Introduction to the Julia and Mandelbrot Sets

Julia and Mandelbrot Sets

Kathryn Lindsey

Introduction Introduction to the Julia and Mandelbrot Julia Set Deﬁnition A dichotomy Sets Mandelbrot Set Preliminaries Uniformization Theorem Poincar´emetric Pick Theorem Kathryn Lindsey degree Bottcher Department of Mathematics coordinates the idea Cornell University coordinates on a neighborhood of ∞ Green’s function external rays land Math 6120, April 24, 2009 proof of hyperbolic case Mandelbrot set is connected Outline

1 Introduction Julia and Julia Set Definition Mandelbrot Sets A dichotomy Kathryn Lindsey Mandelbrot Set Introduction Julia Set Definition 2 Preliminaries A dichotomy Uniformization Theorem Mandelbrot Set Preliminaries Poincarémetric Uniformization Theorem Pick Theorem Poincarémetric Pick Theorem degree degree Bottcher 3 Bottcher coordinates coordinates the idea the idea coordinates on a neighborhood of ∞ coordinates on a neighborhood of ∞ Green’s function external rays land Green’s function proof of hyperbolic case external rays land Mandelbrot set proof of hyperbolic case is connected 4 Mandelbrot set is connected Julia set

Julia and Definition Mandelbrot Sets Kathryn Lindsey Let p : C → C be a polynomial of degree d ≥ 2. The filled-in Julia Set Kp and the Julia Set Jp are Introduction Julia Set Definition A dichotomy n Mandelbrot Set Kp = {z ∈ |p (z) ∞}, Jp = ∂Kp C 9 Preliminaries Uniformization Theorem Poincarémetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set √ is connected 2 1− 5 p(z) = z + 2 2 Kp for p(z) = z − 0.624 + .435i

Julia and Mandelbrot Sets

Kathryn Lindsey

Introduction Julia Set Deﬁnition A dichotomy Mandelbrot Set Preliminaries Uniformization Theorem Poincar´emetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected 2 Kp for p(z) = z + .295 + .55i

Julia and Mandelbrot Sets

Kathryn Lindsey

Julia and Mandelbrot Sets

Kathryn Lindsey

Julia and Mandelbrot Sets

Kathryn Lindsey

Definition Introduction Julia Set Definition z0 is a critical point of p if p is not a local homeomorphism A dichotomy Mandelbrot Set at z0. We will use Ωp to denote the critical points of p. Preliminaries Uniformization Theorem Theorem Poincarémetric Pick Theorem Let p be a polynomial of degree d ≥ 2. If Ωp ⊂ Kp, then Kp degree Bottcher is connected. If Ω ∩ Kp = ∅, then Kp is a Cantor set. coordinates the idea coordinates on a neighborhood of ∞ If p only has one critical point, then this theorem gives a Green’s function external rays land dichotomy. proof of hyperbolic case Mandelbrot set is connected Mandelbrot Set

Julia and Definition Mandelbrot Sets Let Kc be the filled Julia set corresponding to the polynomial Kathryn Lindsey 2 pc (z) = z + c. The Mandelbrot set M is the set Introduction Julia Set Definition A dichotomy M = {c ∈ C : 0 ∈ Kc } = {c ∈ C : Kc is connected}. Mandelbrot Set Preliminaries Uniformization Theorem Poincarémetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Julia and Mandelbrot Sets

Kathryn Lindsey

Introduction Julia Set Deﬁnition A dichotomy Youtube video zoom in on Mandelbrot Set Mandelbrot Set Preliminaries Uniformization Theorem Poincar´emetric Pick Theorem degree Mandelbrot explorer applet Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Wikipedia

Julia and Mandelbrot Sets From Wikipedia’s “Mandelbrot Set:” Kathryn Lindsey

Introduction Julia Set Definition A dichotomy “Mandelbrot studied the parameter space of quadratic Mandelbrot Set polynomials in an article that appeared in 1980. The Preliminaries Uniformization Theorem mathematical study of the Mandelbrot set really began with Poincarémetric Pick Theorem work by the mathematicians Adrien Douady and John degree Hubbard, who established many of its fundamental Bottcher coordinates properties and named the set in honor of Mandelbrot....The the idea coordinates on a work of Douady and Hubbard coincided with a huge increase neighborhood of ∞ Green’s function in interest in complex dynamics and abstract mathematics, external rays land proof of hyperbolic and the study of the Mandelbrot set has been a centerpiece case Mandelbrot set of this field ever since” is connected Uniformization Theorem

Julia and Theorem Mandelbrot Sets Kathryn Lindsey Any simply connected Riemann surface is conformally isomorphic to exactly one of , the open disk , or the Introduction C D Julia Set Definition Riemann sphere ˆ. A dichotomy C Mandelbrot Set Preliminaries Theorem Uniformization Theorem Every Riemann surface S is conformally isomorphic to a Poincarémetric Pick Theorem quotient of the form S˜/Γ, where S˜ is a simply connected degree Riemann surface and Γ =∼ π (S) is a group of conformal Bottcher 1 coordinates automorphisms which acts freely and properly the idea coordinates on a discontinuously on S.˜ neighborhood of ∞ Green’s function external rays land proof of hyperbolic Definition case A hyperbolic Riemann surface is one whose universal cover Mandelbrot set is connected is conformally isomorphic to D Poincarémetric on hyperbolic surfaces

Julia and Mandelbrot Sets Poincar´emetric on D Kathryn Lindsey

is the unique Riemannian metric (up to multiplication by a Introduction Julia Set Definition constant) on D that is invariant under all conformal A dichotomy isomorphisms of D. The Poincarémetric on D is complete Mandelbrot Set Preliminaries and has the property that any two points are connected by a Uniformization Theorem unique geodesic. Poincarémetric Pick Theorem degree Bottcher Poincarémetric on a hyperbolic surface S The universal coordinates the idea cover S˜ is conformally isomorphic to , so S˜ has a Poincaré coordinates on a D neighborhood of ∞ Green’s function metric, which is invariant under deck transformations. The external rays land proof of hyperbolic Poincarémetric on S is the unique Riemannian metric on S case so that the projection S˜ → S is a local isometry. Mandelbrot set is connected Julia and The Schwarz lemma says that if f : D → D is a holomorphic Mandelbrot Sets map that fixes the origin, then |f (z)| ≤ |z| for all z, and if Kathryn Lindsey there exists z0 6= 0 such that |f (z0)| = |z0| then f is a Introduction Julia Set Definition rotation. A dichotomy Mandelbrot Set Preliminaries Uniformization Theorem Poincarémetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Julia and The Schwarz lemma says that if f : D → D is a holomorphic Mandelbrot Sets map that fixes the origin, then |f (z)| ≤ |z| for all z, and if Kathryn Lindsey there exists z0 6= 0 such that |f (z0)| = |z0| then f is a Introduction Julia Set Definition rotation. A dichotomy Mandelbrot Set Preliminaries Uniformization Theorem Poincarémetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Julia and Mandelbrot Sets

Kathryn Lindsey

Introduction Theorem Julia Set Deﬁnition 0 A dichotomy (Pick Theorem) If f : S → S is a holomorphic map between Mandelbrot Set hyperbolic surfaces, then exactly one of the following three Preliminaries Uniformization Theorem statements holds: Poincar´emetric Pick Theorem 1. f is a conformal isomorphism; f is an isometry. degree 2. f is a covering map but is not one-to-one; f is locally but Bottcher coordinates not globally a Poincare isometry. the idea coordinates on a 3. f strictly decreases all nonzero distances. neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected covering maps have constant degree

Julia and Mandelbrot Sets

Kathryn Lindsey

Introduction Julia Set Deﬁnition Theorem A dichotomy Suppose X and Y are Riemann surfaces and f : X → Y is a Mandelbrot Set Preliminaries proper non-constant holomorphic map. Then there exists Uniformization Theorem n ∈ N such that f takes every value c ∈ Y , counting Poincar´emetric Pick Theorem multiplicies, n times. degree Bottcher coordinates A proper non-constant holomorphic maps is called an the idea coordinates on a neighborhood of ∞ n-sheeted holomorphic covering map, where n is as above. Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected the idea of Bottcher coordinates

Julia and Mandelbrot Sets Question: for a polynomial p, how can we understand the Kathryn Lindsey

structure of Kp? Introduction Julia Set Deﬁnition A dichotomy Mandelbrot Set Idea: Find a neighborhood UR of ∞ on which there is an Preliminaries Uniformization analytic homeomorphism Theorem Poincar´emetric Pick Theorem ϕ : UR → DR = {z : |z| > R} degree Bottcher coordinates k which conjugates f and the map z 7→ z , i.e. for z ∈ UR , the idea coordinates on a neighborhood of ∞ k Green’s function ϕ(f (z)) = (ϕ(z)) . external rays land proof of hyperbolic case Then try to extend this to ϕ : − Kp → − . Mandelbrot set C C D is connected the idea of Bottcher coordinates

Julia and Mandelbrot Sets

Kathryn Lindsey

Julia and Mandelbrot Sets Theorem Kathryn Lindsey k Let f (z) = z (1 + g(z)) be an analytic mapping on Cˆ with f (∞) = ∞, k ≥ 2 and |g(z)| ∈ O(1/|z|). Then there exists Introduction Julia Set Definition a neighborhood U of ∞ and an analytic mapping ϕ : U → A dichotomy C Mandelbrot Set k with (ϕ(z)) = ϕ(f (z)). Preliminaries Uniformization Theorem n 1/kn Poincarémetric The idea is to “define” ϕ(z) = limn→∞ (f (z)) , BUT Pick Theorem n degree the problem is we have to specify which k th root is being Bottcher considered. coordinates the idea coordinates on a neighborhood of ∞ n k n+1 k n 1/k n+1 1/k k Green’s function (ϕ(z)) = lim (f (z)) = lim (f (z) ) external rays land n→∞ n→∞ proof of hyperbolic case k 1 Mandelbrot set n+1 n+1 n+1 kn = lim f (z) k = lim f (z) = ϕ(f (z)) is connected n→∞ n→∞ Proof of Bottcher coordinates on nbhd of ∞

n Julia and We will write (f n(z))1/k as a telescoping product in such a Mandelbrot Sets way that in each factor we can use the principal branch of Kathryn Lindsey

the root (i.e. no factor lies on the negative real axis). Introduction Julia Set Deﬁnition A dichotomy 1/k 2 1/k2 n 1/kn n (f (z)) (f (z)) (f (z)) Mandelbrot Set (f n(z))1/k = z · · · ... · . 1/k n−1 1/kn−1 Preliminaries z (f (z)) (f (z)) Uniformization Theorem Poincar´emetric The general term of this product is Pick Theorem degree

m m Bottcher (f m(z))1/k ((f m−1(z))k (1 + g(f m−1(z))))1/k coordinates the idea m−1 = m−1 m−1 1/k m−1 1/k coordinates on a (f (z)) (f (z)) neighborhood of ∞ Green’s function m external rays land m−1 1/k proof of hyperbolic = (1 + g(f (z))) case Mandelbrot set So we want to show that there exists r > 0 so that |z| ≥ r is connected implies |g(f m−1(z))| < 1 for all m. Proof of Bottcher coordinates on nbhd of ∞

Julia and Mandelbrot Sets C Pick r1 > 0 and C > 0 so that |g(z)| < |z| for |z| ≥ r1. Kathryn Lindsey k+1 Let r2 be the greatest positive root of x (1 + Cx) = 1. 1 Introduction Set r = max(r1, r2, ). Julia Set Definition 2C A dichotomy Mandelbrot Set Then for |z| ≥r: Preliminaries Uniformization Theorem Poincarémetric Pick Theorem |f (z)| = |z|k |1 + g(z)| ≥ |z|r k−1|1 + Cr| ≥ |z| degree Bottcher m coordinates ∴ |f (z)| ≥ |z| ∀m the idea coordinates on a m−1 C C 1 neighborhood of ∞ ∴ |g(f (z)| ≤ ≤ = Green’s function |f m−1(z)| 2C 2 external rays land proof of hyperbolic n 1/kn case ∴ the expression (f (z)) is well-defined Mandelbrot set is connected Proof of Bottcher coordinates on nbhd of ∞

Julia and Mandelbrot Sets

Kathryn Lindsey

Introduction n Julia Set Deﬁnition It remains to show that the limit lim (f n(z))1/k A dichotomy n→∞ Mandelbrot Set converges. The maximum value of | ln(1 + w)| for |w| ≤ 1/2 Preliminaries Uniformization is ln(2) and is achieved at w = −1/2. Then Theorem Poincar´emetric Pick Theorem n 1 ln 2 degree ln |1 + g(f m−1(z))|1/k = ln |1 + g(f m−1(z))| ≤ . m m Bottcher k k coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Green’s function

Julia and Mandelbrot Sets

Kathryn Lindsey Lemma k ˆ Introduction Let f (z) = z (1 + g(z)) be an analytic mapping on C with Julia Set Definition A dichotomy f (∞) = ∞, k ≥ 2 and |g(z)| ∈ O(1/|z|) near ∞. Let A∞ Mandelbrot Set be the basin of attraction of ∞ and let Preliminaries n Uniformization Z = {z : f (z) = ∞ for some n ∈ }. Then the sequence of Theorem N Poincarémetric functions Gn : A∞ → [−∞, ∞) defined by Pick Theorem degree Bottcher 1 1 coordinates Gn(z) = n ln n the idea k f (z) coordinates on a neighborhood of ∞ Green’s function external rays land converges uniformly on compact subset of A∞, with poles on proof of hyperbolic Z, and the limit G satisfies G(f (z)) = kG(z). case Mandelbrot set is connected G converges on A∞ because all points in A∞ eventually −n 1 enter U. Gn(f (z)) = k ln | f n+1(z) | =

−(n+1) 1 k · k ln | f n+1(z) | = kGn+1(z).

Julia and Mandelbrot Sets

Kathryn Lindsey

Proof. Introduction n 1/kn Julia Set Deﬁnition G converges on U because |f (z)| converges A dichotomy 1 1/kn −n 1 Mandelbrot Set ⇒ | f n(z) | converges ⇒ k ln | f n(z) | = Gn(z) converges Preliminaries Uniformization on U. Theorem Poincar´emetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected −n 1 Gn(f (z)) = k ln | f n+1(z) | =

−(n+1) 1 k · k ln | f n+1(z) | = kGn+1(z).

Julia and Mandelbrot Sets

Kathryn Lindsey

Proof. Introduction n 1/kn Julia Set Deﬁnition G converges on U because |f (z)| converges A dichotomy 1 1/kn −n 1 Mandelbrot Set ⇒ | f n(z) | converges ⇒ k ln | f n(z) | = Gn(z) converges Preliminaries Uniformization on U. Theorem Poincar´emetric Pick Theorem G converges on A∞ because all points in A∞ eventually degree Bottcher enter U. coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Julia and Mandelbrot Sets

Kathryn Lindsey

Proof. Introduction n 1/kn Julia Set Deﬁnition G converges on U because |f (z)| converges A dichotomy 1 1/kn −n 1 Mandelbrot Set ⇒ | f n(z) | converges ⇒ k ln | f n(z) | = Gn(z) converges Preliminaries Uniformization on U. Theorem Poincar´emetric Pick Theorem G converges on A∞ because all points in A∞ eventually degree −n 1 Bottcher enter U. Gn(f (z)) = k ln | f n+1(z) | = coordinates the idea coordinates on a −(n+1) 1 neighborhood of ∞ k · k ln | n+1 | = kGn+1(z). Green’s function f (z) external rays land proof of hyperbolic case Mandelbrot set is connected Julia and Mandelbrot Sets

Kathryn Lindsey Notation: For 0 < ρ ≤ 1, define Uρ to be the connected Introduction component of {z ∈ U : G(z) < ln ρ} which contains 0, and Julia Set Definition A dichotomy let ρ0 be the supremum of the ρ such that Uρ contains no Mandelbrot Set critical point of f other than 0. Preliminaries Uniformization Theorem Poincarémetric Pick Theorem Proposition degree Bottcher The map ϕ extends to an analytic isomorphism from Uρ0 to coordinates {z : |z| > eρ0 }. In particular, if A contains no critical point the idea ∞ coordinates on a neighborhood of ∞ of f other than ∞, the map ϕ is a conformal map from the Green’s function external rays land immediate basin of 0 to C − D. proof of hyperbolic case Mandelbrot set is connected Julia and Proof. Mandelbrot Sets There exists n ∈ sufficiently large so that U n is contained Kathryn Lindsey N ρ0 in the domain of definition of ϕ. The restriction Introduction f : U n−1 → Uρn is a covering map of degree k ramified only Julia Set Definition ρ0 0 A dichotomy k Mandelbrot Set at ∞, and the map z 7→ z as a map from ˆ − D n−1 to C ρ0 Preliminaries Uniformization ˆ − D n is also a covering map ramified only at ∞ (because Theorem C ρ0 Poincarémetric U n−1 ∩ Ωp = ∅). There is only one such ramified covering Pick Theorem ρ0 degree space (up to automorphisms). Hence there are precisely k Bottcher coordinates different maps gi : U n−1 → D n−1 (the k lifts of ϕ) such ρ0 ρ0 the idea k coordinates on a that ϕ(f (z)) = (gi (z)) . These k lifts of ϕ different by neighborhood of ∞ Green’s function postmultiplication by a kth root of unity. But precisely one external rays land proof of hyperbolic of these g coincides with ϕ on U n . This map is the analytic case i ρ0 Mandelbrot set extension of ϕ to U n−1 . Iterating this process, we can is connected ρ0 extend ϕ successively to U n−1 ⊂ U n−2 ⊂ ... ⊂Uρ0 . ρ0 ρ0 The idea of the proof

Julia and Mandelbrot Sets

The case in which A∞ contains no critical point of f other Kathryn Lindsey 2 than ∞ (ex. polynomials z 7→ z + c). Introduction Julia Set Definition A dichotomy Mandelbrot Set Preliminaries Uniformization Theorem Poincarémetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Theorem Let p be a polynomial of degree d ≥ 2 such that every critical point of p is either 1. attracted to an attracting cycle (not infinity), 2. has a finite orbit containing a repelling cycle, or 3. is attracted to a parabolic cycle. 1 Then ψp extends to S .

Can ψ = ϕ−1 be extended to S 1?

Julia and Mandelbrot Sets Suppose you have extended ϕ to C − Kp. Do external rays actually land on the Julia set? Define ψ = ϕ−1, Kathryn Lindsey 1 ψ : C − D → C − Kp. Can we extend ψ to S ? Introduction Julia Set Definition A dichotomy If we could extend ψ to S1, one immediate consequence Mandelbrot Set Preliminaries would be that Kp is locally connected (since the continuous Uniformization Theorem image of a locally connected set is locally connected). Poincarémetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Can ψ = ϕ−1 be extended to S 1?

Julia and Mandelbrot Sets Suppose you have extended ϕ to C − Kp. Do external rays actually land on the Julia set? Define ψ = ϕ−1, Kathryn Lindsey 1 ψ : C − D → C − Kp. Can we extend ψ to S ? Introduction Julia Set Definition A dichotomy If we could extend ψ to S1, one immediate consequence Mandelbrot Set Preliminaries would be that Kp is locally connected (since the continuous Uniformization Theorem image of a locally connected set is locally connected). Poincarémetric Pick Theorem Theorem degree Bottcher Let p be a polynomial of degree d ≥ 2 such that every coordinates the idea critical point of p is either coordinates on a neighborhood of ∞ 1. attracted to an attracting cycle (not infinity), Green’s function external rays land proof of hyperbolic 2. has a finite orbit containing a repelling cycle, or case 3. is attracted to a parabolic cycle. Mandelbrot set 1 is connected Then ψp extends to S . proof external rays land in the hyperbolic case

Julia and Mandelbrot Sets

Kathryn Lindsey

Let Z be the set of attracting cycles. Let V0 be a Introduction Julia Set Definition neighborhood of Z such that p(V0) is relatively compact in A dichotomy −n Mandelbrot Set V0. Define Vn = p (V0) where n ∈ is the smallest N Preliminaries integer such that Ωp ⊂ Vn. Fix R > 0 sufficiently large that Uniformization −1 Theorem p (U) is a relatively compact subset of U, where Poincarémetric Pick Theorem degree U = − (V¯n ∪ {z ∈ − Kp : |ϕp(z)| ≥ R}). Bottcher C C coordinates the idea 0 −1 0 coordinates on a Let U denote p (U). p : U → U is a covering map, and neighborhood of ∞ 0 0 Green’s function so for (z, w) ∈ TU , |(z, w)|U0 = |(p(z), p (z) · w)|U . external rays land proof of hyperbolic case Mandelbrot set is connected Julia and Mandelbrot Sets 0 Kathryn Lindsey As U is a proper subset of U, |(z, w)|U < |(z, w)|U0 for all (z, w) ∈ TU0. Since |(z,w)|U is continuous on Introduction |(z,w)|U0 Julia Set Definition 0 0 A dichotomy {(z, w) ∈ TU : w 6= 0} and U is relatively compact in U, Mandelbrot Set there exists C < 1 such that Preliminaries Uniformization Theorem Poincarémetric |(z, w)|U ≤ C|(z, w)|U0 Pick Theorem degree 0 0 Bottcher for all (z, w) ∈ TU , w 6= 0. Thus for all (z, w) ∈ TU with coordinates the idea w 6= 0 coordinates on a neighborhood of ∞ 0 Green’s function |(z, w)| ≤ C|(z, w)| 0 = C|p(z), p (z)w| . external rays land U U U proof of hyperbolic case Mandelbrot set is connected Denote by αn,t the arc that is the image of the map

2πit 1/dn+1 1/dn ρ → ψp(ρe ), R ≤ ρ ≤ R

and by l the length of α . Write l = sup l . Since n,t n,t n t∈R/Z n,t d ψp(z ) = p(ψp(z)), we have p(αn,t ) = αn−1,td . Hence ln,t ≤ Cln−1,td for all t, and so ln ≤ Cln−1. Thus the ln form a convergent series (by comparison to the geometric series −n C ). It follows that the family of mappings βp : R/Z → U 2πit given by βρ(t) = ψ(ρe ) converges uniformly as ρ & 1.

Julia and Mandelbrot Sets

Kathryn Lindsey 0 0 |(z, w)|U ≤ C|(z, w)|U = C|p(z), p (z)w|U . Introduction Julia Set Deﬁnition A dichotomy Mandelbrot Set Preliminaries Uniformization Theorem Poincar´emetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected Julia and Mandelbrot Sets

Kathryn Lindsey 0 0 |(z, w)|U ≤ C|(z, w)|U = C|p(z), p (z)w|U . Introduction Julia Set Deﬁnition Denote by α the arc that is the image of the map A dichotomy n,t Mandelbrot Set Preliminaries n+1 n 2πit 1/d 1/d Uniformization ρ → ψp(ρe ), R ≤ ρ ≤ R Theorem Poincar´emetric Pick Theorem and by l the length of α . Write l = sup l . Since degree n,t n,t n t∈R/Z n,t d Bottcher ψp(z ) = p(ψp(z)), we have p(αn,t ) = αn−1,td . Hence coordinates the idea ln,t ≤ Cln−1,td for all t, and so ln ≤ Cln−1. Thus the ln form coordinates on a neighborhood of ∞ a convergent series (by comparison to the geometric series Green’s function −n external rays land C ). It follows that the family of mappings βp : R/Z → U proof of hyperbolic 2πit case given by βρ(t) = ψ(ρe ) converges uniformly as ρ & 1. Mandelbrot set is connected M is connected

Julia and Mandelbrot Sets

Kathryn Lindsey For c ∈ C, let ϕc be the map corresponding to the 2 Introduction polynomial Pc (z) = z + c, and let Kc be the corresponding Julia Set Definition ¯ A dichotomy filled-in Julia set. ϕc : C − Kc → C − D. Define Mandelbrot Set Preliminaries Uniformization Φ: C − M → C by Theorem Poincarémetric Pick Theorem degree Φ(c) = ϕc (c). Bottcher coordinates the idea Theorem coordinates on a neighborhood of ∞ The map Φ is an analytic isomorphism − M → − ¯. In Green’s function C C D external rays land proof of hyperbolic particular, the Mandelbrot set M is connected. case Mandelbrot set is connected Proof that M is connected

Julia and Φ is analytic: Mandelbrot Sets Kathryn Lindsey Φ (:= ϕc (c)) the the composition of the analytic mappings c 7→ (c, c) and (z, c) 7→ ϕc (z), so Φ is analytic. Introduction Julia Set Definition A dichotomy Φ is proper: Mandelbrot Set Preliminaries Uniformization ∞ Theorem −n n X c Poincarémetric −Gc (z) = lim 2 ln |Pc (z)| = ln |z| + ln 1 + . Pick Theorem n→∞ Pn(z) degree n=1 c Bottcher coordinates ∞ the idea X −n c coordinates on a ln |Φ(c)| = −Gc (c) = ln |c| + 2 ln 1 + n . neighborhood of ∞ P (c) Green’s function n=1 c external rays land proof of hyperbolic case c If |c| > 2, then 1 + n ≤ 2, so −Gc (c) − ln |c| is Mandelbrot set Pc (c) is connected bounded, and c 7→ |Gc (c)| is a proper map. Julia and Mandelbrot Sets c 7→ |Gc (c)| is proper implies immediately that Kathryn Lindsey Φ: C − M → C − D¯ is proper, because for any compact set ¯ Introduction in C − D is a closed subset of an annulus Julia Set Definition −1 A dichotomy {z ∈ C : r1 ≤ |z| ≤ r2} for some 1 < r1 < r2 < ∞, and Φ Mandelbrot Set of such an annulus is compact: Preliminaries Uniformization Theorem Poincarémetric {c ∈ C : r1 ≤ |Φ(c)| ≤ r2} = {c ∈ C : ln r1 ≤ |Gc (c)| ≤ ln r2}, Pick Theorem degree Bottcher which is compact. coordinates the idea coordinates on a Φ is surjective: neighborhood of ∞ Green’s function external rays land Φ proper implies its image is closed, and all non-constant proof of hyperbolic analytic mappings are open, so the image is both closed and case ¯ Mandelbrot set open in C − D. Hence Φ is surjective. is connected Julia and Mandelbrot Sets

Kathryn Lindsey

Φ is a bijection: Introduction Julia Set Deﬁnition P∞ −n c A dichotomy ln |Φ(c)| = −Gc (c) = ln |c| + 2 ln 1 + n , Mandelbrot Set n=1 Pc (c) Preliminaries and −G(c) − ln |c| is bounded, in particular bounded near Uniformization Theorem ∞, so Φ(c)/c is bounded near ∞. Hence Φ(c)/c is Poincar´emetric Pick Theorem bounded near ∞, which implies Φ has a simple pole at ∞. degree Thus Φ extends to an analytic map ¯ − M → ¯ − D¯ , and Bottcher C C coordinates the only inverse image of ∞ is ∞, which local degree 1. the idea coordinates on a Hence Φ has degree 1, i.e. Φ is a bijection. neighborhood of ∞ Green’s function external rays land proof of hyperbolic case Mandelbrot set is connected