On the quasiconformal equivalence of dynamical Cantor sets
OON THE QUASICONFORMAL EQUIVALENCE OFDYNAMICAL CANTOR SETS
HIROSHIGE SHIGA
Abstract.
The complement of a Cantor set in the complex plane isitself regarded as a Riemann surface of infinite type. The problem ofthis paper is the quasiconformal equivalence of such Riemann surfaces.Particularly, we are interested in Riemann surfaces given by Cantorsets which are created through dynamical methods. We discuss thequasiconformal equivalence for the complements of Cantor Julia setsof rational functions and random Cantor sets. We also consider theTeichm¨uller distance between random Cantor sets. Introduction
Let E be a Cantor set in the Riemann sphere (cid:98) C , that is, a totally dis-connected perfect set in (cid:98) C . The standard middle one-third Cantor set C is a typical example. We consider the complement X E := (cid:98) C \ E of theCantor set E . It is an open Riemann surface with uncountable boundarycomponents. We are interested in the quasiconformal equivalence of suchRiemann surfaces. In the previous paper [11], we show that the complementof the limit set of a Schottky group is quasiconformally equivalent to X C , thecomplement of C ([11] Theorem 6.2). In this paper, we discuss the quasicon-formal equivalence for the complements of Cantor Julia sets of hyperbolicrational functions and random Cantor sets (see § Theorem I.
Let f be a rational function of degree d > and J be the Juliaset of f . Suppose that f is hyperbolic and J is a Cantor set. Then, thecomplement X J of J is quasiconformally equivalent to X C . As for random Cantor sets, we obtain the followings.
Theorem II.
Let ω = ( q n ) ∞ n =1 and (cid:101) ω = (˜ q n ) ∞ n =1 be sequences with δ -lowerbound. We put (1.1) d ( ω, (cid:101) ω ) = sup n ∈ N max (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) log 1 − ˜ q n − q n (cid:12)(cid:12)(cid:12)(cid:12) , | ˜ q n − q n | (cid:27) . Date : August 30, 2019.2010
Mathematics Subject Classification.
Primary 30C62, Secondary 30F25.
Key words and phrases.
Cantor sets, Quasiconformal maps.The author was partially supported by the Ministry of Education, Science, Sports andCulture, Japan; Grant-in-Aid for Scientific Research (B), 16H03933, 2016–2020. a r X i v : . [ m a t h . C V ] A ug HIROSHIGE SHIGA (1) If d ( ω, (cid:101) ω ) < ∞ , then there exists an exp( C ( δ ) d ( ω, (cid:101) ω )) -quasiconformalmapping ϕ on (cid:98) C such that ϕ ( E ( ω )) = E ( (cid:101) ω ) , where C ( δ ) > is aconstant depending only on δ ; (2) if lim n →∞ log − ˜ q n − q n = 0 , then E ( (cid:101) ω ) is asymptotically conformal to E ( ω ) , that is, there exists a quasiconformal mapping ϕ on (cid:98) C with ϕ ( E ( ω )) = E ( (cid:101) ω ) such that for any ε > , ϕ | U ε is (1+ ε ) -quasiconformalon a neighborhood U ε of E ( ω ) . From above results and a result [11] Theorem 6.2, immediately we obtain;
Corollary 1.1.
Let E be a Cantor set which is a Julia set of a rationalfunction satisfying the conditions in Theorem I. Then, the complement ofthe limit set of a Schottky group G is quasiconformally equivalent to X E . As consequences of Theorem II (1), we obtain;
Corollary 1.2.
Let E ( ω ) be a random Cantor set for ω = ( q n ) ∞ n =1 . Sup-pose that ω has lower and upper bounds. Then, X E ( ω ) is quasiconformallyequivalent to X C . We have also the following.
Corollary 1.3.
Let E be a Cantor set as in Corollaries 1.1 or 1.2. Then, theCantor set E is quasiconformally removable, that is, every quasiconformalmapping on the complement of E is extended to a quasiconformal mappingon the Riemann sphere. It is known ([7] V. 11F. Theorem) that the random Cantor set E ( ω ) for ω is of capacity zero if and only if ∞ (cid:89) n =1 (1 − q n ) − n = 0 . (1.2)Hence if { q n } ∞ n =1 rapidly converges to one as it satisfies (1.2), then X E ( ω ) isnot quasiconformally equivalent to X C because the positivity of the capacityof closed sets in the plane is preserved by quasiconformal mappings (cf. [7]III. Theorem 8 H). In fact, we can say more: Theorem III. If ω does not have an upper bound, then X E ( ω ) is not qua-siconformally equivalent to X C . The proof of Theorem II gives the following.
Corollary 1.4.
Let ω and (cid:101) ω be sequences satisfying the same conditions asin Theorem II (2). Then, the Hausdorff dimension of E ( (cid:101) ω ) is the same asthat of E ( ω ) . Acknowledgement.
The author thanks Prof. K. Matsuzaki for his valu-able comments. This research was partly done during the author’s stay inthe Erwin Schr¨odinger institute at Vienna. He also thanks the institute forits brilliant support to his research.
UASICONFORMAL EQUIVALENCE 3 Preliminaries
Complex dynamics.
We begin with a small and brief introductionof complex dynamics. We may refer textbooks on the topic, e. g. [5] for ageneral theory of complex dynamics.Let f be a rational function of degree d > C . We denote by f n the n times iterations of f . The Fatou set F of f is the set of points in (cid:98) C whichhave neighborhoods where { f n } ∞ n =1 is a normal family. The complement of F , which is denoted by J , is called the Julia set of f .A rational function f is called hyperbolic if it is expanding near J . Moreprecisely, if J (cid:54)(cid:51) ∞ , then f is hyperbolic if there exist a constant A > σ ( z ) | dz | in a neighborhood U of J such that σ ( f ( z )) | f (cid:48) ( z ) | ≥ Aσ ( z ) , z ∈ J (see [5] V. 2). If ∞ ∈ J , the hyperbolicity of f is defined by conjugation ofM¨obius transformations as usual.The hyperbolicity is also characterized in terms of the Euclidean metricand the dynamical behavior of rational functions as well. Proposition 2.1 ([5] V. 2. Lemma 2.1 and Theorem 2.2) . A rational func-tion f is hyperbolic if and only if every critical point belongs to F and isattracted to an attracting cycle. If ∞ (cid:54)∈ J , then the hyperbolicity of f isequivalent to the existence of m ≥ such that | ( f m ) (cid:48) | > on J . Random Cantor sets. (cf. [7] I. 6). Let ω = ( q n ) ∞ n =1 = ( q , q , . . . )be a sequence of real numbers with 0 < q n < n ∈ N . We constructa Cantor set E ( ω ) for ω inductively as follows.First, we remove an open interval J of length q from E := I = [0 , I \ J consists of two closed intervals I , I of the same length. Weput E = ∪ i =1 I i . We remove an open interval of length | I i | q from each I i so that the remainder E consists of four closed intervals of the samelength, where | J | is the length of an interval J . Inductively, we define E k +1 from E k = ∪ k i =1 I ik by removing an open interval of length | I ik | q k +1 from eachclosed interval I ik of E k so that E k +1 consists of 2 k +1 closed intervals of thesame length. The random Cantor set E ( ω ) for ω is defined by E ( ω ) = ∩ ∞ k =1 E k . It is a generalization of the standard middle one-third Cantor set C . In fact, C = E ( ω ) for ω = ( ) ∞ n =1 = ( , , . . . ).We say that a sequence ω = ( q n ) ∞ n =1 as above is of ( δ -)lower bound ifthere exists a δ > q n ≥ δ for any n ∈ N . We also say that asequence ω has a ( δ -)upper bound if q n ≤ − δ for any n ∈ N .2.3. Hausdorff dimension.
Let E be a subset of C and α >
0. We con-sider a countable open covering { U i } i ∈ N of E with diam( U i ) < r for a given HIROSHIGE SHIGA r >
0. Then, we set Λ rα ( E ) := inf (cid:40)(cid:88) i ∈ N (diam( U i )) α (cid:41) , where the infimum is taken over all countable open covering { U i } i ∈ N withdiam( U i ) < r . We put Λ α ( E ) = lim r → Λ rα ( E )and the Hausdorff dimension dim H ( E ) of E bydim H ( E ) = inf { α | Λ α ( E ) = 0 } . The quasiconformal equivalence of open Riemann surfaces.
Wesay that two Riemann surfaces R , R are quasiconformally equivalent ifthere exists a quasiconformal homeomorphism between them. We also saythat they are quasiconformally equivalent near the ideal boundary if thereexist compact subset K j of R j ( j = 1 ,
2) and quasiconformal homeomor-phism ϕ from R \ K onto R \ K .It is obvious that if R , R are quasiconformally equivalent, then they arequasiconformally equivalent near the ideal boundary. On the other hand,we have shown that the converse is not true in general. In fact, we have con-structed two Riemann surfaces which are not quasiconformally equivalentwhile they are homeomorphic to each other and quasiconformally equivalentnear the ideal boundary ([11] Example 3.1). We also give a sufficient condi-tion for Riemann surfaces to be quasiconformally equivalent ([11] Theorem5.1). Proposition 2.2.
Let R , R be open Riemann surfaces which are homeo-morphic to each other and quasiconformally equivalent near the ideal bound-ary. If the genus of R is finite, then R and R are quasiconformallyequivalent. Proof of Theorem I
Let f be a hyperbolic rational function with the Cantor Julia set J . Weshow that X J is quasiconformally equivalent to X C . By Proposition 2.2, itsuffices to show that there exists a compact subset K of F such that F \ K isquasiconformally equivalent to the complement of a compact subset of X C .Considering the conjugation by M¨obius transformations, we may assumethat J does not contain ∞ . Since J is a Cantor set, the Fatou set F isconnected. Furthermore, it follows from Proposition 2.1 that F containsan attracting fixed point z of f . Then, we may find a simply connectedneighborhood Ω of z such that f (Ω ) ⊂ Ω . We may take Ω so that theboundary ∂ Ω is a smooth Jordan curve and it does not contain the forwardorbits of critical points of f .For each k ∈ N , let Ω k be a connected component of f − k (Ω ) containing z . We may assume that Ω is bounded by at least two Jordan curves. Then, UASICONFORMAL EQUIVALENCE 5 each Ω k is bounded by mutually disjoint finitely many smooth Jordan curvesand we have z ∈ Ω ⊂ Ω ⊂ Ω ⊂ Ω · · · ⊂ Ω k ⊂ Ω k +1 ⊂ . . . and F = ∪ ∞ k =0 Ω k . Since f is hyperbolic, the Julia set J does not contain critical points.Moreover, there exists a simply connected neighborhood V z of each z ∈ J such that f | V z is injective on V z (Proposition 2.1). Hence, from compactnessof J there exist disks V , . . . , V n for some n ∈ N such that J ⊂ ∪ nj =1 V j and f | V j is injective (1 ≤ j ≤ n ). Then, we show; Lemma 3.1.
There exists k ∈ N such that for any k ≥ k , each connectedcomponent of Ω k +1 \ Ω k is contained in some V j (1 ≤ j ≤ n ) .Proof. Since f (Ω k +1 \ Ω k ) = Ω k \ Ω k − and Ω k +1 ⊃ Ω k , we see that if everyconnected component of Ω k +1 \ Ω k is contained in some V j , then so is for k ≥ k . Hence, we may find such a k to show the statement of the lemma.Suppose that for any k ∈ N , there exists a connected component ∆ k ofΩ k +1 \ Ω k such that ∆ k is not contained in any V j ( j = 1 , , . . . , n ). Thus, forsufficiently large k , ∆ k is contained in ∪ nj =1 V j but it is not contained in any V j . By taking a subsequence if necessary, we may assume that ∆ k ∩ V j (cid:54) = ∅ and ∆ k ∩ V j (cid:48) (cid:54) = ∅ for some j, j (cid:48) ∈ { , , . . . , n } . Let x be an accumulationpoint of { ∆ k } ∞ k =1 . Then, x has to be in J because F = ∪ ∞ k =1 Ω k .The Julia set J is totally disconnected. Hence, if a sequence { x k m } ∞ m =1 ( x k m ∈ ∆ k m ) converges to x , then { ∆ k m } ∞ m =1 also converges to { x } . In otherwords, for any neighborhood U of x , there exists m ∈ N such that for any m ≥ m , ∆ k m is contained in U . However, x ∈ J is in some V j because J ⊂ ∪ nj =1 V j . Therefore, ∆ k m is contained in V j if m is sufficiently large.Thus, we have a contradiction. (cid:3) We take k ∈ N in the above lemma. Let ω , ω , . . . , ω (cid:96) be the set ofconnected components of Ω k +1 \ Ω k . Each ω j is bounded by a finite number,say L ( j ) + 1, of mutually disjoint simple closed curves. We may assume that L ( j ) > j = 1 , , . . . (cid:96) ). Note that the number of boundary components of ∂ Ω k ∩ ∂ Ω k +1 is equal to (cid:96) . It is because ∂ (Ω k +1 \ Ω k ) consists of mutuallydisjoint simple closed curves in (cid:98) C , and Ω k is compact.For any k > k and for a connected component ∆ of Ω k +1 \ Ω k , we have f k − k (∆) ⊂ Ω k +1 \ Ω k and f k − k is conformal in ∆ since ∆ is contained insome V j . Hence, ∆ is conformally equivalent to ω J for some J ∈ { , , . . . , (cid:96) } .Therefore, if k > k , then Ω k +1 \ Ω k contains at most (cid:96) conformally differentconnected components.Now, we consider the middle one-third Cantor set C and X C := (cid:98) C \ C . It isnot hard to see that X C admits a pants decomposition { P i,j } i ∈ Z \{ } ,j ∈{ ,..., | i |− } as in Figure 1. In fact, we may take all P i,j so that they are conformallyequivalent to each other. Let P N ( N ∈ N ) be a subdomain of X C consisting HIROSHIGE SHIGA P P −1,1 P P P −2,1 P −2,2 P P Figure 1.
The middle one-third Cantor setof P i,j for i = 1 , . . . , N and j = 1 , . . . , i − . We see that P N is bounded by2 N + 1 mutually disjoint simple closed curves.Let N ∈ N be the largest number with 2 N + 1 ≤ (cid:96) . We put K := P N ∪ (cid:96) j =1 P N +1 ,j , where (cid:96) = (cid:96) − N −
1. Then, K is a compact subset of X C bounded by (cid:96) simple closed curves. We denote by C , . . . , C (cid:96) , where C ⊂ ∂P , . We maytake a subdomain G of X C so that G \ K is quasiconformally equivalentto Ω k +1 \ Ω k as follows.We take the largest number L with 2 L ≤ L (1). Then, G , := (cid:16) ∪ L i =1 ∪ j =1 ,..., | i | P − i,j (cid:17) ∪ (cid:16) ∪ j =1 ,...,L (1) − L P − L − ,j (cid:17) is a closed subdomain of X C with L (1) + 1 boundary curves. Hence, G , is quasiconformally equivalent to ω since both of them are planar domainsbounded by the same number of closed curves.Similarly, we may construct subdomains G , , . . . , G ,(cid:96) such that ∂G ,j ∩ ∂K = C j and each G ,j is quasiconformally equivalent to ω j ( j = 1 , , . . . , (cid:96) ).Combining K with G , , . . . , G ,(cid:96) , we obtain a desired subdomain G . UASICONFORMAL EQUIVALENCE 7
By using the same argument as above, we have a subdomain G of X C suchthat G ⊂ G and G \ G is quasiconformally equivalent to Ω k +2 \ Ω k +1 .Moreover, we may use this argument inductively and we obtain a exhaustion { G i } ∞ i =1 of X C such that K ⊂ G ⊂ G ⊂ · · · ⊂ G i ⊂ G i +1 ⊂ . . . , X C = ∪ ∞ i =1 G i , and G i +1 \ G i are quasiconformally equivalent to Ω k + i +1 \ Ω k + i .Now, we note the following. Proposition 3.1.
Let R , R be Riemann surfaces. We consider simpleclosed curves α i in R i with R i \ α i = S ( i )1 ∪ S ( i )2 , where S ( i )1 and S ( i )2 are mu-tually disjoint subsurface of R i ( i = 1 , . Suppose that there exist quasicon-formal mappings f j : S (1) j → S (2) j ( j = 1 , such that f ( α ) = f ( α ) = α .Then, there exists a quasiconformal mapping f : R → R . Moreover, themaximal dilatation of f depends only on those of f , f and the local behaviorof those mappings near α . We may apply this proposition to domains G i +1 \ G i and Ω k + i +1 \ Ω k + i ( i = 1 , , . . . ). Noting that there only finitely many conformal equivalenceclasses in those domains, we verify that X C \ K = ∪ i ∈ N ( G i +1 \ G i ) and F \ Ω k +1 = ∪ i ∈ N (Ω k + i +1 \ Ω k + i ) are quasiconformally equivalent. (cid:3) Proof of Theorem II
Proof of (1).
We divide the proof into several steps.
Step 1 : Analyzing random Cantor sets.
Let ω = ( q n ) ∞ n =1 and (cid:101) ω =(˜ q n ) ∞ n =1 be sequences with δ -lower bound. We take E k = ∪ k i =1 I ik and (cid:101) E k = ∪ k i =1 (cid:101) I ik as in § ω and (cid:101) ω , respectively. In fact, I ik (resp. (cid:101) I ik ) is locatedat the left of I i +1 k (resp. (cid:101) I i +1 k ) for i = 1 , , . . . , k −
1. The set [0 , \ E k (resp. [0 , \ (cid:101) E k ) consists of 2 k − J k , . . . , J k − k (resp. (cid:101) J k , . . . , (cid:101) J k − k ). Each J ik (resp. (cid:101) J ik ) is located between I ik and I i +1 k (resp. (cid:101) I ik and (cid:101) I i +1 k ).Because of the construction, we have | I ik +1 | = 12 (1 − q k ) | I ik | . Therefore, we have(4.1) | I ik | = 2 − k k (cid:89) j =1 (1 − q j ) . Next, we estimate the length of J ik .In construction E k +1 from E k , we obtain open intervals I i − k +1 , I ik +1 andthe closed interval J i − k +1 such that I ik = I i − k +1 ∪ J i − k +1 ∪ I ik +1 for each i, k (Figure 2). HIROSHIGE SHIGA I ik J i −1 k J ik I ik +1 I i +1 k +1 E k J i −1 k +1 J ik +1 J i +1 k +1 E k +1 Figure 2. If i is odd, we have(4.2) | J ik +1 | = | I ik | q k +1 = 2 q k +1 − q k +1 | I k +1 | ≥ δ | I k +1 | , as q k +1 ≥ δ .If i is even, then i = 2 (cid:96) m for an integer (cid:96) with 1 ≤ (cid:96) ≤ k and anodd number m . Since J ik +1 is located between I ik +1 and I i +1 k +1 , we see that J ik +1 = J i/ k = J (cid:96) − mk . Repeating this argument, we have J ik +1 = J mk − (cid:96) +1 .Since m is odd, we conclude from (4.2) that | J ik +1 | = | J mk − (cid:96) +1 | = 2 − k + (cid:96) q k − (cid:96) +1 k − (cid:96) (cid:89) j =1 (1 − q j ) ≥ − k +1 δ k (cid:89) j =1 (1 − q j ) ≥ δ | I k +1 | (4.3)as q k − (cid:96) +1 ≥ δ .Thus, we obtain the following from (4.2) and (4.3). UASICONFORMAL EQUIVALENCE 9
Lemma 4.1.
Let I ik and J ik +1 be the same ones as above for a sequence ω = ( q n ) ∞ n =1 with δ -lower bound. Then, (4.4) | J ik +1 | ≥ δ | I k +1 | hold for i = 1 , , . . . , k +1 − . Step 2: Constructing a pants decomposition.
We draw a circle C ik centered at the midpoint of I ik with radius (1 + δ ) | I k | for each k ∈ N and1 ≤ i ≤ k . From (4.4), we see that C ik ∩ C jk = ∅ if i (cid:54) = j . Since12 · δ | I k +1 | < · δ | I k | , we also see that C ik +1 ∩ C jk = ∅ . Therefore, ∪ ∞ k =1 ∪ k i =1 C ik gives a pantsdecomposition for X E ( ω ) .We draw circles (cid:101) C ik for (cid:101) ω by the same way. Then, we also see that ∪ ∞ k =1 ∪ k i =1 (cid:101) C ik gives a pants decomposition for X E ( (cid:101) ω ) . Step 3: Analyzing a pair of pants.
We denote by P ik a pair of pantsbounded by C ik , C i − k +1 and C ik +1 . We consider the complex structure of P ik so that we may assume that the center of C ik is the origin with radius (1 + δ ) | I k | . Then, the centers of C i − k +1 and C ik +1 are − q k +1 | I k | −
14 (1 + δ ) (1 − q k +1 ) | I k | and 12 q k +1 | I k | + 14 (1 + δ ) (1 − q k +1 ) | I k | , respectively.By applying an affine map z (cid:55)→ αz + β for some α > , β ∈ R to P ik sothat the circle C ik is sent a circle centered at the origin with radius 1 + δ .We denote the circle by C k, . Then, the circle C i − k +1 is sent a circle C k, centered at − x k := − q k +1 −
12 (1 + δ ) (1 − q k +1 ) = − { (1 + δ ) + (1 − δ ) q k +1 } with radius r k := 12 (1 + δ ) (1 − q k +1 )and C ik +1 is sent a circle C k, centered at x k with radius r k . We may con-formally identify P ik with a pair of pants P k bounded by C k, , C k, and C k, .Similarly, we consider a pair of pants (cid:101) P ik bounded by (cid:101) C ik , (cid:101) C i − k +1 and (cid:101) C ik +1 ,and apply an affine map to the pair of pants (cid:101) P ik so that the circle (cid:101) C ik ismapped a circle centered at the origin with radius 1 + δ , which is the samecircle as the image of C ik above. We denote by (cid:101) C k,i the image of (cid:101) C ik ( i =1 , , (cid:101) P ik with a pair of pants (cid:101) P k bounded by (cid:101) C k, , (cid:101) C k, and (cid:101) C k, , where (cid:101) C k, is the same circle as C k, , (cid:101) C k, is centeredat − (cid:101) x k := − { (1 + δ ) + (1 − δ ) (cid:101) q k +1 } wirh radius (cid:101) r k := 12 (1 + δ ) (1 − (cid:101) q k +1 )and (cid:101) C k, is centered at (cid:101) x k with radius (cid:101) r k . Step 4 : Constructing intermediate pairs of pants.
By applying z (cid:55)→ ( x k / (cid:101) x k ) z to (cid:101) P k , we obtain a pair of pants (cid:98) P k . The pair of pants (cid:98) P k is bounded by (cid:98) C k, , (cid:98) C k, and (cid:98) C k, . Each (cid:98) C k,i is corresponding to (cid:101) C k,i ( i = 1 , , i , the center of (cid:98) C k,i is x k , the same as thatof C k,i , and (cid:98) P k is conformally equivalent to (cid:101) P k . The radius of (cid:98) C k, is(1 + δ ) · x k (cid:101) x k = (1 + δ ) (1 + δ ) + (1 − δ ) q k +1 (1 + δ ) + (1 − δ ) (cid:101) q k +1 , and the radius of (cid:98) C k, , (cid:98) C k, is (cid:98) r k := 12 (1 + δ ) (1 − (cid:101) q k +1 ) (1 + δ ) (1 + δ ) + (1 − δ ) q k +1 (1 + δ ) + (1 − δ ) (cid:101) q k +1 . Now, we take an intermediate pair of pants P (cid:48) k bounded by (cid:98) C k, , C k, and C k, . Step 5 : Making quasiconformal mappings, I.
In the following theargument, we use a notation d ( ϕ ) for a quasiconformal mapping ϕ as d ( ϕ ) = log K ( ϕ ) , where K ( ϕ ) is the maximal dilatation of ϕ .We suppose that q k +1 ≥ (cid:101) q k +1 . Then, we have (cid:98) r k ≥ r k = 12 (1 + δ ) (1 − q k +1 ) . In other words, the radius of (cid:98) C k, , (cid:98) C k, is not smaller than that of C k, , C k, .Let C k, + be a circle centered at x k with radius (cid:101) R k := (1 + δ ) x k (cid:101) x k − x k , so that C k, + is tangent with (cid:98) C k, .We consider two circular annuli A k, + bounded by C k, + and (cid:98) C k, , A (cid:48) k, + bounded by C k, + and C k, . Here, we use the following well-known fact. Lemma 4.2.
For annuli A i = { < r i < | z | < R i < ∞} ( i = 1 , ), thereexists a quasiconformal mapping ϕ : A → A such that ϕ ( r e iθ ) = r e iθ ϕ ( R E iθ ) = R e iθ UASICONFORMAL EQUIVALENCE 11 and K ( ϕ ) = e d ( ϕ ) , where d ( ϕ ) = (cid:12)(cid:12)(cid:12)(cid:12) log log R − log r log R − log r (cid:12)(cid:12)(cid:12)(cid:12) . It follows from Lemma 4.2 that there exists a quasiconformal mapping ϕ k, + : A k, + → A (cid:48) k, + such that d ( ϕ k, + ) = log log (cid:101) R k − log r k log (cid:101) R k − log (cid:98) r k , (4.5) ϕ k, + ( z ) = z, for any z ∈ C k, + and(4.6) arg( ϕ k, + ( z ) − x k ) = arg( z − x k )for z ∈ (cid:98) C k, .Since log c − ac − b = log (cid:18) b − ac − b (cid:19) ≤ b − ac − b for 0 < a ≤ b < c , we obtain(4.7) d ( ϕ k, + ) ≤ log (cid:98) r k − log r k log (cid:101) R k − log (cid:98) r k . Moreover, we havelog (cid:101) R k − log (cid:98) r k = log (1 + δ ) − (1 − δ ) (cid:101) q k +1 (1 + δ ) − (1 + δ ) (cid:101) q k +1 (4.8) ≥ log (1 + δ ) − (1 − δ ) δ (1 + δ ) − (1 + δ ) δ > , and log (cid:98) r k − log r k = log 1 − (cid:101) q k +1 − q k +1 + log (1 + δ ) + (1 − δ ) q k +1 (1 + δ ) + (1 − δ ) (cid:101) q k +1 . (4.9)We also see that log (1+ δ )+(1 − δ ) q k +1 (1+ δ )+(1 − δ ) (cid:101) q k +1 (4.10) = log (cid:110) (1 − δ )( q k +1 − (cid:101) q k +1 )(1+ δ )+(1 − δ ) (cid:101) q k +1 (cid:111) ≤ (1 − δ )( q k +1 − (cid:101) q k +1 )(1+ δ )+(1 − δ ) (cid:101) q k +1 ≤ q k +1 − (cid:101) q k +1 , because (1 + δ ) + (1 − δ ) (cid:101) q k +1 > − δ > . From (4.7)–(4.10), we obtain d ( ϕ k, + ) ≤ (cid:16) log (1+ δ ) − (1 − δ ) δ (1+ δ ) − (1+ δ ) δ (cid:17) − (4.11) × (cid:110) log − (cid:101) q k +1 − q k +1 + ( q k +1 − (cid:101) q k +1 ) (cid:111) ≤ C ( δ ) d ( ω, (cid:101) ω ) for some constant C ( δ ) > δ .We may do the same operation, symmetrically; we take a circle C k, − centered at − x k of radius (cid:101) R k and consider two annuli A k, − and A (cid:48) k, − . Theannulus A k, − is bounded by C k, − and (cid:98) C k, , and A (cid:48) k, − is bounded by C k, − and C k, . Then, we obtain a quasiconformal mapping ϕ k, − : A k, − → A (cid:48) k, − such that(4.12) ϕ k, − ( z ) = z for z ∈ C k, − and(4.13) arg( ϕ k, − ( z ) + x k ) = arg( z + x k ) . for z ∈ (cid:98) C k, . Moreover, the mapping satisfies an inequality,(4.14) d ( ϕ k, − ) ≤ C ( δ ) d ( ω, (cid:101) ω ) . We define a homeomorphism ϕ k : (cid:98) P k → P (cid:48) k by ϕ k ( z ) = ϕ k, + ( z ) , z ∈ A k, + ϕ k, − ( z ) , z ∈ A k, − z, otherwise . The homeomorhpism ϕ k is quasiconformal except circles C k, + , C k, − . Hence,it has to be quasiconformal on (cid:98) P k with(4.15) d ( ϕ k ) ≤ C ( δ ) d ( ω, (cid:101) ω ) . Step 6 : Making quasiconformal mappings, II.
In this step, we makea quasiconformal mapping from P (cid:48) k to P k . Recall that P (cid:48) k is a pair of pantsbounded by (cid:98) C k, , C k, and C k, , and P k is bounded by C k, , C k, and C k, .Let C k, be a circle centered at the origin of radius x k + r k , so that C k, is tangent with C k, , C k, . We consider circular annuli B (cid:48) k bounded by C k, and (cid:98) C k, , and B k bounded by C k, and C k, . It follows from Lemma 4.2 thatthere exists a quasiconformal mapping ψ k, : B (cid:48) k → B k such that d ( ψ k, ) = log log(1 + δ ) x k (cid:101) x k − log( x k + r k )log(1 + δ ) − log( x k + r k )and ψ k, | C is the identity.As in Step 5, we have d ( ψ k, ) ≤ log x k − log (cid:101) x k log(1 + δ ) − log( x k + r k ) . Now, we see thatlog (1 + δ ) − log( x k + r k ) = log 1 + δ δ (1 − q k +1 )(4.16) ≥ log 1 + δ δ (1 − δ ) > , UASICONFORMAL EQUIVALENCE 13 and log x k − log (cid:101) x k = log (cid:18) − δ ) q k +1 − (cid:101) q k +1 (1 + δ ) + (1 − δ ) (cid:101) q k +1 (cid:19) (4.17) ≤ (1 − δ ) q k +1 − (cid:101) q k +1 (1 + δ ) + (1 − δ ) (cid:101) q k +1 ≤ q k +1 − (cid:101) q k +1 . From (4.16) and (4.17), we have(4.18) d ( ψ k, ) ≤ (cid:18) log 1 + δ δ (1 − δ ) (cid:19) − ( q k +1 − (cid:101) q k +1 ) . We define a homeomorphism ψ k : P (cid:48) k → P k by ψ k ( z ) = (cid:40) ψ k, ( z ) , z ∈ B (cid:48) k z, otherwise . Then, as in Step 5, we see that ψ k is quasiconformal on P (cid:48) k with(4.19) d ( ψ k ) ≤ C ( δ ) d ( ω, (cid:101) ω ) . In the case where q k +1 ≤ (cid:101) q k +1 , the same argument still works in Steps 5and 6; we obtain the same results. Step 7 : Making a global quasiconformal mapping.
In Steps 5 and6, we have made quasiconformal mappings ϕ k : (cid:98) P k → P (cid:48) k and ψ k : P (cid:48) k → P k .Thus, Φ k := ψ k ◦ ϕ k : (cid:98) P k → P k gives a quasiconformal mapping with d (Φ k ) ≤ C ( δ ) d ( ω, (cid:101) ω )for each k ∈ N .Because of the boundary behaviors (4.5), (4.6), (4.12) and (4.13), we seethat those mappings give a quasiconformal mapping Φ from X E ( ω ) onto X E ( (cid:101) ω ) with d (Φ) ≤ C ( δ ) d ( ω, (cid:101) ω ) . Furthermore, from our construction of the mapping, we see that Φ( H ) = H .Therefore, Φ is extended to a quasiconformal self-mapping of (cid:98) C as desired. (cid:3) Proof of (2).
Take any ε >
0. Since, log − (cid:101) q n − q n → n → ∞ , we also seethat q n → (cid:101) q n →
0. Viewing (4.11) and (4.18), we verify that there exists an N ∈ N such that d ( ϕ k ) <
12 log(1 + ε ) and d ( ψ k ) <
12 log(1 + ε ) , if k > N . Hence, if k > N , then(4.20) d (Φ k ) = d ( ψ k ◦ φ k ) ≤ d ( ψ k ) + d ( ϕ k ) < log(1 + ε ) . Since the pants decompositions in Step 2 of the proof (1) give exhaustions X E ( ω ) and X E ( (cid:101) ω ) , (4.20) implies the maximal dilatation K (Φ) = e d (Φ) is less than (1 + ε ) on the outside of a sufficiently large compact subset of X E ( ω ) .Therefore, Φ : X E ( ω ) → X E ( (cid:101) ω ) is asymptotically conformal. (cid:3) Proof of Theorem III
Suppose that there exists a K -quasiconformal map from X C to X E ( ω ) . Let d > X C .By Wolpert’s formula (cf. [10], [12]), the hyperbolic length of any simpleclosed curve in X E ( ω ) is not less than K − d .Let ε > { q n | n ∈ N } = 1,there exist a sequence { n k } ∞ k =1 in N and N ∈ N such that1 − ε < q n k < , if k > N .Now we look at I q k − of E q k − for k > N . Then, I q k ⊂ E q k is an intervalof length (1 − q k ) | I q k − | < ε | I q k − | . Therefore, we may take an annulus A k in X E ( ω ) bounded by two concentrated circles C k , C k such that the radius of C k is ε | I q k − | and that of C k is ( − ε ) | I q k − | . If we take ε > A k with respect to the hyperbolicmetric on A k becomes smaller than K − d . Since A k ⊂ X E ( ω ) , the lengthof the core curve of A k with respect to the hyperbolic metric of X E ( ω ) isnot greater than the length with respect to the hyperbolic metric of A k .Thus, we find a closed curve in X E ( ω ) whose length is less that K − d . It isa contradiction and we complete the proof of the theorem.6. Proofs of Corollaries
Proof of Corollary 1.1.
Let Λ be the limit set of the Schottky group G .We have shown ([11] Theorem 6.2) that X Λ is quasiconformally equivalent to X C . Hence, it follows from Theorem I that X E is quasiconformally equivalentto X Λ as desired. (cid:3) Proof of Corollary 1.2.
Since C = E ( ω ) for ω = ( ) ∞ n =1 , the statementfollows immediately from Theorem II (1). (cid:3) Proof of Corollary 1.3.
Let ϕ : X Λ → X E be a quasiconformal map givenby Corollary 1.1. Take any quasiconformal map ψ on X E to (cid:98) C . Then, Φ := ψ ◦ ϕ be a quasiconformal map on X Λ . It is known that any quasiconformalmap on X Λ is extended to a quasiconformal map on (cid:98) C (cf. [9]). Hence, both ϕ and Φ are extended to (cid:98) C and so is ψ = Φ ◦ ϕ − . (cid:3) Proof of Corollary 1.4.
Let Ψ : C → C be the quasiconformal mappinggiven in §
4. We put D = dim H ( E ( ω )) and (cid:101) D = dim H ( E ( (cid:101) ω )). We use theargument in the proof of Theorem II (2).For any ε >
0, there exists N ∈ N such that K (Φ k ) < ε UASICONFORMAL EQUIVALENCE 15 if k > N , where Φ k is the quasiconformal mapping given in §
4. Therefore,Φ | U N is a (1 + ε )-quasiconformal mapping on U N := E ( ω ) ∪ (cid:83) k>N (cid:83) k i =1 P ik .Here, we use the following result by Astala [3]. Proposition 6.1.
Let Ω , Ω (cid:48) be planar domains and f : Ω → Ω (cid:48) K -quasiconformalmapping. Suppose that E ⊂ Ω is a compact subset of Ω . Then, (6.1) dim H ( f ( E )) ≤ K dim H ( E )2 + ( K − dim H ( E ) . It follows from (6.1) thatdim H ( E ( (cid:101) ω )) ≤ − ε )dim H ( E ( ω ))2 + ε dim H ( E ( ω )) . Since ε > H ( E ( (cid:101) ω )) ≤ dim H ( E ( ω )) . By considering Φ − , we get the reverse inequality for dim H ( E ( ω )) anddim H ( E ( (cid:101) ω )). Thus, we conclude that dim H ( E ( ω )) = dim H ( E ( (cid:101) ω )) as de-sired. (cid:3) Examples
Example 7.1.
Let f c ( z ) = z + c . Suppose that c is not in the Mandelbrotset. Then, it is well known that f c is hyperbolic and the Julia set J f c is aCantor set. Thus, f c satisfies the condition of Theorem I. Example 7.2.
Let B ( z ) be a Blaschke product of degree d >
1. Supposethat B has an attracting fixed point on the unit circle T := {| z | = 1 } . Sincethe Julia set J B of B is included in T , it has to be a Cantor set. It isalso easy to see that B is hyperbolic. Thus, B satisfies the condition onTheorem I.In Theorem II, we have estimated the maximal dilatations for sequenceswith lower bound. In next example, we may estimate the maximal dilatationfor sequences without lower bound. Example 7.3.
For 0 < a < L ∈ N , we put q n = a n and (cid:101) q n = a n + L and we consider E ( ω ), E ( (cid:101) ω ) for ω = ( q n ) ∞ n =1 , (cid:101) ω = ( (cid:101) q n ) ∞ n =1 . Byusing the same idea as in the proof of Theorem II, we claim that there existsan exp( Ca − L )-quasiconformal mapping ϕ : C → C with ϕ ( E ( ω )) = E ( (cid:101) ω ),where C > ω and (cid:101) ω . Proof of the claim.
We use the same notations for E ( ω ) and E ( (cid:101) ω ) asthose in the proof of Theorem II. Then, E k = ∪ k i =1 I ik , [0 ,
1] = E k ∪ k − (cid:91) i =1 J ik and for i = 1 , , . . . , k , | I ik | = (cid:18) (cid:19) k k (cid:89) j =1 (1 − a j ) . If i is odd, then | J ik +1 | = a k +1 | I k | ≥ a k +1 | I k +1 | . If i = 2 (cid:96) m (1 ≤ (cid:96) ≤ k ; m is odd), then we have | J ik +1 | = | J mk − (cid:96) +1 | ≥ a k +1 | I k +1 | . Thus, we conclude that(7.1) | J ik +1 | ≥ a k +1 | I k +1 | , for i = 1 , , . . . k +1 − C ik centered at the midpoint of I ik with radius (1+ a k ) | I k | for each k ∈ N and 1 ≤ i ≤ k . From (7.1), we see that C ik ∩ C jk = ∅ if i (cid:54) = j .Therefore, ∪ ∞ k =1 ∪ k i =1 C ik gives a pants decomposition of X E ( ω ) . We alsodraw circles (cid:101) C ik for (cid:101) ω by the same way. Then, ∪ ∞ k =1 ∪ k i =1 (cid:101) C ik gives a pantsdecomposition of X E ( (cid:101) ω ) .We denote by P ik a pair of pants bounded by C ik , C i − k +1 and C ik +1 . As inStep 3 of the proof of Theorem II, we may identify P ik with a pair of pants P k bounded by C k, , C k, and C k, , where C k, is a circle centered at theorigin with radius 1 + a k , C k, is centered at − x k := − a k +1 −
12 (1 + a k +1 )(1 − a k +1 )with radius r k := 12 (1 + a k +1 )(1 − a k +1 )and C k, is centered at x k with radius r k .Similarly, we take a pair of pants (cid:101) P ik bounded by (cid:101) C ik , (cid:101) C i − k +1 and (cid:101) C ik +1 ,which is conformally equivalent to a pair of pants (cid:101) P k bounded by (cid:101) C k, , (cid:101) C k, and (cid:101) C k, , where (cid:101) C k, is the same circle as C k, , (cid:101) C k, is centered at − (cid:101) x k := − − a k + L +1 −
12 (1 + a k + L +1 )(1 − a k + L +1 )wirh radius (cid:101) r k := 12 (1 + a k + L +1 )(1 − a k + L +1 )and (cid:101) C k, is centered at (cid:101) x k with radius (cid:101) r k .We also take an intermediate pair of pants, (cid:98) P k similar to that of the proofof Theorem II. Then, by using exactly the same method, we may constructa exp( Ca − L )-quasiconformal mapping from P ik onto (cid:101) P ik , where C > k and i . Since the calculation is a bit long but thesame as in §
4, we may leave it to the reader.
UASICONFORMAL EQUIVALENCE 17
By gluing those quasiconformal mappings together, we get an exp( Ca − L )-quasiconformal mapping ϕ : C → C with ϕ ( E ( ω )) = E ( (cid:101) ω ) as desired. (cid:3) Cantor Julia sets of Blaschke products with parabolic fixed points.
We showed ([11] Example 3.2) that a Cantor set which is the limit set ofan extended Schottky group is not quasiconformally equivalent to the limitset of a Schottky group. We discuss the same thing for Cantor sets definedby non-hyperbolic rational functions.Let B ( z ) be a Blaschke product with a parabolic fixed point on the unitcircle T . Suppose that there exists only one attracting petal at the parabolicfixed point. Then, we see that the Julia set J B is a Cantor set on T (see[5] IV. 2. Example). However, B is not hyperbolic since it has a parabolicfixed point.It follows from Theorem I that two Riemann surfaces X J fc for Example7.1 and X J B for Example 7.2 are quasiconformally equivalent. While theJulia set J B of B is also a Cantor set, it is not hyperbolic. Therefore, wecannot apply Theorem I for B .Now, we consider the Martin compactification of the complement. For ageneral theory of the Martin compactification, we may refer to [6]. Here, wenote the following. Proposition 7.1.
Let B be a hyperbolic Blaschke product of degree d > .Suppose that the Julia set J B is a Cantor set in T . Then, the Martincompactification of X J B is homeomorphic to (cid:98) C . Hence, the same statements as in Proposition 7.1 hold for X J := (cid:98) C \ J and the quasiconformal map ϕ on X J is extended to a homeomorphism ofthe Martin compactification of X J .Next, we consider the Martin compactification of X J , especially the setof the Martin boundary over the parabolic fixed point of B . If the setcontains at least two points, then it follows from Proposition 7.1 that thereexists no quasiconformal map from X J to X J .Indeed, in [9] we observe the Martin compactification of the complementof the limit set of an extended Schottky group and show that the set ofthe Martin boundary over a parabolic fixed point consists of more than twopoints. It is a key fact to show that the limit set of the extended Schottkygroup is not quasiconformally equivalent to that of a Schottky group ([11]).However, by using an argument of Benedicks ([4]) (see also Segawa [8]) onthe Martin compactification, we may show the following. Lemma 7.1.
In the Martin compactification of X J , there is exactly oneminimal point over the parabolic fixed point of B . Remark 7.1.
In the Martin compactification of a Riemann surface, the setcorresponding to a topological boundary component of the Riemann surfaceis connected and the minimal points in the set are regarded as extremepoints of a convex set. Thus, if the set over a boundary component on the
Martin compactification contains only one minimal point, then it consistsof only one point, that is, the minimal point.
Proof.
To prove the lemma, we use a result by Benedicks.We denote by Q ( t, r ) ( t ∈ R , r > (cid:110) x + iy | | x − t | < r , | y | < r (cid:111) . For a fixed α with 0 < α < x ∈ R \ { } , we consider thesolution of the Dirichlet problem on Q ( x, α | x | ) \ E with boundary valuesone on ∂Q ( x, α | x | ) and zero on E ∩ Q ( x, α | x | ). We denote the solution by β Ex . Then, Benedicks showed the following. Proposition 7.2.
On the Martin compactification of (cid:98) C \ E , there exist morethan two points over ∞ if and only if (7.2) (cid:90) | x |≥ β Ex ( x ) | x | dx < ∞ . Let a ∈ T be the parabolic fixed point B . We take a M¨obius transfor-mation γ so that γ ( T ) = R ∪ {∞} and γ ( a ) = ∞ . For (cid:98) B := γB γ − , wesee that ∞ is a parabolic fixed point with a unique attracting petal of (cid:98) B ,and J := γ ( J B ) is contained in R ∪ {∞} .Since z = ∞ is a parabolic fixed point of (cid:98) B with only one attractingpegtal, we may assume that there exists a sufficiently large M > J ∩ { Re z < − M } is empty while J ∩ { Re z > M } is not empty.Hence, J ∩ Q ( x, α | x | ) = ∅ if x < | x | is sufficiently large. Therefore, β J x ( x ) = 1 for such x . Thus, we have (cid:90) | x |≥ β J x ( x ) | x | dx = ∞ and conclude that there exists exactly one point over ∞ from Proposition7.2. (cid:3) Lemma 7.1 implies that we cannot use the argument used for extendedSchottky groups. We exhibit the following conjecture at the end of thisarticle.
Conjecture. X J is not quasiconformally equivalent to X C . References [1] L. V. Ahlfors, Lectures on Quasiconformal Mappings (2nd edition), American Math-ematical Society, Providence Rhode Island, 2006.[2] L. V. Ahlfors and Sario, L., Riemann surfaces, Princeton University Press, Princeton,New Jersey, 1974.[3] K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37–60.[4] M. Benedicks, Positive harmonic functions vanishing on the boundary of certain do-main in R n , Ark. Mat., 18 (1980), 53–71.[5] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext, Springer, 1991. UASICONFORMAL EQUIVALENCE 19 [6] C. Constantinescu and Cornea, A., Ideale R¨ander Riemannscher Fl¨achen, Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1963.[7] L. Sario and Nakai, M., Classification theory of Riemann surfaces, Springer, Berlin-Heidelberg-New York, 1970.[8] S. Segawa, Martin boundaries of Denjoy domains and quasiconformal mappings, J.Math. Kyoto Univ., 30 (1990), 297–316.[9] H. Shiga, On complex analytic properties of limit sets and Julia sets, Kodai Math.J., 28 (2005), 368–381.[10] H. Shiga, On the hyperbolic length and quasiconformal mappings, Complex Variables,50 (2005), 123–130.[11] H. Shiga, The quasiconformal equivalence of Riemann surfaces and the universalSchottky space, arXiv:1807.01096.[12] S. Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. ofMath., 109 (1979), 323–351.
Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamig-amo, Kita-ku Kyoto, Japan
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