Random quadratic Julia sets and quasicircles Krzysztof Lech March 27, 2020 Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 1 / 19
Introduction and notation We will consider compositions of functions f n ( z ) = z 2 + c n . Unlike in normal complex dynamics, c n changes along iteration. Let us denote F n ( z ) = f n ( z ) ◦ f n − 1 ( z ) ◦ ... ◦ f 1 ( z ). We can ask questions about normality of the family { F n } . Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 2 / 19
Non-autonomous definitions The Fatou set is defined by F ( c n ) = { z ∈ � C : { F n } is normal on a neighbourhood of z } The Julia set J ( c n ) is the complement of the Fatou set. In the autonomous case these sets depend on a parameter c , since we investigate the normality of iterations of z 2 + c . In our case these sets depend on a sequence { c n } . Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 3 / 19
Autonomous quadratic iteration The autonomous case where ∀ n c n = c has been studied extensively. The Julia set J c is in this case either connected, or totally disconnected, i.e. every connected component is a single point. The set of points c for which the Julia set is connected is the famous Mandelbrot set. If c is in the interior of the main cardioid of the Mandelbrot set then the Julia set is a quasicircle. Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 4 / 19
Autonomous iteration: Mandelbrot set Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 5 / 19
Quasicircles Definition A quasicircle is the image of the unit circle under a quasiconformal homeomorphism of C onto itself. Theorem (Ahlfors) A Jordan curve γ ⊂ C is a quasicircle if and only if there exists a constant M < ∞ such that | x − y | < M | x − z | holds for any y on the smaller diameter arc between x , z ∈ γ . Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 6 / 19
Some pictures and geometric intuition Figure: Julia set for z 2 + 1 5 Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 7 / 19
Some more pictures and geometric intuition Figure: Not a quasicircle Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 8 / 19
Non autonomous quadratic dynamics Theorem (Rainer Br¨ uck) If ∀ n | c n | < δ < 1 4 then the Julia set J c n is a quasicircle. Theorem (Anna Zdunik, L.) Let V be an open and bounded set such that D (0 , 1 4 ) ⊂ V and 4 ) . Consider the space Ω = V N equipped with the product P of V � = D (0 , 1 uniform distributions on V . Then for P –almost every sequence ω ∈ Ω the Julia set J ω is totally disconnected. Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 9 / 19
Comparison of non-autonomous and autonomous dynamics In the autonomous quadratic dynamics, if the Julia set is disconnected, then it must be totally disconnected. This is not true for non-autonomous iteration. Indeed, it is easy to produce sequences for which the Julia set is disconnected but not totally disconnected. Take c 1 to be some large number, and c n = 0 for n > 1. In the above example, the Julia set has 2 connected components, the preimages of the unit circle under z 2 + c 1 . Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 10 / 19
Comparison of non-autonomous and autonomous dynamics In the autonomous case the Julia set is a quasicircle if c is from the interior of the main cardioid of the Mandelbrot set. In the non-autonomous case if | c n | < δ < 1 4 then the Julia set is a quasicircle. But if c n are chosen from the interior of the main cardioid of the Mandelbrot set, then the sequences for which the Julia set is totally disconnected are of full measure. Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 11 / 19
John domains Before we look at Br¨ uck’s proof we need one more auxilliary result. Definition A domain G ⊂ � C with ∂ G ⊂ C is called a John domain , if there exists a constant b > 0 and a point w 0 ∈ G such that for any z 0 ∈ G , there is an arc γ = γ ( z 0 ) ⊂ G joining z 0 and w 0 and satisfying dist ( z , ∂ G ) � b | z − z 0 | for any z ∈ γ . Theorem (Raimo N¨ akki, Jussi V¨ ais¨ al¨ a) If the two complementary components of a Jordan curve are John domains, then that curve is a quasicircle. Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 12 / 19
The Fatou set has two components From now on let ∀ n | c n | < δ for some δ < 1 4 . For any r let D r = { z : | z | < r } and ∆ r = { z : | z | > r } √ √ Let R � R δ := 1 1 + 4 δ ) and r δ := 1 1 − 4 δ ) > r > 1 2 (1 + 2 (1 + 2 Then ∀ c n ∈ D (0 ,δ ) f c n (∆ R ) ⊂ ∆ R and f c n ( D r ) ⊂ D r The Fatou set has two connected components, one which contains infinity, and one which contains 0, let us denote them by A ( c n ) ( ∞ ) and A ( c n ) (0) respectively. Finally, we have: ∂ A ( c n ) (0) = J ( c n ) = ∂ A c n ( ∞ ). Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 13 / 19
Br¨ uck’s proof of the quasicircle theorem Let us recall again: Theorem (Br¨ uck) If δ < 1 4 and ( c n ) ∈ D N δ then J ( c n ) is a quasicircle. Br¨ uck proves this by showing that both A ( c n ) (0) and A ( c n ) ( ∞ ) are John domains. The remainder of the slides are a presentation of his proof, exactly as in his paper. Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 14 / 19
Br¨ uck’s proof of the quasicircle theorem Lemma Let δ < 1 δ and 1 4 , ( c n ) ∈ D N 2 < r < r δ . Let γ : [0 , 1] → V be a rectifiable curve in V := ∆ r . Let z := γ (0) , w := γ (1) and let F − 1 be an analytic n branch of the inverse function of F n on some disk D ⊂ V with center at z. Finally, we denote the analytic continuation of F − 1 along γ also by F − 1 n . n Then we have | ( F − 1 n ) ′ ( z ) n ) ′ ( w ) | � 1 + αℓ ( γ ) e αℓ ( γ ) ( F − 1 for some constant α . Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 15 / 19
Proof of the lemma Proof. For k = 0 , 1 , ..., n − 1 we set F n , k := f c n ◦ ... ◦ f c k +1 . Since: 1 1 1 ( F − 1 n ) ′ ( z ) = ( z )) = = n ( F − 1 n − 1 n − 1 F ′ n F j ( F − 1 F − 1 2 n � ( z )) 2 n � n , j ( z ) n j =0 j =0 and V is backward invariant we have n ) ′ ( z ) | � q n for z ∈ V | ( F − 1 and even | ( F − 1 n , k ) ′ ( z ) | � q n − k where q := 1 2 r < 1. This yields � | F − 1 n , k ( w ) − F − 1 γ | ( F − 1 n , k ) ′ ( ζ ) || d ζ || � q n − k ℓ ( γ ) n , k ( z ) | � | Finally: n − 1 � n − 1 � n − 1 � F − 1 F − 1 n , k ( w ) − F − 1 | ( F − 1 n , k ( w ) n , k ( z ) ) ′ ( z ) | 1+2 q n − k +1 ℓ ( γ ) n ) ′ ( w ) | = | n , k ( z ) | = | 1+ | � | ( F − 1 F − 1 F − 1 n , k ( z ) n k =0 k =0 k =0 Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 16 / 19
Proof of the lemma Proof. We finish the proof by: n − 1 � n +1 � � ∞ (1 + 2 q n − k +1 ℓ ( γ )) = (1 + 2 q k ℓ ( γ )) � (1 + 2 q k ℓ ( γ )) = k =0 k =2 k =0 � ∞ � ∞ 2 q k ℓ ( γ )) = e αℓ ( γ ) � 1 + αℓ ( γ ) e αℓ ( γ ) log(1 + 2 q k ℓ ( γ ))) � exp( exp( k =0 k =0 Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 16 / 19
The basin of infinity is a John domain Theorem Let δ < 1 4 and c n ∈ D N δ . Then A c n ( ∞ ) is a John domain. Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 17 / 19
Proof of the basin of infinity being a John domain Proof. Let R > R δ such that R 2 + δ − R � 1 2 , ε := R − R δ < 1, and let U k := F − 1 k (∆ R ) for k ∈ N . Then we have U k ⊂ U k +1 ⊂ A ( c n ) ( ∞ ) and ∞ � A ( c n ) ( ∞ ) = U k . For z ∈ A ( c n ) ( ∞ ) let d ( z ) := dist ( z , J ( c n ) ). k =1 Finally let z ∈ U k and set w := F k ( z ). If U is the component of F − 1 k ( D ε ( w )) containing z , then U ⊂ A ( c n ) ( ∞ ). Now let ρ > 0 be such that D ρ ( z ) ⊂ U . Let z ′ ∈ D ρ ( z ) and w ′ := F k ( z ′ ). We begin the proof by finding a lower bound for d ( z ). Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 18 / 19
Proof of the basin of infinity being a John domain Proof. Krzysztof Lech Random quadratic Julia sets and quasicircles March 27, 2020 18 / 19
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