We show that the supremum for $c$ real of the Hausdorff dimension of the
Julia set of the polynomial $z\mapsto z^d+c$ ($d$ is an even natural number) is
greater than $2d/(d+1)$.Comment: 10 page, 4 figure

Levin, Genadi

Zinsmeister, Michel

English

arXiv

10 page, 4 figuresWe show that the supremum for $c$ real of the Hausdorff dimension of the Julia set of the polynomial $z\mapsto z^d+c$ ($d$ is an even natural number) is greater than $2d/(d+1)$

HAL-Polytechnique

On the Hausdorff dimension of Julia sets of some realpolynomialsGenadi Levin, Michel ZinsmeisterTo cite this version:Genadi Levin, Michel Zinsmeister. On the Hausdorff dimension of Julia sets of some realpolynomials. 10 page, 4 figures. 2012. <hal-00662708>HAL Id: hal-00662708https://hal.archives-ouvertes.fr/hal-00662708Submitted on 24 Jan 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.On the Hausdorff dimension of Julia sets of some realpolynomialsGenadi Levin, Michel Zinsmeister25 janvier 2012Supported in part by IMPAN-BC European Community Centre of Excellence andby the Marie Curie European network CODYRe´sume´We show that the supremum for c real of the Hausdorff dimension of the Juliaset of the polynomial z 7→ zd + c (d is an even natural number) is greater than2d/(d + 1).1 Introduction and statement of the resultThe Julia set of a non-linear polynomial P : C→ C is the set of points havingno neighborhood on which the family of iterates (Pn) is normal. It is a compactnon-empty set which is, except for very special polynomials P , a fractal set. Forc ∈ C we denote by fc the polynomialfc(z) = zd + c,where d ≥ 2 is an even integer number, which we fix. Denote by Jc the Julia setof fc.In the quadratic case d = 2, the values of c for which the Hausdorff dimension(HD) of Jc is big has attracted a lot of attention. It has been known since the pio-neering work of Douady-Hubbard [3] that the Hausdorff dimension of the Julia setis less than 2 for every hyperbolic polynomial. Thus HD(Jc) < 2 outside ∂M andthe (hypothetic) non-hyperbolic components of (the interior)M . We recall thatMstands for the Mandelbrot set, that is the compact subset of c ∈ C such that Jc isconnected. Shishikura [9] was the first to find quadratic Julia sets with Hausdorffdimension 2. He indeed proved that this property holds on a dense Gδ subset of ∂Mor even on a dense Gδ subset on the boundary of every hyperbolic component ofM .1More recently, Buff and Che´ritat [12] have found quadratic Julia sets with positiveLebesgue measure (see also http ://annals.math.princeton.edu/articles/3682).Both Shishikura’s and Buff-Che´ritat’s results are based on the phenomenon ofparabolic implosion which has been discovered and studied by Douady-Hubbard[3]. It should be pointed out that Buff-Che´ritat’s result is very involved and thatwe will make no use of it. There is no doubt that if they exist, values of c ∈ Rsuch that the Julia set Jc has positive measure must be as hard to find as Buff-Cheritat’s ones. The aim of present note is much more modest. Its starting pointis the second author re-readind of Shishikura’s result [10] : it states that if oneimplodes a polynomial with a parabolic cycle having q petals, then the dimensionof its Julia set automatically gets bigger than 2q/(q+1). Shishikura’s result followsfrom this by a Baire argument.Very little is known about Hausdorff dimension of Jc for real c. In particular,it is not known if, for a given degree d,sup{HD(Jc), c ∈ R} = 2.Possible candidates for high dimension are of course (just look at them !) infinitelyrenormalizable polynomials but the analysis seems to be very delicate and atleast no result concerning dimension 2 has been proven so far (for results in theopposite direction, see [1] though). It is for example unknown if the Julia set ofthe Feigenbaum polynomial has dimension 2 or not (see [4]- [5] for the Julia set ofthe Feigenbaum universal map though). The only known result about this set is avery general result of Zdunik [14] : it has dimension bigger than 1.If one tries to use the same ideas as in [9] for real polynomials fc, one im-mediately faces the problem that if fc, for c real, has a parabolic cycle that maybe imploded along the real axis then the number of petals is 1 and this does notimply more than Zdunik’s general result. The only trick of this paper is to makeuse of a virtual doubling of petals when the critical point is mapped to a parabolicpoint (by Lavaurs map). It was inspired by Douady et al paper [13] and impliesthe following theorem, which is the main result of this work.Theorem 1 Let fc(z) = zd + c, d even. Let N be the set of parameters c ∈ R,such that fc has a parabolic cycle of period at least 2 and multiplier 1. Then thereexists an open set Y of R whose closure contains N such that Jc is connected andHD(Jc) >2dd+ 1,for every c ∈ Y .Comment 1 In fact, we prove a stronger statement : hyperbolic dimension [9] ofJc is bigger than 2d/(d + 1).Comment 2 By [11] ( [6], [8] for d = 2), the set of real c such that fc ishyperbolic is dense in R. In particular, hyperbolic parameters c are dense in Y .2Acknowledgment. This work was done during the first author’s one month’svisit at the university of Orle´ans in 2008.2 Proof of the theoremWe fix an even integer d ≥ 2 and consider the family fc(x) = xd+ c, for real c.Then the Julia set Jc is connected if and only if c ∈ [a, b], where a < 0 is such thatf2a (0) is a fixed point, and b > 0 is such that fb has a fixed point with multiplier1. It is sufficient to prove that, given c0 ∈ (a, b) such that fc0 has a neutral cycleof period k > 1 with multiplier 1, that is, with one petal, there is an open set Zaccumulating at c0 for which HD(Jc) > 2d/(d + 1) for c ∈ Z.We begin with three pictures. The first two ones illustrate how to choose theparameter (for d = 2) and how the corresponding Julia set looks like. The thirdone has been kindly drawn for us by the referee and shows the case d = 4.Figure 1 – Choice of the parameterSince c0 ∈ (a, b), the set Jc is connected for c in a small neighborhood U of c0.It is known, that fc has an attracting periodic orbit of period k for c ∈ U on theleft side of c0. Since k > 1, the corresponding filled-in Julia set Kc0 is such that3Figure 2 – The corresponding Julia setits interior has a component ∆ containing 0, and infinitely many preimages of ∆.The boundary ∂∆ contains a parabolic point α of period k. In particular, there isa sequence of preimages of ∆, which intersect the real line and accumulate at α.For definicity, one can assume that α > 0. Then F = fkc0 has the following localform :F (z) = z + a(z − α)2 + b(z − α)3 + ..., (1)where a > 0. This implies that a parabolic implosion phenomenon occurs as → 0, > 0, for the maps fc0+.At this point we digress somewhat and describe briefly the theory of parabolicimplosion.Let δ > 0 be very small and D± be the disks centered at α ± δ with radius δ.The map F sends D− into itself while D+ ⊂ F (D+). This defines, after iden-tification of z with F (z) at the boundary, two cylinders U− = D−\F (D−) andU+ = F (D+)\D+. The fact that U± are actual cylinders is best seen in the ap-proximate Fatou coordinate I : z 7→ −1/(a(z − α)) which sends α to ∞ and4Figure 3 – The case d=4conjugates F to a map which is asymptotically the translation by 1 at ∞ :F∞(w) = I ◦ F ◦ I−1(w) = w + 1 +Aw+O(1|w|2), (2)where A = 1 − b/a2. The real number A is a conformal invariant. In the case ofreal polynomial F which has a parabolic fixed point α with multiplier 1 and witha single critical point in its immediate basin ∆, it is known [9] that A > 0.By Riemann mapping theorem these two cylinders may be uniformized by”straight” cylinders. In other words there exists ϕ± mapping the cylinders U± tovertical strips V± of width 1 conjugating F to the translation by 1. For further usewe notice that, by symmetry, c0 being real, we may assume that ϕ±(z) = ϕ±(z).We also notice that these maps are unique up to post-composition by a real trans-lation. These two maps are called respectively repelling (+) and attracting (-)Fatou coordinates. We normalize them as follows. For every κ > 0 and R > 0,consider two sectors Σ−(κ,R) = {w : Re(w) > R − κ|Im(w)|}, Σ+(κ,R) = {w :Re(w) < −R + κ|Im(w)|}. Then for any κ > 0 there is a big enough R(κ), such5Figure 4 – Fatou coordinatesthat, if we introduce two sectors Σ±(κ) = Σ±(κ,R(κ)), then ϕ± = Φ± ◦ I, whereΦ±(w) = w −A log±(w) + C± + o(1) (3)as w → ∞ within Σ±(κ) respectively. We specify the constants C± and the log-branches in such a way, that Φ± are real for real w. Namely, we choose C− = 0and log− to be the standard log-branch in the slit plane C \ {x ≤ 0}. In turn,let C+ = iApi and log+ to be a branch of log in C \ {x ≥ 0} so that log+(w) =log |w| + ipi for w < 0.Now, we extend ϕ± in the following way. Sinceϕ−(F (z)) = T1(ϕ−(z)) (4)where Tσ denotes the translation by σ, and since every orbit converging to α passesthrough U− exactly once, ϕ− extends uniquely to ∆ to an holomorphic functionstill satisfying (4).It is seen from (4), that the map ϕ− : ∆ → C is a branched covering, withthe critical points at 0 and all its preimages in ∆ by Fn, n > 0, and the criticalvalues at the real numbers ϕ−(0) − n, n ≥ 0. In particular, there exists a simply-connected domain Ω− ⊂ ∆ ∩ H+, where H+ is the upper half-plane, such thatϕ− : Ω− → H+ is a holomorphic homeomorphism. Moreover, the intersecrion ofthe boundary of Ω− with R is the interval (0, α).6Concerning the repelling Fatou coordinate it is best to consider ψ+ = ϕ−1− :V+ → U+. The functionnal relation is nowψ+(T1(z)) = F (ψ+(z)) (5)and we can extend ψ+ to an entire function by putting, for n ∈ Z, ψ+(Tn(z)) =Fn(ψ+(z)) for z ∈ V+. There exists a simply connected domain Ω+ ⊂ H+, suchthat ψ+ : Ω+ → H+ is a homeomorphism.Let now σ be a real number. We define the Lavaurs map gσ on the component ∆of the interior of the filled-in Julia set of fc0 bygσ = ψ+ ◦ Tσ ◦ ϕ−.The ”raison d’eˆtre” of this definition is the following theorem due to Douady andLavaurs ([2]) :Theorem 2 :There exists a sequence of positive n converging to 0 and a sequenceof positive integers Nn such thatgσ(z) = limn→∞fkNnc0+n(z) (6)uniformly on compact sets of ∆.Using (3) with the constants C± and the log-branches specified as above, it is easyto get, that, for every κ, if w tends to ∞ in Σ(κ) := Σ−(κ) ∩ Σ+(κ) ∩H+, theng∞(w) := I ◦ gσ ◦ I−1(w) = w + (σ − iApi) +O( 1|w|), where A is real and positive.Therefore, for every real σ and every κ > κ(σ), the inverse map g−1∞ leaves thesector Σ(κ) invariant and g−n∞ → ∞ as n → ∞. Coming back to the z-plane, weconclude that the branch G = I−1 ◦ g−1∞ ◦ I of g−1σ leaves the set S(κ) = I−1(Σ(κ))invariant, and Gn(z) → α as n →∞, for z ∈ S(κ) and every κ > κ(σ). We have,for w ∈ Σ(κ) :G∞(w) := I ◦G ◦ I−1(w) = w + (−σ + iApi) +O(1|w|), (7)Now, from the definition of gσ(z) and the global properties of the maps ϕ−and ψ+, it follows the existence of a simply-connected domain Ω0 ⊂ Ω−, which ismapped by gσ homeomorphically onto H+ and such that α ∈ Ω¯0. Moreover, fromthe above description, S(κ) ⊂ Ω0, for every κ > κ(σ). Therefore, the branch G ofg−1 which is defined above, extends to a global univalent branch G : H+ → Ω0 ofg−1σ . Since Ω0 ⊂ H+, the iterates Gn : H+ → Ω0, n > 0, converge uniformly oncompact sets in H+ to a unique fixed point in Ω¯0, which must be α.Let us consider the continuous map σ 7→ gσ(0) = ψ+(ϕ−(0) + σ) : if σ runs inthe interval I = {ϕ+(x)−ϕ−(0) : x ∈ U+ ∩R}, then gσ(0) runs over U+ ∩R. It isthus clear, and this is the key point in the proof, that we can choose σ in such away that gσ(0) is a preimage of α : there is j ≥ 1, such that fjc0 ◦ gσ(0) = α. Since0 is a critical point for gσ, taking the inverse image by fjc0 ◦ gσ has the same effectas multiplying the number of petals by d and we may state :7Lemma 2.1 There exists an infinite iterated function system defined on a smallcompact neighborhood B0 of zero and generated by some holomorphic branches off−1c0 and g−1σ such that its limit set has Hausdorff dimension bigger than 2d/(d+1).Comment 3 In fact, the limit set is a subset of a so-called Julia-Lavaurs setdenoted by Jc0,σ. It is defined as follows. The map gσ can be extended in a naturalway from ∆ to the interior of the filled-in Fatou set of fc0 : if fkc0(z) ∈ ∆, setgσ(z) = gσ ◦fkc0(z). Then Jc0,σ is simply the closure of the set of points z for whichthere exists m ∈ N such that gmσ (z) is defined and belongs to J(f0).This lemma together with the above discussion implies the theorem. Indeed, by ageneral property of iterated function systems [7], there exists its finite subsystemwith the Hausdorff dimension of its limit set bigger than 2d/(d+1). On the otherhand, the finite iterated function system persists for fc0+ by (6). To be moreprecise, if {Ij : B0 → Xj , 1 ≤ j ≤ j0} is this finite iterated function system, theneach Ij can be extended to a univelent map to a fixed neighborhood Y of B0 as Ij :Y → Yj . Consider the inverse univalent map I−1j : Yj → Y . Since the convergencein (6) is uniform on compacts in ∆, for every n small enough there is some integerNj > 0 and a compact set Xj,n, so that fNjc0+n : Xj,n → B0 is univalent, too. Now itis clear, that, for every n small enough, the non-escaping set Kn of the dynamicalsystem which consists of a finitely many maps fNjc0+n : Xj,n → B0, 1 ≤ j ≤ j0,has the Hausdorff dimension which is bigger than 2d/(d+ 1). On the other hand,Kn must lie in the Julia set of fc0+n because some iterate of the map fN1N2...Nj0c0+nleaves the set Kn invariant and is expanding on it.Proof of Lemma 2.1. As the first step, let us fix a small enough closed ballB0 around zero, so that it does not contain points of the postcritical set of fc0 .There exists its preimage B′ by F−1 in ∆ ∩ H+. Then we can apply to B′ themaps Gn, n > 0. By the above, B′n = Gn(B′) are pairwise disjoint, compactlycontained in ∆, and B′n → α as n → ∞. Now, for every n ≥ n0, so that B′nlies in a small enough neighborhood U of α, we make “clones” of B′n in U ∩ ∆applying to it F r, r ∈ Z, where F r for r < 0 is a well-defined in U ∩ ∆ branchwhich fixes α . We obtain the sets B′n,r = Fr(B′n). On the second step, we considerthe map f jc0 ◦ gσ from a neighborhood V of 0 onto U , This map is a remifiedcover with the only remification point at 0 of order d. Let U∗ = U \ {x ≥ α},and V ∗ = V ∩ {z : Arg(z) ∈ (0, 2pi/d)}. Denote by h a branch of (f jc0 ◦ gσ)−1from U∗ onto V ∗. Let Bn,r = h(B′n,r). We obtain a system of holomorphic mapsΨ = {ψn,r : B0 → Bn,r}, where ψn,r = h ◦ Fr ◦ Gn ◦ F−1, r ∈ Z, n ≥ n0. If theneighborhood U is chosen small enough, the maps ψn,r extend to univalent mapsin a fixed neighborhood B˜ of B0 into itself. In particular, the compact sets Bn,r arepairwise disjoint and compactly contained in B0. Now, it is quite standard to checkthatΨ form a conformal infinite iterated function system in the sense of [7] (strictlyspeaking, in the hyperbolic metric of B˜, which is equivalent to the Eucledian one8on B0 though). Let us calculate the parameter θ = inf{t : ψ(t) <∞} of Ψ, whereψ(t) =∑(n,r)maxz∈B0 |ψ′n,r(z)|t. The map ψn,r = h◦I−1 ◦F r∞ ◦Gn∞ ◦I ◦F−1. HereF−1 is a univalent map of a neighborhood B˜ of B0 into ∆. Now, routine and well-know calculations based on (2)-(7) show (see e.g. [10]), that, for all r ∈ Z, n ≥ n0and some C, which depends only on a compact set in H+ ∩ ∆, from which w istaken, C−1|r + (−σ + ipiA)n| ≤ |F r∞ ◦Gn∞(w)| ≤ C|r+ (−σ + ipiA)n|, and C−1 ≤|(F r∞◦Gn∞)′(w)| ≤ C. On the other hand, the map h is a composition of a univalentmap with an inverse branch of z1/d. This gives us : C−11 |r+(−σ+ ipiA)n|−1−1/d ≤|ψ′n,r(z)| ≤ C1|r + (−σ + ipiA)n|−1−1/d, for some C1 and every z ∈ B0. It followsthat the series for ψ(t) converges if and only if t > θ = 2d/(d+1), and ψ(θ) =∞.Hence [7], the Hausdorff dimension of the limit set of Ψ is strictly bigger than2d/(d + 1).Inst. of Math., Hebrew University, Jerusalem 91904, Israel,MAPMO, Universite´ d’Orle´ans, BP 6759 45067 Orle´ans Cedex, FranceRe´fe´rences[1] A.Avila and M.Lyubich. Examples of Feigenbaum Julia sets with small Haus-dorff dimension. Dynamics on the Riemann Sphere, in honor of B.Branner,1 :71–87, 2006.[2] A.Douady. Does a Julia set depend continuously on the Polynomial ? Procee-dings of Symposia in Applied Mathematics, 49 :91–135, 1994.[3] A.Douady and J.Hubbard. Etude dynamique des Polynoˆmes complexes :Premie`re partie. Publications mathe´matiques d’Orsay, 84-02 :1–75, 1984.[4] G.Levin and G.Swiatek. Hausdorff dimension of Julia sets of Feigenbaumtype. Communications in Mathematical Physics, 258 :135–148, 2005.[5] G.Levin and G.Swiatek. Measure of the Julia set of the Feigenbaum mapwith high criticality. Ergodic Theory and Dynamical Systems, 30 :855–875,2009.[6] J.Graczyk and G.Swiatek. Generic Hyperbolicity in the Logistic Family. An-nals of Mathematics, 146 :1–52, 1997.[7] R.D. Mauldin and M. Urbanski. Dimension and measure in infinite iteratedfunction systems. Proc. London Math. Soc., 73 :105–154, 1996.[8] M.Lyubich. Dynamics of quadratic polynomials. Acta Mathematica, 178 :185–297, 2007.[9] M.Shishikura. The Hausdorff Dimension of the Boundary of the Mandelbrotset and Julia Sets. Annals of Mathematics, 147 :225–267, 1998.[10] M.Zinsmeister. Fleur de Leau-Fatou et dimension de Hausdorff. Comptes-rendus de l’Acade´mie des Sciences, 326 :1227–1232, 1998.9[11] O.Kozlovsky, W.Shen, and S. van Strien. Density of Hyperbolicity in Dimen-sion one. Annals of mathematics, 166 :145–182, 2007.[12] X.Buff and A.Che´ritat. Ensembles de Julia quadratiques de mesure stric-tement positive. Comptes-rendus de l’Acade´mie des Sciences, 341 :669–674,2005.[13] X.Buff, R.Devaney, A.Douady, and P.Sentenac. Baby Mandelbrot sets areborn in cauliflowers. London Mathematical Society Lecture Notes Series,274 :19–36, 2000.[14] Anna Zdunik. Parabolic orbifolds and the dimension of the maximal measurefor rational maps. Inventiones Mathematicae, 99 :627–649, 1990.10

On the Hausdorff dimension of Julia sets of some real polynomials

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