ON THE HAUSDORFF DIMENSION OF THE SIERPIŃSKI JULIA SETS

We estimate the Hausdorff dimension of hyperbolic Julia sets of maps from the well-known family Fλ,n(z) = z n + λ/z, n ≥ 2, λ ∈ C \ {0}. In particular, we show that dimH J(Fλ,n) = O(1/ ln |λ|) for large |λ|, and dimH J(Fλ,n) = 1 + O(1/ lnn) for large n in the three cases: when J(Fλ,n) is a Cantor set, a Cantor set of quasicircles and a Sierpiński curve.


Introduction and results
This paper studies the Hausdorff dimension of hyperbolic Julia sets of rational maps of the form F λ,n (z) = z n + λ z n , λ ∈ C \ {0}, n = 2, 3, . . . .
The family has been extensively studied in the last ten years, starting from [1,4,5,6,11,13] by Devaney and his collaborators, and continued in e.g.[8,9,10,12,14,15,16,25,28] (see also a recent survey [7] and references therein).Note that for small |λ|, the map F λ,n can be regarded as a singular perturbation of the polynomial z → z n .The Julia set J(f ) of a rational map f : C → C of degree greater than 1 is defined as the set of points z in the Riemann sphere C such that for every neighborhood U of z the family of iterations {f k | U } k>0 is not normal in the sense of Montel.The Julia set is an finvariant, closed perfect set in C. The connected components of C \ J(f ) are called Fatou components.Periodic Fatou components of a rational map (i.e.Fatou components U with f p (U ) = U for some p ∈ N) can be classified into the following four types: components of basins of an attracting periodic point (then f pk | U → z 0 as k → ∞, where z 0 ∈ U , f p (z 0 ) = z 0 , |(f p ) (z 0 )| < 1), basins of a parabolic periodic point (then f pk | U → z 0 as k → ∞, where z 0 ∈ ∂U , f p (z 0 ) = z 0 , |(f p ) (z 0 )| = 1), and rotation domains (Siegel discs or Herman rings), where f p | U is conformaly conjugated to an irrational rotation on a disc or annulus, respectively.It is well-known (see [29]) that for a rational map, every Fatou component is eventually periodic under the iteration of f .For more information on the general theory of the dynamics of rational maps we refer to [3,23].
The dynamics of a rational map is strongly influenced by the behavior of its critical points.For the map F λ,n , we have n(z 2n − λ) z n+1 .Hence, the set of critical points of F λ,n consists of the superattracting fixed point at infinity (i.e.F λ,n (∞) = ∞, F λ,n (∞) = 0), its preimage 0, and 2n "free" critical points, which are 2n-th complex roots of λ.Denote by B λ,n the immediate basin of attraction to ∞, i.e. the connected component of the set {z ∈ C : F k λ,n (z) → ∞ as k → ∞} containing ∞, and let c λ,n = 2n √ λ be one of 2n "free" critical points.The map F λ,n has only two finite critical values Note that due to the symmetry F λ,n (−z) = (−1) n F λ,n (z), the dynamical behavior of both critical values is the same.We are particularly interested in the case, when the critical value v λ,n tends to infinity under the iteration of F λ,n (then −v λ,n also tends to infinity).Notice that this condition implies that the map F λ,n is hyperbolic, i.e. all its critical points are contained in the basins of attracting periodic points.Moreover, we have F k 0 λ,n (v λ,n ) ∈ B λ,n for some k 0 ≥ 0. In [13, Theorem 0.1], the following classification was established.
Then exactly one of the three following possibilities holds.
Note that in the first case we have T λ,n = B λ,n , where T λ,n is the component of F −1 λ,n (B λ,n ) containing 0, while in the two remaining cases there holds T λ,n = B λ,n .See Figure 1 for examples.For given n ≥ 2, a domain of parameters λ, where one of the three types of behavior appear, is called, respectively, Cantor locus, McMullen domain (since the phenomenon was first observed by McMullen in [21]) and Sierpiński hole.The Cantor locus forms an external region in the λ-parameter plane and for n ≥ 3 there is a unique McMullen domain containing a punctured neighborhood of λ = 0 (for n = 2 McMullen domains do not appear).See e.g.[6,28].More precisely, the following estimates were proved in [6].
In this paper we improve the estimates from Theorem 1.2, showing the following.(a) (c) , and J(F λ,n ) is a Sierpiński curve.In particular, the estimates (a) and (b) show that for large n, the "inner" McMullen domain and the "outer" Cantor locus occupy the whole punctured complex plane C \ {0} outside a thin annulus around the circle {|z| = 1/4}.
Despite a number of results on topological and combinatorial aspects of the dynamics of the maps F λ,n , their dimensional properties seem to be almost not described.Recently, Wang and Yang proved in [31] that for n ≥ 3 we have (1) dim where dim H denotes the Hausdorff dimension and ∂ is the boundary.Note that by [25], ∂B λ,n is a Jordan curve unless J(F λ,n ) is a Cantor set.
In this paper we estimate the Hausdorff dimension of hyperbolic Julia sets of F λ,n .A special attention is given to the three cases described in Theorem 1.1 for large parameter n.The first result is the following general estimate.
Theorem B together with Theorem A implies immediately the following corollary.
Note that if J(F λ,n ) is a Cantor set of quasicircles or a Sierpiński curve, then we have dim H J(F λ,n ) ≥ 1 by topological reasons.Indeed, in this case J(F λ,n ) contains non-trivial continua (curves) of topological dimension 1, so by Szpilrajn's Theorem (see [30]), its Hausdorff dimension is at least 1.Therefore, the estimate in Theorem B can be non-trivial only outside the McMullen domain and Sierpiński holes.In fact, in the case of a Cantor set of quasicircles we have a better lower estimate, which follows from a result by Haissïnsky and Pilgrim [18].
We include the proof of this theorem in Section 4. Note that Theorem 1.4 and (1) give dim H J(F λ,n ) > dim H ∂B λ,n for λ close to 0.
The main results of the paper are gathered in the following theorem.
) is a Cantor set of quasicircles and ) is a Sierpiński curve and Remark 1.5.The cases (b) and (c) occur, respectively, for n ≥ 39 and n ≥ 63018.
Theorem C implies immediately the following two corollaries.
In particular, Theorem C provides examples of the Julia set of F λ,n being a Cantor set, a Cantor set of quasicircles or a Sierpiński curve of Hausdorff dimension arbitrarily close to 1.
Remark 1.6.For given n ≥ 3, the assumption in the Theorem C part (b) does not hold when λ is close enough to 0. In [9], the authors proved that for n = 2 the Julia sets of F λ,2 converge to the closed unit disc in the Hausdorff metric as λ → 0 and show that this is not the case for n > 2. It would be interesting to determine the behavior of the Hausdorff dimension of J(F λ,n ) when λ → 0.
Remark 1.7.By [26], if a rational map f = f λ depends analytically on a parameter λ ∈ Λ for an open domain Λ ⊂ C, such that f λ is hyperbolic for λ ∈ Λ (i.e.Λ is an hyperbolic component), then the Hausdorff dimension depends real-analytically on λ ∈ Λ.This implies that for given n, dim H J(F λ,n ) depends real-analytically on λ within parameter components being Cantor locus, McMullen domain or Sierpiński holes.In particular, this holds for the sets of parameters λ described in the three parts of Theorem C.
For the definition and basic properties of the Hausdorff dimension we refer to [17,20].To estimate the Hausdorff dimension of the Julia set of F λ,n we use methods of the thermodynamic formalism [2,26,27].A standard tool is the classical Bowen's formula (see e.g.[24, Sections 9.1 and 12.5]).
Theorem 1.8 (Bowen's formula).Let f : C → C be a rational map of degree greater than 1.If f is hyperbolic (i.e.all critical points of f are in the basins of attracting periodic points), then the Hausdorff dimension of the Julia set of f is equal to the unique zero of the pressure function for t > 0, where z is an arbitrary point in J(f ).
It should be noticed that there exist a number of effective algorithms for calculating the Hausdorff dimension of hyperbolic Julia sets numerically, see e.g.[19,22].
The plan of the paper is as follows.In Section 2 we describe a convenient change of coordinates and prove a lemma on the location of the Julia set.In Section 3 we show Theorem A, while Theorem B and Theorem 1.4 are proved in Section 4. The three cases of Theorem C are dealt with, respectively, in Sections 5-7.
Acknowledgment.The authors would like to thank the referees for useful remarks and drawing their attention to the papers [18] and [31].

Change of coordinates
for some branch of the 2n-th root, and consider the map F λ,n in the new coordinates, i.e. the map Note that the map G λ,n exhibits dynamical symmetries We have ), so the dynamical behavior of both critical values ±w λ,n is the same.
By Bλ,n and Tλ,n we will denote, respectively, the immediate basin of attraction to ∞ for the map G λ,n and the component of its preimage under G λ,n containing 0 (note that we may have Bλ,n = Tλ,n ).By (2), the set Tλ,n is the image of Bλ,n under the map z → 1/z.Let (3) The following lemma will be used frequently throughout the paper.
Lemma 2.1.We have In particular, Proof.We check for which z ∈ C there holds for t > 0. We will show that g(t) > 0 for every t > r λ,n .First, note that if We conclude that in both cases g(r λ,n ) ≥ 0.Moreover, if t > r λ,n , then For the map F λ,n , Lemma 2.1 implies the following estimate, which strengthens the escape criterion from [6, Section 2].
In particular, for a given λ and large n, the Julia set of F λ,n is contained in a thin annulus near the unit circle.

Proof of Theorem A
To prove Theorem A, we use Theorem 1.1 together with the following three propositions.First, we consider the Cantor set case.
Proof.In view of Lemma 2.1, it is sufficient to show |w λ,n | > r λ,n for r λ,n from (3).We have To deal with the other two cases, we use the following lemma.
Recall that in the proof of Lemma 2.1 we showed and Bλ,n ⊃ {z : |z| > r λ,n }, Tλ,n ⊃ {z : |z| < 1/r λ,n }.This together with (2) implies Hence, to prove the lemma, it suffices to show Tλ,n = Bλ,n .Let Since G λ,n (U 0 ) ⊂ U 0 , we can define inductively a sequence of sets ) containing U j for j ≥ 0 and Bλ,n = ∞ j=0 U j .Moreover, by (5), the boundary of U j does not contain the critical values ±w λ,n , so G λ,n maps U j+1 onto U j properly.
By (5), ±w λ,n / ∈ U k 0 .Suppose that U j 0 contains a critical value (say w λ,n ) for some (minimal) j 0 > k 0 .Note that G −1 λ,n (∞) = {0, ∞} and the local degree of the map near 0 and near ∞ is equal to n.Hence, by the Riemann-Hurwitz formula, the sets U 1 , . . ., U j 0 are topological discs not containing 0 and the degree of G λ,n on U j 0 is equal to n.
We have (with the convention 1/0 = ∞) are contained in U 0 ⊂ U j 0 −k 0 and are different from λ,n (w λ,n ), so the point G k 0 +1 λ,n (w λ,n ) ∈ U j 0 −k 0 −1 has more than n preimages in U j 0 −k 0 (counting with multiplicities).Therefore, the degree of G λ,n on U j 0 −k 0 (and hence on U j 0 ) is greater than n, which gives a contradiction.We conclude that ±w λ,n / ∈ ∞ j=0 U j = Bλ,n , so Tλ,n = Bλ,n .
The next two propositions consider, respectively, the McMullen and Sierpiński cases.
Let D ⊂ B λ,n be a topological disc in C containing ∞, with a Jordan boundary, such that and F λ,n | ∂D j for j = 1, 2 is a covering of degree d.(The set D can be defined by the use of the Böttcher coordinates on B λ,n .)Let D 3 ⊂ D 2 be a topological disc with a Jordan boundary, such that By the Riemann-Hurwitz formula, the set , such that E is a topological annulus with the boundary consisting of two Jordan curves, E contains all 2n "free" critical points of F λ,n and the degree of F λ,n on E is equal to 2d.Denote, respectively, by D + and D − the "outer" and "inner" component of C \ E and let Then A, A + , A − are topological annuli with boundaries consisting of Jordan curves and the following properties hold.
• the "outer" (resp."inner") component of ∂A + and the "inner" (resp."outer") component of ∂A − are mapped by F λ,n onto the "outer" (resp."inner") component of ∂A as coverings of degree n.Under the above assumptions on F λ,n | A + ∪A − we can use [18, Corollary 2.1], which gives the following estimate on the Hausdorff dimension of the set of points remaining in A under the iteration of F λ,n : ln 2 ln n .

Cantor locus: Proof of Theorem C (a)
The assertion (a) of Theorem C is implied directly by Theorem B and the following proposition.
Cantor set and Proof.The first part follows by Proposition 3.1 and Theorem 1.1.To estimate the Hausdorff dimension, note that by assumption, we have Hence, if w ∈ J(G λ,n ), then by Lemma 2.1, Thus, by (6), for z ∈ J(G λ,n ) and t > 0 we have By the assumption of the proposition, ln n + n−1 2n ln |λ| − n+1 2n ln 2 > 0. Using (7), we complete the proof.

McMullen domain: Proof of Theorem C (b)
The assertion (b) of Theorem C is equivalent the following fact.
) is a Cantor set of quasicircles and Proof.The first part is due to Proposition 3.3 and Theorem 1.1.The lower estimate of the Hausdorff dimension of J(G λ,n ) follows from Theorem 1.4.To prove the upper one, note that by assumption, we have For every 0 < r = 1, the function H(z) = z + 1/z maps the circle of radius r centered at 0 onto the ellipse symmetric with respect to 0, with horizontal and vertical semi-axes of lengths, respectively, r + 1/r and |r − 1/r|.This together with (8) implies . Hence, by Lemma 2.1, we have 2) and contradicts the assumption of the theorem.Hence, for every w ∈ J(G λ,n ).Using this together with (6) and Lemma 2.1 we have, for z ∈ J(G λ,n ) and t > 0, By the assumption of the theorem, we have ln n+ ln |λ| − 2n n−1 ln 2 > 0. The proof is completed by (7).

Sierpiński holes: Proof of Theorem C (c)
In this section we prove the following fact, which is equivalent to the assertion (c) of Theorem C.
To prove the theorem, we will use a number of lemmas.Since the "free" critical points of G λ,n are 2n-th complex roots of the unity, for every z ∈ C \ {0} there exists a "free" critical point u z of G λ,n (not necessarily unique), for which Proof.Take z ∈ C with |z| ≥ 1 and connect z 2n to 1 by a semicircle γ, such that its diameter is the segment joining z 2n to 1. Then the length of γ is equal to π|z 2n − 1|/2 and |ζ| ≥ 1 for every ζ ∈ γ.By (9), on γ there exists a well-defined branch g of (2n)-th root, such that g(z 2n ) = z, g(1) = u z .Then Proof.By trigonometry, Throughout this section, suppose , where λ 1 , . . ., λ n−1 are (n − 1)-th complex roots of −4 −n .First we show that under this assumption, we have To prove (10), note that since It is obvious that it is sufficient to prove the left hand side inequality.By the Mean Value Theorem for the function x → x , which proves (11) and (10).In particular, (10) implies Lemma 7.6.For n ≥ 7, Proof.First we show (13) r λ,n < 4 3 .
To see it, note that by (12), it is sufficient to check We have .
By the Mean Value Theorem for the function x → x n−1 and (11), Using ( 14), ( 15) and (10), we obtain This together with (13) ends the proof.
Proof of Theorem Moreover, if additionally .
The latter formula implies that for k ≥ 2 and every j 1 , .

Figure 1 .
Figure 1.Three types of the Julia sets of F λ,n described by the escape trichotomy.

Figure 2 .
Figure 2. The λ-parameter plane for n = 6.The central region is the Mc-Mullen domain, surrounded by five large Sierpiński holes and an infinite number of smaller ones.The outer region is the Cantor locus.