ASYMPTOTIC EXPANSION FOR THE INTEGRAL MIXED SPECTRUM OF THE BASIN OF ATTRACTION OF INFINITY FOR THE POLYNOMIALS z(z + δ)

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1 ASYMPOIC EXPANSION FOR HE INEGRAL MIXED SPECRUM OF HE BASIN OF ARACION OF INFINIY FOR HE POLYNOMIALS + δ ILIA BINDER Abstract. In this paper we establish asymptotic expansion for the integral mixed spectrum of the basin of attraction of infinity for the polynomials + δ. It allows us to give another prove of the Ruelle s expansion for the dimension of the Julia set, as well to estimate the prevalent rate of rotation with respect to the Hausdorff measure of maximal dimension.. Introduction he asymptotic expansion for the Hausdorff dimension of the Julia set of + δ was obtained by Ruelle in [0], who also proved that it depends analytically on δ. In his proof Ruelle used the dynamical ζ-function. It was further improved by Collet, Dobbertin, and Moussa in [6] by the utiliation of thermodynamic formalism. Some additional estimates on the asymptotic expansion were given by Baker and Stallard in []. hey used the estimates of the spherical derivative of the polynomial + δ. Let us now recall the definition of the Integral mixed spectrumsee [3] for motivation and properties of the spectrum. Let Ω be a simply connected domain, 0 Ωandφ : D Ω be the Riemann map with φ0 = 0, φ 0 > 0. Definition. he integral mixed spectrum of the domain Ω is m Ω = lim sup r log r φ ζ d ζ log r In this paper we obtain the asymptotic expansion for the integral mixed spectrum of quadratic Julia sets. heorem. m Aδ = δ 6 log + Rδ δ δ cos argδ + O δ 4, 64 log 3 log where A δ is the basin of attraction of infinity for the polynomial f δ = + δ. Combined with the Bowen s formula it allows us to estimate the Hausdorff dimension of the Julia sets, re obtaining the asymptotic from [6].. Corollary. d =dim A δ =+ δ 6 log δ 3 cos arg δ + O δ 4 64 log Date: August 3, 008. Key words and phrases. Harmonic measure, Boundary behavior of conformal maps, Polynomial Julia Sets, Prevalent rate of rotation. Supported in part by NSERC Discovery grant

2 heorem also has an interesting geometric consequence. By [3], we can rewrite the dimension spectrum see the definition below in terms of the integral means spectrum. hen the asymptotic of the previous theorem allows us to describe the prevalent rotation on the Julia set. heorem. Let κ denote the Hausdorff measure of the full dimension on the Julia set. hen at κ-a.e. the Julia set behaves like a logarithmic spiral with parameter γ = δ 3 64 log sin arg δ + O δ 4, in the sense that the radial behavior of the Riemann map is arg φ δrζ γ log. Proofs r. Let us start with the proof of the corollary. We need to start with a few definitions and statements from [3]. Let x Ω, δ>0, Ω δ be the connected component of the set {y Ω: x y >δ} containing x 0. Define the rotation of the domain Ω near x at distance δ as ρx, δ=exp inf argy x, y Ω δ, y x =δ where the branch of the arg is selected so that π <argx 0 x π. Definition. Let Ω be a domain with the boundary invariant under hyperbolic dynamics. hen the dimension mixed spectrum of Ω is defined as { } log ωbx, δ log rhox, δ fα, γ=dim x Ω : lim = α; : lim = γ. δ 0 log δ δ 0 log δ Here Bx, δ is a disk of radius δ around x, dim is the Hausdorff dimension. heorem 3 [3]. Let Ω be a Jordan repeller. hen Integral and dimension mixed spectra are related by a Legendre type transform: f Ω α, γ=inf m Ω =sup α,γ αm Ω+ αr γi+α fω α, γ αr+γi α f Ω α, γ and m Ω are real-analytic. 3 Either the boundary Ω is real analytic and all the mixed spectra are trivial, or m Ω is a strictly convex function, f Ω α, γ is strictly concave on the domains where it is not equal to. In particular, the rotational part of the integral mixed spectrum is nontrivial iff the real part is nontrivial. Lemma. Let Ω be the basin of attraction of infinity of a hyperbolic polynomial of degree d. hen m Ω = log d P R log F φ log B. α.

3 Proofofthecorollary. A δ is a quasi circle. So, by heorem 3. from [8] see also [9], m Aδ d =d. Using the asymptotic from heorem, it can be rewritten as d δ 6 log + d3 δ 3 64 log d δ cos argδ + Od δ 4 =d. 3 log he statement of the corollary is just the asymptotic expansion for the solution of the last equation. Let us now prove heorem. Proof of heorem. As established in [3], the dimension spectrum is a strictly concave function because m Aδ 0, so 0. So there is a unique point α 0,γ 0 with fα 0,γ 0 =dima δ =:d. So, by the definition of the dimension spectrum, for any ɛ { dim A δ : there exists a sequence r n 0 such that } ρ, r n lim γ 0 >ɛ <d. n log r n So to prove the theorem, we need to establish the asymptotic for γ 0. But fα 0,γ 0 =α 0 md α 0 d + α + d. So, the function ld 0,d =α 0 md 0 + id α 0 d 0 γ 0 d + d 0 + α 0 reaches its minimum at d 0 = d, d =0. Itmeansthat d l 0 d, 0 = d l d, 0 = 0. hus i m d γ 0 = d, 0 m d 0 d, 0. By Corollary, d =+O δ. Hence, by the expansion in heorem, i m d, 0 = δ d sin arg δ + Od δ 4 = δ sin arg δ d 64 log 64 log + O δ 4 and m = O δ. d 0 Now the theorem follows from the last two equations. We can now turn to the proof of the. o get the asymptotic for the m Aδ, let us first get the asymptotic for the Riemann map of the A δ. Let φ δ be the Riemann map φ δ : D A δ conjugating + δ and : φ δ+δφ =φ δ, φ =. For the existence of such a map, see, for example, [5], Chapter VIII. Since the map φ δ satisfies the functional equation, it is analytic in both and δ. By the theorems of Mañé, Sad, and Sullivan see [7],Sullivan and hurston see [] and Bers and Royden see [], the family φ δ can be extended to an analytic family of quasiconformal homeomorphisms of Ĉ. So we can write φ δ = + δg +δ g +δ 3 g 3, δ, 3

4 where the maps g,g are independent on δ, and g 3, δ is bounded independent on δ in the α-hölder metric for any α<. Let us draw a few consequences of. Lemma. 3 Proof. By the equation, g = +g g =g +g +g. φ δ = + δg +δ g +Oδ 3 =φ δ φ δ +δ = + δg +δ g +Oδ 3 + δ + g + δ g +Oδ 3 = + δg + + g + δ g +g +g + Oδ 3 Now all the statements of the lemma may be obtained by comparing the coefficients of δ k k =,. Let W = k k be the Weierstrass function. Lemma 3. 4 g = W. Proof. Let g = c k k. hen, by, c k k = + Hence we have c k k. c 0 = c k =0, k c k+ = c k k 0. So, by induction we get c l = l l 0 and all other coefficients are equal to 0. We say that two α-hölder functions f, f on are equivalent, f f if f =f + u u forsomeα-hölder function u. he main use of this equivalence is the fact that if f f,thenp f =P f. Here the pressure is taken with respect to. Let us derive a few equivalences for the coefficients in the expansion of φ δ. Lemma g g 4g 3 + g g

5 Proof. he relation 5 follows from, which can be rewritten as g g = + g. o get 6 we write in the form g g = 3g + + 4g. If we write 3 in the form we get, using 6, which gives us 7. g g g g g = g + g + g, g g 3 o compute the asymptotic of m Aδ, we will need the asymptotic of ψ δ =log φ δ + δ. Lemma 5. 8 ψ δ =ψ 0 +O δ 3, where ψ 0 δ + δ W 34. Proof. ψ δ =log +δ g + + δ g +O δ 3 = δ g + + δ g δ g + + O δ 3. By 5, δ g + =. By equations 6 and 7 δ g δ g + 3 δ g = δ 4 W. his proves the lemma. Now we can start the estimation of m Aδ. By Lemma, 9 m Aδ w = log P R w log F δ φ δ = log P [ R w log φ δ + δ ] = log P wψ δ. Let us note that if f f, then R wf R wf. So, by Lemma 5 5

6 0 P wψ δ = P R wψ 0 + O δ 3 = P δw R + δ ww 34 w Let α =argδ, β =argw, and denote hen, ψ = Reiα β ψ = R e iα β W 3 4 ψ 3 = R wψ δ ψ 0 δ 3. P wψ δ = P w δ ψ + δ ψ + δ 3 ψ 3. + O w δ 3. Nowweusethefactthat P tγ = log + lim e tsnγ, n n where S n Γ= n Γk. It follows from the first description of the pressure in [4]. hus, since the convergence in is uniform, we can write 3 P tγ = log + t lim S n Γ+ t n n lim S n Γ n n Applying 3 to we get + t3 6 lim n S n Γ 3 + O n t 4 sup S n Γ 4. 4 P wψ δ = log + lim S n ψ δ + lim S n ψ δ n n n n + 3 lim S n ψ δ 3 + O 4 sup S n ψ δ 4. 6 n n Let us note that So we can rewrite 4 as 5 P wψ δ = log + lim S n ψ δ =logm 0 = 0. n n [ δ lim S n ψ + δ 3 lim n n n n 6 3 δ 3 lim n he desired asymptotic for the m follows now from the next lemma. 6 n ] S n ψ ψ S n ψ 3 + O δ 4.

7 Lemma 6. 6 n ψ S = n 4 7 lim S n ψ S n ψ = n n 3 cos α 8 S n ψ 3 = 3 n cosα β. 3 Proof. Let us rewrite ψ and ψ hen ψ e iθ = cosθ α + β ψ e iθ = c k cos k +θ + β α c k = S n ψ e i θ= n cos k θ + β α. cosaθ + bcoscθ + d 0ifandonlyifa = ±c for integer a, c. { 8, k =0 k, k > 0. his implies that n and S n ψ 3 = 8 S n ψ = 4 n k,k,k 3 =0 cos k θ + β α = n 4 cos k θ + β αcos k θ + β αcos k 3 θ + β α. Since cos x cos y = cosx + y+cosx y, the integral cosk θ + β αcos k θ + β αcos k 3 θ+β α is not equal to 0 if and only if k = k + k 3,or k = k + k 3,or k 3 = k + k. So S n ψ 3 = 3 n 8 cos k+ θ + β αcos k θ +β α = 3 n cosβ α. 3 o prove 7, let us recall that by the ergodic theorem, lim S n ψ ψ = ψ ψ + ψ ψ k + n n ψ ψ k. Letusnotethatallthecoefficientsinfrontofθ in the expansions of ψ and ψ are odd, so all the integrals ψ ψ k and ψ ψ k areequalto0.itmeansthat lim S n ψ ψ = ψ ψ = cosθ + β αcosθ + β α = cos α. n n 6 3 Now we plug the expansion from the last lemma in equation 5, and then in equation 9 to get the asymptotic expansion in heorem. 7

8 References [] I. N. Baker and G. M. Stallard. Error estimates in a calculation of Ruelle. Complex Variables heory Appl., 9:4 59, 996. [] Lipman Bers and H. L. Royden. Holomorphic families of injections. Acta Math., 573-4:59 86, 986. [3] I. Binder. Harmonic measure and rotation of planar domains. Preprint, 998. [4] Rufus Bowen. Equilibrum States and the Ergodic heory of Anosov Diffeomorphisms, volume 470 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 975. [5] Lennart Carleson and heodore W. Gamelin. Complex dynamics. Universitext: racts in Mathematics. Springer- Verlag, New York, 993. [6] P. Collet, R. Dobbertin, and P. Moussa. Multifractal analysis of nearly circular Julia set and thermodynamical formalism. Ann. Inst. H. Poincaré Phys.héor., 56:9, 99. [7] R. Mañé, P. Sad, and D. Sullivan. On the dynamics of rational maps. Ann. Sci. École Norm. Sup. 4, 6:93 7, 983. [8] N. G. Makarov. Fine structure of harmonic measure. Algebra i Anali, 0: 6, 998. [9] Ch. Pommerenke. On boundary sie and conformal mapping. Complex Variables heory Appl., -4:3 36, 989. [0] David Ruelle. Spectral properties of a class of operators associated with conformal maps in two dimensions. Comm. Math. Phys., 443: , 99. [] Dennis P. Sullivan and William P. hurston. Extending holomorphic motions. Acta Math., 573-4:43 57, 986. Department of Mathematics, University of oronto, oronto, ON M5S E4, CANADA address: iliamath.toronto.edu 8