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 1 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 Poincar´emetric Uniformization Theorem Pick Theorem Poincar´emetric Pick Theorem degree degree Bottcher 3 Bottcher coordinates coordinates the idea the idea coordinates on a neighborhood of 1 coordinates on a neighborhood of 1 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 = fz 2 jp (z) 1g; Jp = @Kp C 9 Preliminaries Uniformization Theorem Poincar´emetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of 1 Green's function external rays land proof of hyperbolic case Mandelbrot set p 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 1 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 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 1 Green's function external rays land proof of hyperbolic case Mandelbrot set is connected 2 Kp for p(z) = z + 0:285 + 0:01i 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 1 Green's function external rays land proof of hyperbolic case Mandelbrot set is connected 2 Kp for p(z) = z − 0:70176 − 0:3842i 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 1 Green's function external rays land proof of hyperbolic case Mandelbrot set is connected 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 Poincar´emetric 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 1 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 = fc 2 C : 0 2 Kc g = fc 2 C : Kc is connectedg: Mandelbrot Set Preliminaries Uniformization Theorem Poincar´emetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of 1 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 1 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 Poincar´emetric 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 1 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 ^. A dichotomy C Mandelbrot Set Preliminaries Theorem Uniformization Theorem Every Riemann surface S is conformally isomorphic to a Poincar´emetric 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 1 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 Poincar´emetric 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 Poincar´emetric on D is complete Mandelbrot Set Preliminaries and has the property that any two points are connected by a Uniformization Theorem unique geodesic. Poincar´emetric Pick Theorem degree Bottcher Poincar´emetric on a hyperbolic surface S The universal coordinates the idea cover S~ is conformally isomorphic to , so S~ has a Poincar´e coordinates on a D neighborhood of 1 Green's function metric, which is invariant under deck transformations. The external rays land proof of hyperbolic Poincar´emetric 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 jf (z)j ≤ jzj for all z, and if Kathryn Lindsey there exists z0 6= 0 such that jf (z0)j = jz0j then f is a Introduction Julia Set Definition rotation. A dichotomy Mandelbrot Set Preliminaries Uniformization Theorem Poincar´emetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of 1 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 jf (z)j ≤ jzj for all z, and if Kathryn Lindsey there exists z0 6= 0 such that jf (z0)j = jz0j then f is a Introduction Julia Set Definition rotation. A dichotomy Mandelbrot Set Preliminaries Uniformization Theorem Poincar´emetric Pick Theorem degree Bottcher coordinates the idea coordinates on a neighborhood of 1 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 1 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 2 N such that f takes every value c 2 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 1 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 1 on which there is an Preliminaries Uniformization analytic homeomorphism Theorem Poincar´emetric Pick Theorem ' : UR ! DR = fz : jzj > Rg degree Bottcher coordinates k which conjugates f and the map z 7! z , i.e.