Deepness of the boundary of bounded type quadratic Siegel disks following C. McMullen

Transcription

1 Deepness of the boundary of bounded type quadratic Siegel disks following C. McMullen At the advent of the 70 aniversary of Adrien Douady Carsten Lunde Petersen IMFUFA, Roskilde University Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 1/27

2 Introduction This will be a talk in the Bourbaki tradition. All the results I will be presenting are due to Curt. T McMullen. They can be found in the paper: Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Math. Vol 180, Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 2/27

3 Basic definitions Define P θ (z) = e i2πθ z + z 2 : C C, θ R. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 3/27

4 Basic definitions Define P θ (z) = e i2πθ z + z 2 : C C, θ R. J θ the Julia set of P θ and c θ the critical point. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 3/27

5 Basic definitions Define P θ (z) = e i2πθ z + z 2 : C C, θ R. J θ the Julia set of P θ and c θ the critical point. K θ = {z C n : Pθ n (z) 4} the filled in Julia set. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 3/27

6 Basic definitions Define P θ (z) = e i2πθ z + z 2 : C C, θ R. J θ the Julia set of P θ and c θ the critical point. K θ = {z C n : P n θ (z) 4} the filled in Julia set. For any Brjuno Number θ B: The Siegel disk θ of P θ and Ω θ := C\ θ its complement. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 3/27

7 Basic definitions Define P θ (z) = e i2πθ z + z 2 : C C, θ R. J θ the Julia set of P θ and c θ the critical point. K θ = {z C n : P n θ (z) 4} the filled in Julia set. For any Brjuno Number θ B: The Siegel disk θ of P θ and Ω θ := C\ θ its complement. For θ B the ɛ-neighbourhood θ (ɛ) of θ and K θ (ɛ) = {z C n : P n θ (z) θ(ɛ)} Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 3/27

8 Main Theorem Theorem 1 (McMullen). For every bounded type θ BT and for every ɛ > 0 the boundary θ is uniformly measureably deep in K θ (ɛ). Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 4/27

9 Main Theorem Theorem 1 (McMullen). For every bounded type θ BT and for every ɛ > 0 the boundary θ is uniformly measureably deep in K θ (ɛ). That is α, C > 0 depending on θ and ɛ such that z θ, r 1 : Area(B r (z)\k θ (ɛ)) Cr 2+α, where B r (z) denotes the euclidean ball of center z and radius r. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 4/27

10 Main Theorem Theorem 1 (McMullen). For every bounded type θ BT and for every ɛ > 0 the boundary θ is uniformly measureably deep in K θ (ɛ). That is α, C > 0 depending on θ and ɛ such that z θ, r 1 : Area(B r (z)\k θ (ɛ)) Cr 2+α, where B r (z) denotes the euclidean ball of center z and radius r. In particular every point of θ is a Lebesgue density point of K θ (ɛ). Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 4/27

11 Auxillary definitions 1 Definition 2. A point z 0 Λ C, where Λ is a compact subset, is called a deep point of Λ, iff δ > 0 and C > 0 such that r 1: B s (z) B r (z 0 )\Λ s Cr 1+δ Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 5/27

12 Auxillary definitions 1 Definition 2. A point z 0 Λ C, where Λ is a compact subset, is called a deep point of Λ, iff δ > 0 and C > 0 such that r 1: B s (z) B r (z 0 )\Λ s Cr 1+δ Definition 3. A compact subset Λ C is called porous iff C > 0 such that: z 0 C, r > 0, z : B Cr (z) (B r (z 0 )\Λ). We also say that Λ is C porous. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 5/27

13 Auxillary definitions 1 Definition 2. A point z 0 Λ C, where Λ is a compact subset, is called a deep point of Λ, iff δ > 0 and C > 0 such that r 1: B s (z) B r (z 0 )\Λ s Cr 1+δ Definition 3. A compact subset Λ C is called porous iff C > 0 such that: z 0 C, r > 0, z : B Cr (z) (B r (z 0 )\Λ). We also say that Λ is C porous. The two notions deep and porous are in some sense opposite of each other. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 5/27

14 Comments on Auxillary definitions 1 Deepness of a point z 0 Λ means that round holes in the complement of Λ become exponentially small relative to the scale when we pass to small scales near z. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 6/27

15 Comments on Auxillary definitions 1 Deepness of a point z 0 Λ means that round holes in the complement of Λ become exponentially small relative to the scale when we pass to small scales near z. Poroussity of Λ means that Λ has definite size holes relative to the scale everywhere. It is like a uniform Swiss cheese. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 6/27

16 Comments on Auxillary definitions 1 Deepness of a point z 0 Λ means that round holes in the complement of Λ become exponentially small relative to the scale when we pass to small scales near z. Poroussity of Λ means that Λ has definite size holes relative to the scale everywhere. It is like a uniform Swiss cheese. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 6/27

17 General definitions Definition 4. For Λ C a compact subset. The upper box-dimension of Λ is the real number dim box (Λ) := lim sup r 0 log(n(λ, r)), log 1/r where N(Λ, r) is the minimal number of squares of side length r needed to cover Λ. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 7/27

18 General definitions Definition 4. For Λ C a compact subset. The upper box-dimension of Λ is the real number dim box (Λ) := lim sup r 0 log(n(λ, r)), log 1/r where N(Λ, r) is the minimal number of squares of side length r needed to cover Λ. Trivially the Hausdorff dimension dim H (Λ) satisfies: dim H (Λ) dim box (Λ) 2. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 7/27

19 General poroussity results Proposition 5. For C > 0 let N = N(C) N satisfy N Then Any C-porous subset Λ C satisfies: 2 C. dim H (Λ) dim box (Λ) log(n 2 1) log N := d N < 2. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 8/27

20 General poroussity results Proposition 5. For C > 0 let N = N(C) N satisfy N Then Any C-porous subset Λ C satisfies: 2 C. dim H (Λ) dim box (Λ) log(n 2 1) log N := d N < 2. Moreover for d N < d < 2 there exists K 0 N such that for any K K 0, for any square Q of side length r 1: The partition of Q into sub squares q of equal side lengths r/k has the property that the number N((Λ Q), K) of small squares q needed to cover (Λ Q) satisfies N((Λ Q), K) K d. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 8/27

21 ", N D H L \ _^ T p v t n z ˆ µ Proof lim n log (N 2 1) n log N n log r = log(n 2 1) log N = d N Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 9/27

22 General results 2 Proposition 6. Any deep point z 0 of a compact subset Λ C with pourous boundary is a measureable deep point. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 10/27

23 General results 2 Proposition 6. Any deep point z 0 of a compact subset Λ C with pourous boundary is a measureable deep point. Moreover if z 0 is a δ, C D > 0 deep point and Λ is C P > 0 porous. Then r 1 Area(B r (z 0 )\Λ) C MD r 2+α, where α, C MD > 0 depends only on the constants δ, C D, C P > 0. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 10/27

24 Proof of Proposition 6 It suffices to prove it for small r > 0. Let d N < d < 2 and K 0 1 be as in Proposition 5. For r write (2CK 0 ) 1/δ K 0 K 1 2Cr < K + 1 N. δ Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 11/27

25 Proof of Proposition 6 It suffices to prove it for small r > 0. Let d N < d < 2 and K 0 1 be as in Proposition 5. For r write (2CK 0 ) 1/δ K 0 K 1 2Cr < K + 1 N. δ 2 K squares q z 0 2r d K covers q Q Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 11/27

26 Proof of Proposition 6 cont. Hence B r (z 0 )\Λ is contained in K d small squares and we have the estimate: Area(B r (z 0 )\Λ) K d ( 2r K ) 2 (as 1 (K + 1) 2Cr δ ) 4r 2 ( K + 1 K C MD r 2+α ) 2 d (2Cr δ ) 2 d ) with C MD = 4( K d(2c) 2 d K 0 and α = δ(2 d). Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 12/27

27 Main Technical Theorems Theorem 7 (McMullen). For every bounded type θ BT and for every ɛ > 0 the boundary θ is uniformly deep in K θ (ɛ). Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 13/27

28 Main Technical Theorems Theorem 7 (McMullen). For every bounded type θ BT and for every ɛ > 0 the boundary θ is uniformly deep in K θ (ɛ). That is δ, C > 0 depending on θ and ɛ such that r 1 : B s (z) θ (r)\k θ (ɛ) s Cr 1+δ. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 13/27

29 Main Technical Theorems Theorem 7 (McMullen). For every bounded type θ BT and for every ɛ > 0 the boundary θ is uniformly deep in K θ (ɛ). That is δ, C > 0 depending on θ and ɛ such that r 1 : B s (z) θ (r)\k θ (ɛ) s Cr 1+δ. Theorem 8 (McMullen). For every bounded type θ BT the Julia set J θ is porous. Moreover so is K θ (ɛ) for every ɛ > 0. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 13/27

30 Main Technical Theorems Theorem 7 (McMullen). For every bounded type θ BT and for every ɛ > 0 the boundary θ is uniformly deep in K θ (ɛ). That is δ, C > 0 depending on θ and ɛ such that r 1 : B s (z) θ (r)\k θ (ɛ) s Cr 1+δ. Theorem 8 (McMullen). For every bounded type θ BT the Julia set J θ is porous. Moreover so is K θ (ɛ) for every ɛ > 0. Combining Theorem 7 and Theorem 8 with Proposition 6 we obtain the Main Theorem. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 13/27

31 Golden Siegel disk Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 14/27

32 Approx K θ (ɛ) Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 15/27

33 Zoom 1 Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 16/27

34 Zoom 2 Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 17/27

35 Reducing notation In the following we shall fix a bounded type irrational θ and drop all subscripts θ to simplify writting and reading. Recall that Ω = C\ and that c denotes the critical point. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 18/27

36 Nearby critical visits Theorem 9 (McMullen). There exists C > 0 such that for every z J and every r > 0 there is a univalent iterate between pointed disks P n : (U, y) (V, c), n 0, such that y z r and B Cr (y) U. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 19/27

37 Nearby critical visits Theorem 9 (McMullen). There exists C > 0 such that for every z J and every r > 0 there is a univalent iterate between pointed disks P n : (U, y) (V, c), n 0, such that y z r and B Cr (y) U. The Koebe distortion theorems for univalent maps and the fact that is a quasi disk easily implies that: Corollary 10 (McMullen). The Julia set J of P is porous. Elaborating a bit more one also obtain that K(ɛ) is porous. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 19/27

38 Small hyperbolic balls Theorem 11. For each ɛ > 0 there exists α, C > 0 depending on ɛ such that for any z 0 C\K(ɛ) with d = d(z 0, ) 1 d Ω (z 0, K(ɛ)) C d α. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 20/27

39 Small hyperbolic balls Theorem 11. For each ɛ > 0 there exists α, C > 0 depending on ɛ such that for any z 0 C\K(ɛ) with d = d(z 0, ) 1 d Ω (z 0, K(ɛ)) C d α. The uniform deepness of in K(ɛ) Theorem 12. McMullen For every bounded type θ BT and for every ɛ > 0 the boundary θ is uniformly deep in K θ (ɛ). is an easy Corollary as the coeficient function of the hyperbolic metric λ Ω (z) is uniformly comparabel to d = d(z, ): Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 20/27

40 Hyperbolic expansion Let Ω = P 1 (Ω) and let = Ω P 1 ( ) denote the co-preimage of. Then P : C\( ) = Ω Ω = C\ is a local hyperbolic isometry. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 21/27

41 Hyperbolic expansion Let Ω = P 1 (Ω) and let = Ω P 1 ( ) denote the co-preimage of. Then P : C\( ) = Ω Ω = C\ is a local hyperbolic isometry. Hence z Ω : DP (z) Ω := P (z) λ Ω (P (z)) λ Ω (z) > 1 Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 21/27

42 Hyperbolic expansion Let Ω = P 1 (Ω) and let = Ω P 1 ( ) denote the co-preimage of. Then P : C\( ) = Ω Ω = C\ is a local hyperbolic isometry. Hence z Ω : DP (z) Ω := P (z) λ Ω (P (z)) λ Ω (z) > 1 and C > 0 Λ > 1 such that z Ω with d Ω (z, ) C : DP (z) Ω Λ. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 21/27

43 Critical Visits Proposition 13. C, C 1, C 2, 1/β > 1 s. t. (z k ) k 0, P (z k ) = z k+1, d = d(z 0, ) 1: k 0 : d β /C 2 z k0 c C 2 d 1/β Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 22/27

44 Critical Visits Proposition 14. C, C 1, C 2, 1/β > 1 s. t. (z k ) k 0, P (z k ) = z k+1, d = d(z 0, ) 1: k 0 : d β /C 2 z k0 c C 2 d 1/β Moreover if z 0 c 1 then: Either k 1 : z k c z 0 c Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 22/27

45 Critical Visits Proposition 15. C, C 1, C 2, 1/β > 1 s. t. (z k ) k 0, P (z k ) = z k+1, d = d(z 0, ) 1: k 0 : d β /C 2 z k0 c C 2 d 1/β Moreover if z 0 c 1 then: Either k 1 : z k c z 0 c Or k 1 : z k c C z 0 c d Ω (z k, ) C 1. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 22/27

46 Critical Visits Proposition 16. C, C 1, C 2, 1/β > 1 s. t. (z k ) k 0, P (z k ) = z k+1, d = d(z 0, ) 1: k 0 : d β /C 2 z k0 c C 2 d 1/β Moreover if z 0 c 1 then: Either k 1 : z k c z 0 c Or k 1 : z k c C z 0 c d Ω (z k, ) C 1. so that DP (z k ) Ω Λ > 1. where Λ = Λ(C 1 ) is the associated hyperbolic expansion coefficient. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 22/27

47 Sketch of proof of small hyperbolic balls. Fix 0 < ɛ 1, let z 0 B ɛ (c)\k(ɛ), write d = z 0 c < ɛ and denote by (z k ) k 0 the orbit of z 0. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 23/27

48 Sketch of proof of small hyperbolic balls. Fix 0 < ɛ 1, let z 0 B ɛ (c)\k(ɛ), write d = z 0 c < ɛ and denote by (z k ) k 0 the orbit of z 0. Let k 1,... k n+1 denote the critical visits until z kn+1 c > ɛ. Then DP k n+1 (z 0 ) Ω Λ n and d Ω (z kn+1, ) C 1 and in the worst case C n d ɛ < C n+1 d so that n log(ɛ/d) log C < n + 1, Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 23/27

49 more small hyperbolic balls Hence pulling back a geodesic arc of length at most C 1 from z kn+1 to w kn+1 we obtain a point w 0 K(ɛ) with d Ω (z 0, w 0 ) C 1 Λ n C 1 Λ exp((log d log ɛ) log Λ log C ) = C d α, where C = C 1Λ ɛ α and α = log Λ log C. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 24/27

50 more small hyperbolic balls Hence pulling back a geodesic arc of length at most C 1 from z kn+1 to w kn+1 we obtain a point w 0 K(ɛ) with d Ω (z 0, w 0 ) C 1 Λ n C 1 Λ exp((log d log ɛ) log Λ log C ) = C d α, where C = C 1Λ and α = log Λ ɛ α log C. As you may have noticed the sketch has a slight oversight. The closest point in may bee outside (ɛ) or one of the iterates w k may bee outside (ɛ). Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 24/27

51 more small hyperbolic balls Hence pulling back a geodesic arc of length at most C 1 from z kn+1 to w kn+1 we obtain a point w 0 K(ɛ) with d Ω (z 0, w 0 ) C 1 Λ n C 1 Λ exp((log d log ɛ) log Λ log C ) = C d α, where C = C 1Λ and α = log Λ ɛ α log C. As you may have noticed the sketch has a slight oversight. The closest point in may bee outside (ɛ) or one of the iterates w k may bee outside (ɛ). Also what about the points near, but not near c? Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 24/27

52 rectifying the sketch of proof. For the first obstacle as the λ Ω (z) is comparable to d(z, ) on (ɛ). There exists a constant C 3 > 1 such that d Ω ( (ɛ/c 3 ), (ɛ)) > C 1. Replacing ɛ by ɛ/(c C 3 ) in the estimates only changes C and ensures that w kn+1 ( (ɛ)). Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 25/27

53 rectifying the sketch of proof. For the first obstacle as the λ Ω (z) is comparable to d(z, ) on (ɛ). There exists a constant C 3 > 1 such that d Ω ( (ɛ/c 3 ), (ɛ)) > C 1. Replacing ɛ by ɛ/(c C 3 ) in the estimates only changes C and ensures that w kn+1 ( (ɛ)). Replacing further ɛ by ɛ β /C 2 also only changes C and ensures that the points z j and their companions w j with k n < j < k n+1 belongs to (ɛ) as d(z j, ) β /C 2 d(z kn+1, c) ɛ β /C 2 Hence the full orbit of w 0,..., w kn+1 (ɛ) and thus in K(ɛ). Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 25/27

54 final estimate Finally an arbitrary z near with d = d(z, ) has an iterate z k0 with d = d(z k0, c) C 2 d 1/β. Let w k0 K(ɛ) be a point with d Ω (w k0, z k0 ) C (d ) α. Then there is a point w 0 K(ɛ) with d Ω (w 0, z 0 ) C (d ) α C d α, where α = α /β and C = C C2 α = C 1ΛC2 2α ( CC3 ɛ ) βα. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 26/27

55 final comments McMullen uses a very nice idea to prove the hyperbolic estimate and the theorem on nearby critical visits. The idea being: The boundary of is a quasi circle so the linearizer φ : D of P extends to a quasi conformal homeomorphism also denoted φ : C C. The conjugate degree 2 map f = φ P φ 1 coincides on D with the corresponding rigid rotation R. Its iterates are uniformly quasi regular and tends to be close to corresponding iterates of R near D. With the aid of this he maneges to prove all the above mentioned theorems without making explicit reference to the usual Blaschke model described by Douady in his 1987 Bourbaki seminar. Disques de Siegel, implosion parabolique et aire des ensembles de Julia p. 27/27