ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM

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1 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM KARI ASTALA, OLEG IVRII, ANTTI PERÄLÄ, AND ISTVÁN PRAUSE Abstract. We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a k-quasicircle is at most + k, it is natural to expect that the maximum asymptotic variance Σ =. In this paper, we prove.8793 Σ. For the lower bound, we give examples of polynomial Julia sets which are k-quasicircles with dimensions k for k small, thereby showing that Σ The key ingredient in this construction is a good estimate for the distortion k, which is better than the one given by a straightforward use of the λ-lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating Σ in terms of nearly circular polynomial Julia sets.. Introduction In his work on the Weil-Petersson metric [7], McMullen considered certain holomorphic families of conformal maps ϕ t : D C, ϕ (z) = z, where D = {z : z > }, that naturally arise in complex dynamics and Teichmüller theory. For these special families, he used thermodynamic formalism to relate a number of different dynamical features. For instance, he showed that the infinitesimal growth of the Hausdorff dimension of the Jordan curves ϕ t (S ) is connected Mathematics Subject Classification. Primary 3C6; Secondary 3H3. Key words and phrases. Quasiconformal map, Beurling transform, Asymptotic variance, Bergman projection, Bloch space, Hausdorff dimension, Julia set. A.P. was supported by the Vilho, Yrjö and Kalle Väisälä Foundation and the Emil Aaltonen Foundation. I.P. and A.P. were supported by the Academy of Finland (SA) grants 668 and All authors were supported by the Center of Excellence Analysis and Dynamics, SA grants 7566 and Part of this research was performed while K.A. and I.P. were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation.

2 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE to the asymptotic variance of the first derivative of the vector field v = dϕ t dt t= by the formula d dt H. dim ϕ t (S ) = σ (v ), (.) t= where the asymptotic variance of a Bloch function g in D is given by σ (g) = lim sup g(z) dz. (.) π R + log(r ) z =R This terminology is justified by viewing g as a stochastic process Y s (ζ) = g(( e s )ζ), ζ S, s <, with respect to the probability measure dζ /π, in which case σ (g) = lim sup s s σ Y s. For further relevance of probability methods to the study of the boundary distortion of conformal maps, we refer the reader to [7, ]. For arbitrary families of conformal maps, the identity (.) may not hold. For instance, Le and Zinsmeister [9] constructed a family {ϕ t } for which σ (v ) is zero, while t M. dim ϕ t (S ) (with Hausdorff dimension replaced by Minkowski dimension) is equal to for t < but grows quadratically for t >. Nevertheless, it is natural to enquire if McMullen s formula (.) holds on the level of universal bounds. As will be explained in detail in the subsequent sections, for general holomorphic families of conformal maps ϕ t parametrised by a complex parameter t D, one can combine the work of Smirnov [4] with the theory of holomorphic motions [3, 4] to show that H. dim ϕ t (S ) + ( t ) t = + t 4 + O( t 4 ), t D. (.3) It is conjectured that the equality in (.3) holds for some family, but this is still open. On the other hand, the derivative of the infinitesimal vector field v = dϕt dt t= can be represented in the form v = Sµ where µ(z) χ D and S is the Beurling transform, the principal value integral Sµ(z) = π C µ(w) dm(w). (.4) (z w) (Since the support of µ is contained in the unit disk, v is a holomorphic function on the exterior unit disk.)

3 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 3 In this formalism, McMullen s identity describes the asymptotic variance σ (Sµ) for a dynamical Beltrami coefficient µ, which is invariant under a co-compact Fuchsian group or a Blaschke product. In this paper, we study the quantity Σ := sup{σ (Sµ) : µ χ D } (.5) from several different perspectives. In addition to the problem of dimension distortion of quasicircles, Σ is naturally related to questions on integral means of conformal maps, which we discuss later in the introduction. The first result in this work is an upper bound for Σ : Theorem.. Suppose µ is measurable in C with µ χ D. Then, σ (Sµ) := lim sup π R + log(r ) π Sµ(Re iθ ) dθ. (.6) We give two different proofs for (.6), one using holomorphic motions and quasiconformal geometry in Section 4, and another based on complex dynamics and fractal approximation in Section 6. In view of McMullen s identity and the possible sharpness of Smirnov s dimension bounds, it is natural to expect that the bound (.6) is optimal with Σ =, and in the first version of this paper we formulated a conjecture to that extent. However, after having read our manuscript, Håkan Hedenmalm managed to show [] that actually Σ <. For lower bounds on Σ, we produce examples in Section 5 showing: Theorem.. There exists a Beltrami coefficient µ χ D such that σ (Sµ) > In fact, our construction gives new bounds for the quasiconformal distortion of certain polynomial Julia sets: Theorem.3. Consider the polynomials P t (z) = z d + t z. For t <, the Julia set J (P t ) is a Jordan curve which can be expressed as the image of the unit circle by a k-quasiconformal map of C, where k = d d 4 t + O( t ).

4 4 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE In particular, when d = and t is small, k.585 t and J (P t) is a k-quasicircle with H. dim J (P t ) k. (.7) Note that the distortion estimates in Theorem.3 are strictly better (for d 3) than those given by a straightforward use of the λ-lemma. For a detailed discussion, see Section 5. In terms of the dimension distortion of quasicircles, Theorem.3 improves upon all previously known examples. For instance, the holomorphic snowflake construction of [8] gives a k-quasicircle of dimension +.69 k. In order to further explicate the relationship between asymptotic variance and dimension asymptotics, consider the function D(k) = sup{h. dim Γ : Γ is a k-quasicircle}, k <. The fractal approximation principle of Section 6 roughly says that infinitesimally, it is sufficient to consider certain quasicircles, namely nearly circular polynomial Julia sets. As a consequence, we prove: Theorem.4. Σ lim inf k Together with Smirnov s bound [4], D(k) k. (.8) D(k) + k, (.9) Theorem.4 gives an alternative proof for Theorem.. We note that the function D(k) may be also characterised in terms of several other properties in place of Hausdorff dimension, see [3]. It would be interesting to know if the reverse inequality in Theorem.4 holds. In Section 7, we study the fractal approximation question in the Fuchsian setting. One may expect that it may be possible to approximate Σ using Beltrami coefficients invariant under co-compact Fuchsian groups. However, this turns out not to be the case. To this end, we show: Theorem.5. Σ F := sup σ (Sµ) < /3. µ M F, µ χ D

5 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 5 Theorem.5 may be viewed as an upper bound for the quotient of the Weil-Petersson and Teichmüller metrics, over all Teichmüller spaces T g with g. (To make the bound genus-independent, one needs to normalise the hyperbolic area of Riemann surfaces to be.) The proof follows from simple duality arguments and the fact that there is a definite defect in the Cauchy-Schwarz inequality. Finally, we compare our problem with another method of embedding a conformal map f into a flow: log f t(z) = t log f (z), t D. (.) In this case, the derivative of the infinitesimal vector field at t = is just the Bloch function log f (z). However, even if f itself is univalent, the univalence of f t is only guaranteed for t /4, see [3]. One advantage of the notion (.5) and holomorphic flows parametrised by Beltrami equations is that they do not suffer from this univalency gap. In the case of domains bounded by regular fractals and the corresponding equivariant Riemann mappings f(z), we have several interrelated dynamical and geometric characteristics: The integral means spectrum of a conformal map: β f (τ) = lim sup r log z =r (f ) τ dθ, τ C. (.) log r The asymptotic variance a Bloch function g B: σ (g) = lim sup g(z) dθ. (.) r π log( r) z =r The LIL constant of a conformal map is defined as the essential supremum of C LIL (f, θ) over θ [, π) where C LIL (f, θ) = lim sup r log f (re iθ ). (.3) log r log log log r Theorem.6. Suppose f(z) is a conformal map, such that the image of the unit circle f(s ) is a Jordan curve, invariant under a hyperbolic conformal dynamical system. Then, d dτ β f (τ) = σ (log f ) = CLIL(f), (.4) τ=

6 6 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE where β(τ) is the integral means spectrum, σ is the asymptotic variance of the Bloch function log f, and C LIL denotes the constant in the law of the iterated logarithm (.3). We emphasise that the above quantities are not equal in general, but only for special domains Ω that have fractal boundary. For these domains, the limits in the definitions of β f (τ) and σ (log f ) exist, while C LIL (f, θ) is a constant function (up to a set of measure ). The equalities in (.4) are mediated by a fourth quantity involving the dynamical asymptotic variance of a Hölder continuous potential from thermodynamic formalism. The equality between the dynamical variance and C LIL is established in [35, 36], while the works [, ] give the connection to the integral means β(τ). The missing link, it seems, is the connection between the dynamical variance and σ, which can be proved using a global analogue of McMullen s coboundary relation. Details will be given in Section 8. We note that an alternative approach connecting β(τ) and σ directly has been considered in the special case of polynomial Julia sets, see [7]. With these connections in mind, we relate our quantity Σ to the universal integral means spectrum B(τ) = sup f β f (τ): Theorem.7. lim inf τ B(τ) τ /4 Σ. In view of the lower bound for Σ given by Theorem., this improves upon the previous best known lower bound [3] for the behaviour of the universal integral means spectrum near the origin. The proof of Theorem.7 along with additional numerical advances is presented in Section 8. While the two approaches above for constructing flows of conformal maps are somewhat different, there is a relation: singular conformal maps via welding-type procedures [3]. singular quasicircles lead to The parallels are summarised in Table below, where exact equalities hold only in the dynamical setting.. Bergman projection and Bloch functions In this section, we introduce the notion of asymptotic variance for Bloch functions and discuss some of its basic properties.

7 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 7 Holomorphic motion ϕ t = t µ ϕ t log f t = t log f Bloch function v Sµ log f Univalence µ f conformal σ (v ) = c H. dim ϕ t (S ) = + c t / β f (τ) = c τ / Examples Lacunary series Table. Singular conformal maps and the growth of Bloch functions.. Asymptotic variance. The Bloch space B consists of analytic functions g in D which satisfy g B := sup ( z ) g (z) <. z D Note that B is only a seminorm on B. A function g B belongs to the little Bloch space B if lim z ( z ) g (z) =. To measure the boundary growth of a Bloch function g B, we define its asymptotic variance by σ (g) := lim sup π r log( r) π g(re iθ ) dθ. (.) Lacunary series provide examples with non-trivial (i.e. positive) asymptotic variance. For instance, for g(z) = n= zdn with d, a quick calculation based on orthogonality shows that σ (g) = log d. (.) Following [3, Theorem 8.9], to estimate the asymptotic variance, we use Hardy s identity which says that ( ) ( d r d ) π 4r dr dr π g B ( r g(re iθ ) dθ = π ( d 4r dr ) = g B From (.3), it follows that σ (g) g B. π ) ( r d dr g (re iθ ) dθ (.3) ) log r. In particular, the asymptotic variance of a Bloch function is finite. It is also easy to see that adding an element from the little Bloch space does not affect the asymptotic variance, i.e. σ (g + g ) = σ (g).

8 8 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE.. Beurling transform and the Bergman projection. For a measurable function µ with µ χ D, the Beurling transform g = Sµ is an analytic function in the exterior disk D = {z : z > } which satisfies a Bloch bound of the form g B := g (z) ( z ) C. Note that we use the notation B for functions in D we reserve the symbol B for the standard Bloch space in the unit disk D. By passing to the unit disk, we are naturally led to the Bergman projection P µ(z) = π D µ(w)dm(w) ( zw) (.4) and its action on L -functions. Indeed, comparing (.4) and (.4), we see that P µ(/z) = z Sµ (z) for µ (w) = µ(w) and z D. From this connection between the Beurling transform and the Bergman projection, it follows that Σ = sup µ χ D σ (Sµ) = sup µ χ D σ (P µ). (.5) In view of the above equation, the Beurling transform and the Bergman projection are mostly interchangeable. Due to natural connections with the quasiconformal literature, we mostly work with the Beurling transform. However, in this section on a priori bounds, it is preferable to work with the Bergman projection to keep the discussion in the disk..3. Pointwise estimates. According to [9], the seminorm of the Bergman projection from L (D) B is 8/π. Integrating (.3), we get π ( ) 8 P µ(re iθ ) dθ log π π r, r, which implies that Σ (8/π). One can also equip the Bloch space with seminorms that use higher order derivatives f B,m = sup( z ) m f (m) (z), (.6) z D where m is an integer. Very recently, Kalaj and Vujadinović [6] calculated the seminorm of the Bergman projection when the Bloch space is equipped with (.6). According to their result, P B,m = Γ( + m)γ(m) Γ (m/ + ). (.7)

9 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 9 It is possible to apply the differential operator in (.3) m times and use the pointwise estimates (.7). In this way, one ends up with the upper bounds σ (Sµ) = σ (P µ) Γ( + m) Γ(m) Γ(m)Γ 4 (m/ + ). (.8) Putting m = in (.8), one obtains that σ (Sµ) 6, which is a slight improvement to (8/π) and is the best upper bound that can be achieved with this argument. Using quasiconformal methods in Section 4, we will show the significantly better upper bound σ (Sµ)..4. Césaro integral averages. In Section 6 on fractal approximation, we will need the Césaro integral averages from [7, Section 6]. Following McMullen, for f B, m and r [, ), we define σm(f, r) = [ r ds π ] Γ(m) log( r) s π ( s ) m f (m) (se iθ ) dθ and σm(f) = lim sup r σm(f, r). (.9) We will need [7, Theorem 6.3] in a slightly more general form, where we allow the use of limsup instead of requiring the existence of a limit: Lemma.. For f B, σ (f) = σ (f) = σ 4(f) = σ 6(f) =... (.) Furthermore, if the limit as r in σm (f) exists for some m, then the limit as r exists in σm (f) for all m. The original proof from [7] applies in this setting. 3. Holomorphic families Our aim is to understand holomorphic families of conformal maps, and the infinitesimal change of Hausdorff dimension. The natural setup for this is provided by holomorphic motions [3], maps Φ : D A C, with A C, such that For a fixed a A, the map λ Φ(λ, a) is holomorphic in D. For a fixed λ D, the map a Φ(λ, a) = Φ λ (a) is injective. The mapping Φ is the identity on A, Φ(, a) = a, for every a A.

10 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE It follows from the works of Mañé-Sad-Sullivan [3] and Slodkowski [4] that each Φ λ can be extended to a quasiconformal homeomorphism of C. In other words, each f = Φ λ is a homeomorphic W, loc (C)-solution to the Beltrami equation f(z) = µ(z) f(z) for a.e. z C. Here the dilatation µ(z) = µ λ (z) is measurable in z C, and the mapping f is called k-quasiconformal if µ k <. As a function of λ D, the dilatation µ λ is a holomorphic L -valued function with µ λ λ, see []. In other words, Φ λ is a λ -quasiconformal mapping. Conversely, as is well-known, homeomorphic solutions to the Beltrami equation can be embedded into holomorphic motions. For this work, we shall need a specific and perhaps non-standard representation of the mappings which quickly implies the embedding. For details, see Section Quasicircles. Let us now consider a holomorphic family of conformal maps ϕ t : D C, t D such as the one in the introduction. That is, we assume ϕ(t, z) = ϕ t (z) is a D D C holomorphic motion which in addition is conformal in the parameter z. By the previous discussion, each ϕ t extends to a t -quasiconformal mapping of C. Moreover, by symmetrising the Beltrami coefficients like in [8, 4], we see that ϕ t (S ) is a k-quasicircle, where t = k/( + k ). More precisely, ϕ t (S ) = f(r { }) for a k-quasiconformal map f : Ĉ Ĉ of the Riemann sphere Ĉ, which is antisymmetric with respect to the real line in the sense that µ f (z) = µ f (z) for a.e. z C. Smirnov used this antisymmetric representation to prove (.9). In terms of the conformal maps ϕ t, Smirnov s result takes the form mentioned in (.3). 3.. Heuristics for σ (Sµ). An estimate based on the τ = case of [3, Theorem 3.3] tells us roughly that for R >, ϕ πr t(z) dz C( t ) (R ) t. (3.) z =R (The precise statement is somewhat weaker but we are not going to use this.) A natural strategy for proving σ (Sµ) is to consider the holomorphic

11 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM motion of principal mappings ϕ t generated by µ, ϕ t = tµ ϕ t, t D; ϕ t (z) = z + O(/z) as z. For the derivatives, we have the Neumann series expansion: ϕ t = ϕ t = + tsµ + t SµSµ +..., z D. (3.) In view of this, taking the limit t in (3.), one obtains a growth bound (as R ) for the integrals z =R Sµ dz. However, in order to validate this strategy, one needs to have good control on the constant term C( t ) in (3.). Namely, one would need to show that C( t ) as t fast enough, for instance at a quadratic rate C( t ) C t. Unfortunately, while the growth exponent in (3.) is effective, the constant is not. In order to make this strategy work, we need two improvements. First, we work with quasiconformal maps that are antisymmetric with respect to the unit circle; and secondly, we use normalised solutions instead of principal solutions. One of the key estimates will be Theorem 4.4 which is the counterpart of (3.) for antisymmetric maps, but crucially with a multiplicative constant of the form C(δ) k. measure estimates of [33]. This naturally complements the Hausdorff 3.3. Interpolation. Let (Ω, σ) be a measure space and L p (Ω, σ) be the usual spaces of complex-valued σ-measurable functions on Ω equipped with the (quasi)norms ( ) Φ p = Φ(x) p p dσ(x), < p <. Ω In the papers [4] [7], holomorphic deformations were used to give sharp bounds on the distortion of quasiconformal mappings. In [6], the method was formulated as a compact and general interpolation lemma: Lemma 3.. [6, Interpolation Lemma for the disk] Let < p, p and {Φ λ ; λ < } M (Ω, σ) be an analytic and non-vanishing family of measurable functions defined on a domain Ω. Suppose M := Φ p <, where M := sup Φ λ p < λ < = r p r + r + r p + r. p and M r := sup Φ λ pr, λ =r

12 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE Then, for every r <, we have r +r M r M M r +r <. (3.3) To be precise, in the lemma we consider analytic families Φ λ of measurable functions in Ω, i.e. jointly measurable functions (x, λ) Φ λ (x) defined on Ω D, for which there exists a set E Ω of σ-measure zero such that for all x Ω \ E, the map λ Φ λ (x) is analytic and non-vanishing in D. For the study of the asymptotic variance of the Beurling transform, we need to combine interpolation with ideas from [4] to take into account the antisymmetric dependence on λ, see Proposition 4.3. In this special setting, Lemma 3. takes the following form: Corollary 3.. Suppose {Φ λ ; λ D} is an analytic family of measurable functions, such that for every λ D, Φ λ (x) and Φ λ (x) = Φ λ (x), for a.e. x Ω. (3.4) Let < p, p. Then, for all k < and exponents p k defined by = k p k + k + k p + k, p we have k ( ) k +k Φ k pk Φ p sup { λ <} Φ λ p +k, assuming that the right hand side is finite. Proof. Consider the analytic family λ Φ λ (x) Φ λ (x). The non-vanishing condition ensures that we can take an analytic square-root. Since the dependence with respect to λ gives an even analytic function, there is a (singlevalued) analytic family Ψ λ such that Ψ λ (x) = Φ λ (x) Φ λ (x). Observe that Φ λ (x) = Ψ λ (x) for real λ by the condition (3.4). By the Cauchy-Schwarz inequality, Ψ λ satisfies the same L p -bounds: Ψ λ p Φ λ / p Φ λ / p sup { λ <} Φ λ p, λ D.

13 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 3 We can now apply the Interpolation Lemma for the non-vanishing family Ψ λ with r = k to get k ( ) k +k Φ k pk = Ψ k pk Ψ p +k sup { λ <} Ψ λ p k ( ) k +k Φ p sup { λ <} Φ λ p +k. 4. Upper bounds In this section, we apply quasiconformal methods for finding bounds on integral means to the problem of maximising the asymptotic variance σ (Sµ) of the Beurling transform. Our aim is to establish the following result: Theorem 4.. Suppose µ is measurable with µ χ D. Then, for all < R <, π Sµ(Re iθ ) dθ ( + δ) log + c(δ), < δ <, (4.) π R where c(δ) < is a constant depending only on δ. The growth rate in (4.) is interesting only for R close to : For z = R >, we always have the pointwise bound Sµ(z) = µ(ζ) π (ζ z) dm(ζ) (R ). (4.) D It is clear that Theorem 4. implies Σ, i.e. the statement from Theorem. that σ (Sµ) = lim sup π R + log(r ) whenever µ χ D. π Sµ(Re iθ ) dθ (4.3) The proof of Theorem 4. is based on holomorphic motions and quasiconformal distortion estimates. In particular, we make strong use of the ideas of Smirnov [4], where he showed that the dimension of a k-quasicircle is at most + k. We first need a few preliminary results.

14 4 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE 4.. Normalised solutions. The classical Cauchy transform of a function ω L p (C) is given by Cω(z) = π C For us it will be convenient to use a modified version C ω(z) := [ ω(ζ) π C z ζ ζ = ( z) ω(ζ) π (z ζ)( ζ) dm(ζ) C ω(ζ) dm(ζ). (4.4) z ζ ] dm(ζ) (4.5) defined pointwise for compactly supported functions ω L p (C), p >. Like the usual Cauchy transform, the modified Cauchy transform satisfies the identities (C ω) = ω and (C ω) = Sω. Furthermore, C ω is continuous, vanishes at z = and has the asymptotics C ω(z) = ω(ζ) dm(ζ) + O(/z) as z. π C ζ We will consider quasiconformal mappings with Beltrami coefficient µ supported on unions of annuli A(ρ, R) := {z C : ρ < z < R}. Typically, we need to make sure that the support of the Beltrami coefficient is symmetric with respect to the reflection in the unit circle. Therefore, it is convenient to use the notation A R := A(/R, R), < R < and (4.6) A ρ,r := A(/R, /ρ) A(ρ, R), < ρ < R <. (4.7) For coefficients supported on annuli A R, the normalised homeomorphic solutions to the Beltrami equation f(z) = µ(z) f(z) for a.e. z C, f() =, f() =, (4.8) admit a simple representation: Proposition 4.. Suppose µ is supported on A R with µ < and f W, loc (C) is the normalised homeomorphic solution to (4.8). Then f(z) = z exp(c ω(z)), z C, (4.9) where ω L p (C) for some p >, has support contained in A R and ω(z) µ(z)sω(z) = µ(z) z for a.e. z C. (4.)

15 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 5 Proof. First, if ω satisfies the above equation, then ( ) µ(z) ω = (Id µs) = µ(z) ( ) ( ) µ(z) µ(z) + µs + µsµs + z z z z with the series converging in L p (C) whenever µ S L p <, in particular for some p >. The solution, unique in L p (C), clearly has support contained in A R. If f(z) is as in (4.9), then f W, loc (C) and satisfies (4.8) with the required normalisation. To see that f is a homeomorphism, note that where α = exp f(z) = α[z + β + O(/z)] as z, (4.) ( ) ω(ζ) π C ζ dm(ζ) and β = ω(ζ)dm(ζ) (4.) π C which shows that f is a composition of a similarity and a principal solution to the Beltrami equation. Since every principal solution to a Beltrami equation is automatically a homeomorphism [5, p.69], we see that f must be a homeomorphism as well. The proposition now follows from the uniqueness of normalised homeomorphic solutions to (4.8). 4.. Antisymmetric mappings. If the Beltrami coefficient in (4.8) satisfies µ(z) = µ(z), then by the uniqueness of the normalised solutions, we have f(z) = f(z) and f preserves the real axis. For normalised solutions preserving the unit circle, the corresponding condition for f is f(/z) = /f(z) which asks for the Beltrami coefficient to satisfy µ( z ) z = µ(z) for a.e. z C. In this case, we say that the Beltrami z coefficient µ is symmetric (with respect to the unit circle). Following [4], we say that µ is antisymmetric if ( ) z µ = µ(z) for a.e. z C. (4.3) z z Given an antisymmetric µ supported on A R with µ =, define µ λ (z) = λ µ(z), λ D, and let f λ be the corresponding normalised homeomorphic solution to (4.8) with µ = µ λ. It turns out that in case of mappings antisymmetric with

16 6 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE respect to the circle, the expression Φ λ (z) := z f λ(z) f λ (z) has the proper invariance properties similar to those used in [4]: Proposition 4.3. For every λ D and z C, [ f λ (/z) = z f ( λ ) (z) z f λ (/z) f ( λ ) (z) In particular, Proof. Let f λ (z) f λ (z) = f ( λ ) (z) f ( λ ) (z) g λ (z) = ]. whenever z =., z C. (4.4) f λ (/z) By direct calculation, g λ has complex dilatation λµ( z ) z which by our z assumption on antisymmetry is equal to λµ(z). Since g λ and f λ are normalised solutions to the same Beltrami equation, the functions must be identical. Differentiating the identity (4.4) with respect to / z, we get f ( λ) (z) = z f λ (/z) f λ (/z) = f ( λ) (z) f λ (/z) z f λ (/z). Rearranging and taking the complex conjugate gives the claim Integral means for antisymmetric mappings. For < R <, consider a quasiconformal mapping f whose Beltrami coefficient is supported on A R,. Since f is conformal in the narrow annulus { R < z < R}, it is reasonable to study bounds for the integral means involving the derivatives of f on the unit circle. We are especially interested in the dependence of these bounds on R as R +. Theorem 4.4. Suppose µ is measurable, µ(z) ( δ)χ AR, (z), and that µ is antisymmetric. Let k. If f = f k W, loc (C) is the normalised homeomorphic solution to f(z) = kµ(z) f(z), then π z = f (z) f(z) where C(δ) < is a constant depending only on δ. dz C(δ) k (R ) k +k, (4.5)

17 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 7 The assumption µ(z) δ above, where δ > is fixed but arbitrary, is made to guarantee that we have uniform bounds in (4.5) for all k <. To estimate the asymptotic variance of the Beurling transform, we will study the nature of these bounds as k, but we need to keep in mind the dependence on the auxiliary parameter δ >. Proof of Theorem 4.4. We embed f in a holomorphic motion by setting µ λ (z) = λ µ(z), λ D. Let f λ denote the normalised solution to the Beltrami equation f z = µ λ f z, with the representation (4.9) described in Proposition 4.. The uniqueness of the solution implies that f k = f. We now apply Corollary 3. to the family Φ λ (z) := z (f λ) (z) f λ (z), λ D, z S. (4.6) By [5, Theorem 5.7.], the map is well-defined, nonzero and holomorphic in λ for each z S. By the antisymmetry of the dilatation µ, we can use Proposition 4.3 to get the identity Φ λ (z) = f λ (z) f λ (z) = f ( λ ) (z) f ( λ ) (z) = Φ λ (z), z S. (4.7) We first find a global L -bound, independent of λ D. For this purpose, we estimate π A R f λ (z) f λ (z) dm(z). Recall that < R < by assumption. Since all f λ s are normalised +δ δ - quasiconformal mappings, we have together with f λ (z) = f λ(z) f λ () f λ () f λ () /ρ δ, /R < z < R, f λ (A R ) f λ B(, ) B(, ρ δ ). Therefore, π A R f λ (z) f λ (z) dm(z) π ρ δ f λa R ρ 4 δ / (4.8)

18 8 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE for some constant < ρ δ < depending only on δ. In particular, (R ) f λ (z) π f λ (z) dz c(δ) <, λ D, z = where the bound c(δ) depends only on < δ <. We now use interpolation to improve the L -bounds near the origin. We choose p = p =, Ω = S and dσ(z) = R π dz. Applying Corollary 3. gives (R ) π z = which implies Theorem 4.4. f k (z) f k (z) dz (R ) k +k c(δ) k +k, 4.4. Integral means for the Beurling transform. We now use infinitesimal estimates for quasiconformal distortion to give bounds for the integral means of Sµ. We begin with the following lemma: Lemma 4.5. Given < R <, suppose µ is an antisymmetric Beltrami coefficient with supp µ A R, and µ. Then, µ (z) := µ(z) z satisfies Sµ (z) dz ( + δ) log + log C(δ/4), < δ <, π (R ) z = where C(δ) is the constant from Theorem 4.4. Proof. First, observe that if h is any L -function vanishing in the annulus {z : /R < z < R}, by the theorems of Fubini and Cauchy, z(sh)(z) dz = (Sh)(z)dz π z = πi S = [ ] h(ζ) π C πi S (ζ z) dz dm(ζ) =. To apply Theorem 4.4, take < k < and solve the Beltrami equation f(z) = kν(z) f(z) for the coefficient ν(z) = ( δ)µ(z). Let f k W, loc (C) be the normalised homeomorphic solution in C. Recall from (4.9) that f k has the representation f k (z) = z exp(c ω(z)) where ω = (Id k νs) ( k ν(z) z ) = k( δ) µ (z) + k ( δ) νsµ (z) + and the series converges in L p (C) for some fixed p = p(δ) >. From this representation, we see that z f k (z) f k (z) = + k( δ)zsµ (z) + k ( δ) zsνsµ (z) + O(k 3 ) (4.9)

19 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 9 holds pointwise in the annulus {z : /R < z < R}, where ν and ω vanish. It follows that f k (z) π f k (z) z = z = dz = +k ( δ) Sµ (z) dz +O(k 3 ). (4.) π z = Finally, combining (4.) with Theorem 4.4, we obtain + k ( δ) Sµ (z) dz + O(k 3 ) π z = ) exp (k log C(δ) + k + k log (R ) = + k log C(δ) + k log (R ) + O(k4 ). Taking k, we find that Sµ (z) dz ( δ) log π (R ) + ( δ) log C(δ). As ( δ/4) + δ, replacing δ by δ/4 proves the lemma. Corollary 4.6. Given < R <, suppose µ is a Beltrami coefficient with supp µ A(/, /R) and µ. Then, Sµ(z) dz ( + δ) log π (R ) + log C(δ/4), < δ <, z = where C(δ) is the constant from Theorem 4.4. Proof. Define an auxiliary Beltrami coefficient ν by requiring ν(z) = zµ(z) for z and ν(z) = z z ν(/z) for z. Then ν is supported on A R,, ν and ν is antisymmetric, so that with help of Lemma 4.5 we can estimate the integral means of Sν, where ν (z) = ν(z) z. On the other hand, the antisymmetry condition (4.3) implies C(χ D ν )(/z) = C(χ C\D ν )(z) C(χ C\D ν )() for the Cauchy transform. Differentiating this with respect to / z gives ( ) z S(χ Dν ) = zs(χ z C\D ν )(z). In particular, for z on the unit circle S, zs(ν )(z) = zs(χ D ν )(z) + zs(χ C\D ν )(z) = i Im [ z S(χ D ν )(z) ] = i Im [ z (Sµ)(z) ].

20 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE In other words, the estimates of Lemma 4.5 take the form Im [ z (Sµ)(z) ] dz = Sν (z) dz π 4 π z = z = 4 ( + δ) log (R ) + log C(δ/4), < δ <. 4 By replacing µ with iµ, we see that the same bound holds for the integral means of Re [ z (Sµ)(z)]. Therefore, Sµ(z) dz = Re [ z (Sµ)(z) ] Im [ ] + z (Sµ)(z) dz π z = π z = ( + δ) log R + log C(δ/4) for every < δ < Asymptotic variance. With these preparations, we are ready to prove Theorem 4.. We need to show that if µ is measurable with µ(z) χ D, then for all < R <, π π Sµ(Re iθ ) dθ ( + δ) log where c(δ) < is a constant depending only on δ. + c(δ), < δ <, R Proof of Theorem 4.. For a proof of this inequality, first assume that additionally µ(z) = for z < 3/4; < R < 3. (4.) Then ν(z) := µ(rz) has support contained in B(, /R) \ B(, /) so that we may apply Corollary 4.6. Since Sν(z) = Sµ(Rz), π Sµ(Re iθ ) dθ = Sν(z) dz π π which is the desired estimate. ( + δ) log z = R + log C(δ/4), For the general case when (4.) does not hold, write µ = µ + µ where µ (z) = χ B(,3/4) µ(z). As Sµ (z) dm(ζ) = π log(8), z =, 4 < z ζ < ζ z we have π Sµ (Re iθ ) + Sµ (Re iθ ) dθ π

21 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM ( + δ) π ( Sµ (Re iθ ) dθ + + ) π Sµ (Re iθ ) dθ π δ π ( + δ) log R + + δ log C(δ/4) + + δ 4π log (8) δ for < δ < and < R < 3 ; while for R 3, we have the pointwise bound (4.). Finally, replacing δ by δ/3, we get the estimate in the required form, thus proving the theorem. Consider the family of polynomials 5. Lower bounds P t (z) = z d + t z, t <, for d. According to [7, Theorem.8] or [, 39], the Hausdorff dimensions of their Julia sets satisfy H. dim J (P t ) = + t (d ) 4d log d + O( t 3 ). (5.) Moreover, each Julia set J (P t ) is a quasicircle, the image of the unit circle by a quasiconformal mapping of the plane. A quick way to see this is to observe that the immediate basin of attraction of the origin contains all the (finite) critical points of P t. (From general principles, it is clear that the basin must contain at least one critical point, but by the (d )-fold symmetry of P t, it must contain them all.) If A Pt ( ) denotes the basin of attraction of infinity, for each t < there is a canonical conformal mapping conjugating the dynamics: ϕ t : D = A P ( ) A Pt ( ) (5.) ϕ t P (z) = P t ϕ t (z), z D. (5.3) By Slodkowski s extended λ-lemma [4] and the properties of holomorphic motions, ϕ t extends to a t -quasiconformal mapping of the plane, see e.g. [5, Section.3]. In particular, the extension maps the unit circle onto the Julia set J (P t ).

22 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE While the extensions given by the λ-lemma are natural, surprisingly it turns out that the maps ϕ t have extensions with considerably smaller quasiconformal distortion, smaller by a factor of when t. c d := d d, d N, (5.4) Theorem 5.. Let P t (z) = z d + tz with t <. Then the canonical conjugacy ϕ t : D A Pt ( ), defined in (5.), has a µ t -quasiconformal extension with µ t = c d t + O( t ). Here c =, but c d < for d 3. Hence for every degree 3 we have an improved bound for the distortion. Furthermore, when representing J (P t ) as the image of the unit circle by a map with as small distortion as possible, one can apply Theorem 5. together with the symmetrisation method described in Section 3. to show that each J (P t ) is a k(t)-quasicircle, where k(t) = c d t + O( t ). By the dimension formula (5.), H. dim J (P t ) = + 4d d (d ) d k(t) + O( k(t) 3 ). (5.5) log d In particular, when d =, we get k-quasicircles whose Hausdorff dimension is greater than k, for small values of k. Therefore, Theorem.3 follows from Theorem 5.. The numerical values for the second order term of (5.5) are presented in Table below. These provide lower bounds on the asymptotic variance (or equivalently, on the quasicircle dimension asymptotics). For comparison, we also show the values for the second order term of (5.) which correspond to the estimate on quasiconformal distortion provided by the λ-lemma. Note that the first explicit lower bound on quasicircle dimension asymptotics [9] is exactly the degree case of the upper-left corner. For the proof of Theorem 5., we find an improved representation for the infinitesimal vector field determined by ϕ t. Differentiating (5.3), we get a

23 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 3 Degree λ-lemma Bounds from (5.5) d = d = d = d = Table. Comparison of lower bounds for Σ functional equation v(z d ) = d z d v(z) + z (5.6) for the vector field v = dϕt dt t=, which in turn forces the lacunary series expansion, see [7, Section 5], v(z) = z z (d )dn d d n, z >. (5.7) n= Our aim is to represent the lacunary series (5.7) as the Cauchy transform (or v as the Beurling transform) of an explicit bounded function supported on the unit disk. We will achieve this through the functional equation (5.6). For this reason, we will look for Beltrami coefficients with invariance properties under f(z) = z d, requiring that f µ = µ in some neighbourhood of the unit circle, where (f µ)(z) := µ(f(z)) f (z) f (z). (5.8) We first observe that the Cauchy transform (4.4) behaves similarly to a vector field under the pullback operation: Lemma 5.. Suppose µ is a Beltrami coefficient supported on the unit disk. Then, { } ( ) dz d Cµ(z d ) Cµ() = C (z d ) µ (z), z C. (5.9) Proof. From [5, p. 5], it follows that the Cauchy transform of a bounded, compactly supported function belongs to all Hölder classes Lip α with exponents < α <. In particular, near the origin, the left hand side of (5.9) is O( z ε ) for every ε >. This implies that the two quantities in (5.9) have the same ( / z)-distributional derivatives. As both vanish at infinity, they must be identically equal on the Riemann sphere.

24 4 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE Remark 5.3. Since the left hand side in (5.9) vanishes at, we always have C ( (z d ) µ ) () =. This can also be seen by using the change of variables z ζ z where ζ is a d-th root of unity. We will use the following basic Beltrami coefficients as building blocks: Lemma 5.4. Let µ n (z) := ( z/ z ) n χa(r,ρ) n N. Then with < r < ρ < and Cµ n (z) = n (ρn r n ) z (n ), z >, and Cµ n () =. Proof. We compute: µ n (w) w n dm(w) = D Hence, by orthogonality Cµ n (z) = πz = πz A(r,ρ) w n dm(w) = π n (ρn r n ). µ n (w)dm(w) D ( w/z) z j µ n (w)w j dm(w) j= = πz z (n ) π n (ρn r n ) = n z (n ) (ρ n r n ) D as desired. The claim Cµ n () = follows similarly. To represent power series in z, we sum up µ n s supported on disjoint annuli: Lemma 5.5. For d 3 and ρ (, ), let n j = (d ) d j, and define the Beltrami coefficient µ by r j = ρ /n j, j =,,,... µ(z) = ( z/ z ) n j, rj < z < r j+, j N, while for z < ρ /n and for z >, we set µ(z) =. With these choices, (i) µ = (z d ) µ + µ χ A(r,r ) and (ii) Cµ(z d ) = dz d Cµ(z) d d [ρ/d ρ ] z, z >. In particular, for z > we have

25 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 5 (iii) Cµ(z) = d d [ρ/d ρ ] v(z), with (iv) Sµ(z) = d d [ρ/d ρ ] v (z), where v = v d is the lacunary series in (5.7). Proof. Claim (i) is clear from the construction. Inserting (i) into (5.9) and using Lemma 5.4 gives (ii). This agrees with the functional equation (5.6) up to a constant term in front of z which leads to (iii). Finally, (iv) follows by differentiation. Remark 5.6. The d = case of Lemma 5.5 is somewhat different since the vector field v does not vanish at infinity, so v is not the Cauchy transform of any Beltrami coefficient. With the choice n j = j+, (ii) and (iii) hold up to an additive constant, while (iv) holds true as stated. Differentiating (5.7), we see that v (z) = z (d )dn (d )dn d n+ n = (d ) z (d )dn + b d n for some function b B, which implies σ (v (z)) = (d ) d log d. Therefore, the Beltrami coefficient µ = µ d from Lemma 5.5 satisfies σ (Sµ) = 4[ρ/d ρ ]. log d Fixing d and optimising over ρ (, ), simple calculus reveals that the maximum is obtained when ρ = d d d. For this choice of ρ, where c d is the constant from (5.4). Moreover, v (z) = c d Sµ(z) (5.) σ (Sµ) = 4d (d ) d d log d (5.) obtains its maximum (over the natural numbers) at d =, in which case σ (Sµ ) >.8793, with µ = χ D.

26 6 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE This construction proves Theorem.. One can proceed further from these infinitesimal bounds and use (5.) to produce quasicircles with large dimension. This takes us to Theorem 5.. Proof of Theorem 5.. By the extended λ-lemma, the conformal maps ϕ t : D A Pt ( ), admit quasiconformal extensions H t : C C, which depend holomorphically on t D. Since the Beltrami coefficient µ Ht function of t, the vector-valued Schwarz lemma implies that µ Ht = tµ + O(t ) is a holomorphic L -valued for some Beltrami coefficient µ χ D. Neumann series in Sµ Ht, c.f. (3.), we get By developing ϕ t = z H t as a Sµ (z) = v (z), z D, for the infinitesimal vector field v = dϕt dt t=. On the other hand, if µ d is the Beltrami coefficient from Lemma 5.5, it follows from (5.) that µ # d := c d µ d also satisfies Sµ # d (z) = v (z) in D. Then the Beltrami coefficient µ µ # d is infinitesimally trivial, and by [, Lemma V.7.], we can find quasiconformal maps N t which are the identity on the exterior unit disk and have dilatations µ Nt = t(µ µ # d ) + O(t ), t <. Therefore, we can replace H t with H t Nt of ϕ t with dilatation µ Ht N t as desired. This concludes the proof. to obtain an extension = tµ # d + O(t ) (5.) Remark 5.7. (i) One can show that for d, the Beltrami coefficient µ # d constructed in Lemma 5.5 is not infinitesimally extremal which implies that the conformal maps ϕ t (with t close to ) admit even more efficient extensions (i.e. with smaller dilatations). One reason to suspect that this may be the case is that µ # q d is not of the form q for some holomorphic quadratic differential q on the unit disk; however, this fact alone is insufficient. It would be interesting to find the dilatation of the most efficient extension, but this may be a difficult problem. For more on Teichmüller extremality, we refer the reader to the survey of Reich [38].

27 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 7 (ii) Let M shell be the class of Beltrami coefficients of the form ( ) z nj χ z A(ri,r i+ ), r < r < r < <. j= One can show that Σ > sup σ (Sµ) = max 4d (d ) d µ M shell d> d log d.8794 where the maximum is taken over all real d >. 6. Fractal approximation In this section, we present an alternative route to the upper bound for the asymptotic variance of the Beurling transform using (infinitesimal) fractal approximation. We show that in order to compute Σ = sup µ χd σ (Sµ), it suffices to take the supremum only over certain classes of dynamical Beltrami coefficients µ for which McMullen s formula holds, i.e. d dt H. dim ϕ t (S π ) = lim v t= R + π log(r ) µ(re iθ ) dθ (6.) where ϕ t is the unique principal homeomorphic solution to the Beltrami equation ϕ t = tµ ϕ t and v µ := dϕt dt t= is the associated vector field. By using the principal solution, we guarantee that v µ vanishes at infinity which implies that v µ = Cµ. We will use this identity repeatedly. (In general, when ϕ t does not necessarily fix, v µ and Cµ may differ by a quadratic polynomial Az + Bz + C.) Consider the following classes of dynamical Beltrami coefficients, with each subsequent class being a subclass of the previous one: M B = f M f (D) consists of Beltrami coefficients that are eventuallyinvariant under some finite Blaschke product f(z) = z d i= z a i a i z, i.e. Beltrami coefficients which satisfy f µ = µ in some open neighbourhood of the unit circle. M I = d M I(d) consists of Beltrami coefficients that are eventuallyinvariant under z z d for some d. M PP = d M PP(d) consists of µ M I for which v µ arises as the vector field associated to some polynomial perturbation of z z d, again for some d. For details, see Section 6.3.

28 8 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE Theorem 6.. [7] If µ belongs to M B, then the function t H. dim ϕ t (S ) is real-analytic and (6.) holds. While McMullen did not explicitly state the relation between Hausdorff dimension and asymptotic variance for M B, the argument in [7] does apply to conjugacies ϕ t induced by this class of coefficients. Note that the class of polynomial perturbations is explicitly covered in McMullen s work, see [7, Section 5]. We show: Theorem 6.. Σ = sup σ (Sµ) = sup σ (Sµ). µ M I, µ χ D µ M PP, µ χ D In view of Theorem 6., the first equality in Theorem 6. is sufficient to deduce Theorem.4. With a bit more work, the second equality also gives the following consequence: Corollary 6.3. For any ε >, there exists a family of polynomials z d + t(a d z d + a d 3 z d a ), t ( ε, ε ), such that each Julia set J t is a k(t)-quasicircle with H. dim(j t ) + (Σ ε)k(t). 6.. Bounds on quadratic differentials. To prove Theorem 6., we work with the integral average σ 4 rather than with σ. The reason for shifting the point of view is due to the fact that the pointwise estimates for v µ (z) = 6 µ(w) dm(w) (6.) π (w z) 4 D are more useful than the pointwise estimates for v, as we saw in Section when we invoked Hardy s identity. According to Lemma., σ (v µ) = σ4(v µ) = 8 v lim sup µ 3 R + A(R,) ρ (z) ρ (z)dm (6.3) where ρ (z) = /( z ) is the density of the hyperbolic metric on D and ffl f(z) ρ (z)dm denotes the integral average with respect to the measure ρ (z)dm. (Note that we are not taking the average with respect to the hyperbolic area ρ (z)dm.) We will need two estimates for v /ρ. To state these estimates, we introduce some notation. For a set E C, let E denote its reflection in

29 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 9 the unit circle. The hyperbolic distance between z, z D is denoted by d D (z, z ). The following lemma is based on ideas from [5, Section ] and appears explicitly in [5, Section ]: Lemma 6.4. Suppose µ is a measurable Beltrami coefficient with µ χ D and v is given by (6.). Then, (a) v /ρ 3/ for z D. (b) If d D (z, supp(µ) ) L, then (v /ρ )(z) Ce L, for some constant C >. Proof. A simple computation shows that if γ is a Möbius transformation, then γ (z )γ (z ) (γ(z ) γ(z )) = (z z ), for z z C. (6.4) The above identity and a change of variables shows that v µ (γ(z)) γ (z) = v γ µ(z), (6.5) analogous to the transformation rule of a quadratic differential. In view of the Möbius invariance, it suffices to prove the assertions of the lemma at infinity. From (6.), one has v lim µ z ρ (z) = 3 π µ(w)dm(w), D which gives (a). For (b), recall that d D (, z) = log( z ) + O() for z <. Then, v lim µ z ρ (z) 3 π as desired. { Ce L < w <} dm(w) = O(e L ) Remark 6.5. Loosely speaking, part (b) of Lemma 6.4 says that to determine the value of v µ /ρ at a point z D, one needs to know the values of µ in a neighbourhood of z. More precisely, for any ε >, one may choose L > sufficiently large to ensure that the contribution of the values of µ outside {w : d D (z, w) < L} to (v µ /ρ )(z) is less than ε. In particular, if µ and µ are two Beltrami coefficients, supported on the unit disk and bounded by that agree on {w : d D (z, w) < L}, then (v µ /ρ )(z) (v µ /ρ )(z) < ε. This simple localisation principle will serve as the foundation for the arguments in this section.

30 3 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE Lemma 6.6. Given an ε >, there exists an < R(ε) <, so that if < z d < R(ε), then v (z d ) µ z ρ (z) z v d µ ρ Note that R(ε) is independent of the degree d. (z d ) < ε. (6.6) Proof. Differentiating (5.9) three times yields d z d v µ (z d ) v (z d ) µ (z) d z ω(z d ), where ω(z)/ρ (z) as z +. The lemma follows in view of the convergence (/d) (ρ (z)/ρ (z d )) as z d +, which is uniform over d. Alternatively, one can use a version of Koebe s distortion theorem for maps which preserve the unit circle, see [5, Section ]. 6.. Periodising Beltrami coefficents. We now prove the first equality in Theorem 6. which says that Σ = sup µ MI, µ χ D σ (Sµ). In view of Lemma., given a Beltrami coefficient µ with µ χ D and ε >, it suffices to construct an eventually-invariant Beltrami coefficient µ d which satisfies µ d χ D and σ 4(v µ d ) σ 4(v µ) ε. (6.7) Proof of Theorem 6., first equality. Given an integer d, we construct a Beltrami coefficient µ d M I (d). We then show that µ d satisfies (6.7) for d sufficiently large. Step. Using the definition of asymptotic variance (6.3), we select an annulus for which A = A(R, R ) D, R = R /d, R, σ 4(v µ) ε/3 8 3 A v µ ρ (z) ρ (z)dm. Let A = A(r, r ) be the reflection of A in the unit circle. We take µ d = µ on A and then extend µ d to {z : r < z < } by z d -invariance. On z < r, we set µ d =.

31 ASYMPTOTIC VARIANCE OF THE BEURLING TRANSFORM 3 Step. The estimate (6.7) relies on an isoperimetric feature of the measure ρ (z)dm, which we now describe. It is easy to see that the ρ (z)dm-area of an annulus A(S, S ), < S < S <, is π times the hyperbolic distance between its boundary components. In particular, the ρ (z)dm-area of A is roughly π log d. By contrast, for a fixed L >, the ρ (z)dm-area of its periphery L A := {z A, d D (z, A ) < L} is 4πL (provided that log d L). We conclude that the ratio of ρ (z)dmareas of L A and A tends to as d. Step 3. By part (b) of Lemma 6.4, v µ d ρ (z) v µ ρ (z) Ce L, z A \ L A, (6.8) while v µ d ρ (z) v µ ρ (z) 3, z LA. (6.9) Putting the above estimates together gives (for large degree d) 8 v µ d ρ (z) ρ (z)dm 8 v µ 3 A ρ (z) ρ (z)dm < ε/3. (6.) 3 A Step 4. Set R k := R /dk and A k = A(R k+, R k ). By Lemma 6.6, 8 v µ d (z) ρ (z)dm 8 v µ d 3 (z) ρ (z)dm < ε/3, 3 A k ρ A ρ which implies that σ 4 (v µ d ) σ 4 (v µ) ε as desired. Remark 6.7. (i) The isoperimetric property used above does not hold with respect to the hyperbolic area ρ (z)dm. In fact, as we explain in Section 7, periodisation fails in the Fuchsian case. (ii) Refining the above argument shows that one can take ε = C/ log d in (6.7), but we will not need this more quantitative estimate.

32 3 K. ASTALA, O. IVRII, A. PERÄLÄ, AND I. PRAUSE 6.3. Polynomial perturbations. To show the second equality in Theorem 6., we need a description of vector fields which arise from polynomial perturbations of z z d, d. Lemma 6.8. [7, Section 5] Consider the family of polynomials P t (z) = z d + t Q(z), deg Q d, t < ε. (6.) Let ϕ t : D = A P ( ) A Pt ( ) denote the conjugacy map and v = dϕt dt t= be the associated vector field as before. Then, v(z) = v k (z) = z Q(z dk ) d d k z, z D. (6.) dk+ k= k Let V PP (d) be the collection of holomorphic vector fields of the form (6.), with deg Q d. From this description, it is clear that each V PP (d), d is a vector space, but the union V PP = d V PP(d) is not. Observe that two consecutive terms in (6.) satisfy the periodicity relation v k+ (z) = dz d v k(z d ), (6.3) which is of the form (5.9) provided that Cµ() =. Similarly, we define M PP = d M PP(d) as the class of Beltrami coefficients that give rise to polynomial perturbations. More precisely, M PP (d) consists of eventually-invariant Beltrami coefficients µ M I (d) for which v µ = Cµ V PP (d) A truncation lemma. In order to approximate infinite series by finite sums, we need some kind of a truncation procedure. To this end, we show the following lemma: Lemma 6.9. Suppose µ is a Beltrami coefficient satisfying µ and supp µ A(ρ, ρ ), with < ρ < ρ <. Given a slightly larger annulus A(ρ, r ) and an ε >, there exists a Beltrami coefficient µ satisfying (i) supp µ A(ρ, r ), (ii) µ µ < ε, (iii) v µ () = v µ (), (iv) v µ is a polynomial in z.