Fractals/Iterations in the complex plane/julia

right|thumb|Various types of dynamics needs various algorithms Types Classification of Julia sets according to : topology of filled Julia set's interior shapeblog scientific american : fractal-kitties-illustrate-the-endless-possibilities-for-julia-sets By Evelyn Lamb on September 26, 2012 local dynamics near periodic points connectedness position of parameter c on the parameter plane math.stackexchange question : classification-of-points-in-the-mandelbrot-set Filled Julia set can have : an non-empty interior ( Julia set is connected ) parabolic: filled Julia set have parabolic cycle ( c is on boundary of hyperbolic component ) Siegel : filled Julia sets containing Siegel disc. Julia set can be locally connected or not. That depends on the rotation number. ( c is on boundary of hyperbolic component ) attracting : filled Julia set have attracting cycle ( c is inside hyperbolic component ) superattracting : filled Julia set have superattracting cycle( c is in the center of hyperbolic component ). Examples : Airplane Julia set, Douady's Rabbit, Basillica. empty interior disconnected ( c is outside of Mandelbrot set )images of disconnected Julia sets connected Cremer Julia sets ( c is on boundary of hyperbolic component , Julia set is connected but not locally connected) dendrits ( Julia set is connected and locally connected ) Misiurewicz Julia sets (c is a Misiurewicz point ) Feigenbaum Julia sets ( ) others which have no Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle by Mikhail Lyubich topology Julia set of root point is topologically the same as the Julia set of the child period center, but the center ( nucleus) Julia set is very easy to draw ( superattracting basin = very fast dynamics because critical point is also periodic point) while the root Julia set ( parabolic) is hard to draw ( parabolic basin and lasy dynamics) Examples: t = 1/2 Julia set of the root point = fat Basilica Julia set: c = -3/4 = - 0.75 Julia set of period 2 center = (slim) Basicica Julia set: c = -1 t = 1/3 Julia set of the root point = fat Douady Rabbit: c = -0.125000000000000 +0.649519052838329i Julia set of period 3 center = (slim) Doudy Rabbit Julia set: c = -0.122561166876654 +0.744861766619744i period = 3 Models of Julia set right|thumb|Lamination of rabbit Julia set Thurston’s invariant lamination of Julia set Simple topological models of Julia sets by L. Oversteegen graph of Julia set Combinatorial Julia Sets (1) By Jim BelkJacek Skryzalin: On Quadratic Mappings With and Attracting Cycle Schreier graphs spine of Julia set disked tree model ( Hubbard trees serve as a model for Julia sets of post-critically finite polynomialsEugene1806's Blog) Douady’s pinched disk model Adrien Douady, Descriptions of compact sets in C. , in: Topological Methods in Modern Mathematics, Publish or Perish (1993), 429– 465. Milnor’s orbit portraits dendrite Dendritic model for Julia sets Hausdorff dimension of biaccessible angles of quadratic Julia sets and of the Mandelbrot set by Henk Bruin and Dierk Schleicher components of Julia set's interior Douady’s pinched disk model Dynamic rays and their landing properties are a key tool to understanding (the topology of) Julia sets of polynomials. In particular, the structure of the Julia set is determined by rays that land at a common point: at least in good cases (under the assumption of local connectivity), the knowledge of which rays land together gives a homeomorphic model for the Julia set that is known as Douady’s pinched disk model HAUSDORFF DIMENSION OF BIACCESSIBLE ANGLES FOR QUADRATIC POLYNOMIALS. VERSION OF May 8, 2017 by HENK BRUIN AND DIERK SCHLEICHER Algorithms "... a single algorithm for computing all quadratic Julia sets does not exist."Computability of Julia sets by Mark Braverman, Michael Yampolsky Examples: rosetta code: Julia_set Algorithms Types Escape time ( attraction time to infinity ( attractor for all polynomials ) attraction time to finite attractor inside filled Julia set ) estimation of distance to Julia set ( DEM/J ) Inverse Iteration Method = IIM/J Testing equicontinouty by Michael Becker Some Julia sets by Michael Becker, 6/2003. Last modification: 2/2004. orbit traps orbit traps example videoorbit traps making finding periodic repelling points using numerical methods. They are dense in the Julia set. ( Newton method Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees by Robin Stoll, Dierk Schleicher ) equicontinouty "The Julia set of f then is the set of all points of G, at which this sequence of iterated functions is not equicontinous. The Fatou set is its complement. Laxly said the action of the iterated functions on near points is examined. Places, where points, which are near enough, remain near during iterations, belong to the Fatou set. Places, where points, as near they may be, are teared apart, belong to the Julia set. In the following I only consider functions, which map the Riemann sphere, i.e. the complex plane with an ideal point "infinity" added, to itself. The Julia sets are white, the Fatou sets black." Michael Becker accuracy Accuracy of Computer Generated Approximations to Julia Sets by John W. Hoggard plane scanning shadowing lemma Numerical Precision points "We know the periodic points are dense in the Julia set, but in the case of weird ones (like the ones with Cremer points, or even some with Siegel disks where the disk itself is very 'deep' within the Julia set, as measured by the external rays), the periodic points tend to avoid certain parts of the Julia set as long as possible. This is what causes the 'inverse method' of rendering images of Julia sets to be so bad for those cases." ( answered Oct 26 '14 at 14:52 by Jacques Carette )Clustering of periodic points for a polynomial iteration of C thumb|right|Distribution of points of inverse orbit of repelling fixed point of complex quadratic polynomial. thumb|right|Periodic points of f(z) = z*z-0.75 for period =6 as intersections of 2 implicit curves compare with: virtual math museum: julia set Special cases "The Julia set is nonempty when d ≥ 2, but the Fatou set may be empty (as demonstrated by Latt`es examples)."Lucjan Emil Böttcher and his mathematical legacy Stanisław Domoradzki, Małgorzata Stawiska Names cauliflower = Julia set for c = 1/4 (c is the cusp of main cardioid), parabolic Julia set an imploded cauliflower is a Julia set for with [6] Douady A., Buff X., Devaney RL. and Sentenac P. Baby Mandelbrot sets are born in cauliflowers. In: Lei T. editor. The Mandelbrot set, theme and variations. Cambridge: University Press, 2000, p. 19-36. , disconnected Julia set airplane Julia set. C is the center of period 3 component on the real axis: c = -1.75487766624669276 helicopter z → z^3 − 0.2634 − 1.2594i (Douady) rabbit. C is a the center period 3 hyperbolic component of Mandelbrot set for complex quadratic map Cubic Rabbit z → z^3 + 0.545 + 0.539i dendrite. C is a tip the Kokopelli Julia set The Thurston Algorithm for quadratic matings by Wolf Jung The angle 3/15 = p0011 = 0.(0011) has preperiod = 0 and period = 4. The conjugate angle on the parameter plane is 4/15 or p0100. The kneading sequence is AAB* and the internal address is 1-3-4. The corresponding parameter rays are landing at the root of a primitive component of period 4. Basilica = = Julia set for c = -1 (c is the center of period 2 component) See also How_to_move_on_parameter_plane_? How_to_choose_a_point_from_parameter_plane_? References