

To put this rigidity in perspective, note that a single quadratic-like map f1 : U1 ->V1 is never rigid; an invariant complex structure for f1 can be specified at will in the fundamental domain V1 – [bar.U]1. In a tower with inf S = 0, on the other hand, f1 is embedded deep within the dynamics of fs for s near zero (since f1/ss = f1). The rigidity of towers makes precise the intuition that a high renormalization of a quadratic-like map should be nearly canonical.

Chapter 7 presents a two-step process to construct fixed points of renormalization. The procedure is analogous to that used to find a fixed-point of ψ. For renormalization, the initial data is a real number c such that the critical point of z [??] z2 + c is periodic with period p. The first step is to construct an infinite sequence of superstable points c*n in the Mandelbrot set by iterating the tuning map x [??] c*x. The iterated tunings c*n converge to a point c∞ such that f(z) = z + c∞ is infinitely renormalizable. In the classical Feigenbaum example, c = -1 and c*n gives the cascade of period doublings converging to the Feigenbaum polynomial f(z) = z2 + (-1)∞ = z2 -1.4101155 ···. The maps f and Rp(f) are quasiconformally conjugate near their Julia sets.

The second step is to iterate the renormalization operator Rp, starting with the point f. By Sullivan’s a priori bounds, we can pass to a subsequence such that Rnp(f) converges to a quadratic-like map F. This F can be embedded in a tower

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

such that f1 = F and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In the limit we also have a quasiconformal mapping φ : C -> C conjugating fps to fs for each s. By the Tower Rigidity Theorem, φ is a conformal map (in fact φ(z) = αz), and thus F is a fixed point of renormalization. Just as for 3-manifolds, we use rigidity of the geometric limit T to conclude the dynamics is self-similar.

Much of the construction also works when cis complex, but some steps at present require c real.

Deep points and uniform twisting. Chapters 8 and 9 develop results leading to the proof that renormalization converges exponentially fast.

Chapter 8 exploits the tower theory further to study the geometry and dynamics of infinitely renormalizable maps f(z) = z2 + c. Assume f has bounded combinatorics and definite moduli. Then we show:

1. The complement of the postcritical Cantor set P(f) is a Riemann surface with bounded geometry.

2. The critical point z = 0 is a deep point of the Julia set J(f). In particular, blowups of J(f) about z = 0 converge to the whole plane in the Hausdorff topology.

3. There are small Julia sets everywhere in J(f). More precisely, for any z [member of] J(f) and 0 <r<1, there is a quadratic-like map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in the dynamics generated by f, such that diam J(g) [??] r and d(z, J(g)) = O(r).

Chapter 9 lays the foundations for a general theory of holomorphic dynamical systems F and their geometric limits. With these foundations in place, it is possible to prove inflexibility results generalizing those for hyperbolic manifolds. The notion of bounded injectivity radius is replaced by that of uniform twisting.

Roughly speaking, (F, Λ) is uniformly twisting if the geometric limits of F as seen from within the convex hull of Λ in H3 are very nonlinear. Condition (3) above implies that (F(f), J(f)) is uniformly twisting, where F(f) is the full dynamics generated by f and f-1.

The Deep Conformality Theorem asserts that a quasiconformal conjugacy between uniformly twisting systems is C1+α-conformal at the deep points of Λ. By (2) above, we conclude that any conjugacy from f to another quadratic-like map is differentiable at the critical point z = 0. Exponentially fast convergence of renormalization then follows.

To state the final result, we remark that a fixed point renormalization F has a canonical maximal analytic continuation [??] : W ->C. The domain W of [??] is an open, dense subset of the plane. Then there is a λ <1 such that for any compact K [subset] W

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all f with the same inner class as F, and all n [much greater than] 0.

The fixed point constructions for mapping classes and for renormalization operators are compared retrospectively in Chapter 10. We conclude with some open problems; among them, the conjectural self-similarity of the boundary of Teichmüller space, as observed in computer experiments conducted jointly with Dave Wright. These parallels and open questions are summarized in Table 1.2.

Harmonic analysis. Two appendices develop the analytic foundations of our results. Appendix A is devoted to quasiconformal flows and Reimann’s theorem, that a vector field with strain in L∞ generates a unique quasiconformal isotopy. The proof is streamlined by showing a function with first derivatives in BMO satisfies a Zygmund condition.

Appendix B is devoted to the visual extension of deformations from Sn-1∞ to Hn. This extension has been studied by Ahlfors, Reimann, Thurston and others.

A key role for us is played by the visual distortion Mυ(p) of a vector field υ on Sn-1∞, as seen from p [member of] Hn. By definition Mυ(p) is the minimum, over all conformal vector fields w, of the maximum visual length of (υ-w) as seen from p. The quantity Mυ depends only on the strain Sυ (the Beltrami differential μ = [bar.[partial derivative]]υ when n = 3), but in a subtle way. Our inflexibility results are all proved by bounding Mυ.

An illustration. The intuitive link between rigidity, deep points in the Julia set and convergence of renormalization is the following. Let f be the Feigenbaum polynomial, and let φ be a quasiconformal conjugacy between f and f ο f. The dilatation of φ is specified by a field of infinitesimal ellipses supported outside the Julia set J(f) and invariant under the action of f. To visualize a typical invariant ellipse field, first consider a family of ellipses in the plane of constant eccentricity whose major axes are along rays through the origin. This ellipse field is invariant under the mapping z [??] z2. Now the Riemann mapping from the outside of the unit disk to the outside of the Feigenbaum Julia set transports the dynamics of z2 to that of f. Since the Julia set is quite dense near the postcritical set, the argument of the derivative of the Riemann mapping varies wildly, and so the resulting ellipse field is more or less random. The stretching in different directions approximately cancels out, so that φ, while fluctuating at very small scales, is close to a conformal map near the critical point. The Julia set of a high renormalization of f also resides in a small neighborhood of the critical point. Since φ also conjugates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], these two mappings are nearly conformally conjugate, and in the limit we obtain a fixed point of renormalization.

A blowup of the Feigenbaum Julia set near the critical point appears in Figure 1.3. The tree-like black regions are the points outside the Julia set. The thin postcritical Cantor set, lying on the real axis and evidently well-shielded from the ellipse field, is also Figure 1.3. Asymptotic rigidity near the postcritical set. shown in black. The white region is a 1-pixel neighborhood of the Julia set itself (which is nowhere dense).

Notes and references. Thurston’s construction of hyperbolic structures on 3-manifolds which fiber over the circle appears in [Th5]; an early account is in [Sull]. Recently Otal has given a self-contained presentation of Thurston’s theorem, using the method of R-trees to establish the double limit theorem [Otal].

Feigenbaum’s work appears in [Feig]; similar discoveries were made independently by Coullet and Tresser [CoTr]. Many additional contributions to the theory of renormalization are collected in [Cvi]. Sullivan’s a priori bounds and a proof of convergence of renormalization using Riemann surface laminations appear in [Sul5] and [MeSt].

The dictionary between rational maps and Kleinian groups was introduced in [Sul4]. See [Mc5] for an illustrated account, centering on classification problems for conformal dynamical systems.

A brief discussion of the results presented here and other iterations on Teichmiiller space appeared in [Mc3]. This work is a sequel to [Mc4], which discussed the foundations of renormalization, and to [Mc2], which began the program of using analytic estimates in an “effective” deformation theory for Kleinian groups.

Parts of this work were presented at CUNY and IHES in 1989 and 1990, at the Boston University Geometry Institute in 1991 and in the Keeler Lectures at University of Michigan in 1993. A preliminary version of this manuscript was circulated in Fall of 1994. Many useful corrections and suggestions were provided by G. Anderson and the referee. This research was funded in part by the Miller Institute for Basic Research and the NSF. I would like to thank all for their support.

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