Fractals/Iterations in the complex plane/def cqp

Definitions Order is not only alphabetical but also by topic so use find (Ctrl-f) See also Pictures_of_Julia_and_Mandelbrot_Sets - Terminology Index of Mu-Ency from Robert Munafo's home pages on HostMDS   © 1996-2020 Robert P. Munafo. fractalNotes by perianney Category: Book Fractals , something like index of pages Address "Internal addresses encode kneading sequences in human-readable form, when extended to angled internal addresses they distinguish hyperbolic components in a concise and meaningful way. The algorithms are mostly based on Dierk Schleicher's paper Internal Addresses Of The Mandelbrot Set And Galois Groups Of Polynomials (version of February 5, 2008) http://arxiv.org/abs/math/9411238v2." Claude Heiland-Allenhaskell package: ruff-0.2by Claude Heiland-Allen types finite / infinite accessible/non-accessible on the parameter plane / on th edynamic plane simple/ angled for Crossed RenormalizationsOn the Locus of Crossed Renormalization (Problems on complex dynamical systems) by Riedl, Johannes; Schleicher, Dierk Internal the internal address of a hyperbolic component A lists the periods of certain components that are “on the way” from the main cardioid to hyperbolic component ATrees of visible components in the Mandelbrot set by Virpi K a u k o Internal addresses describe the combinatorial structure of the Mandelbrot set.Rational Maps with Clustering and the Mating of Polynomials by Thomas Joseph Sharland It is one of the Analytical Naming Systemsanalytical naming system From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020.  math.stackexchange question: naming-bulbs-on-the-mandelbrot-set the ancestral route of a hyperbolic component is the ordered sequence of all its ancestors Internal address: is not constant within hyperbolic component. Example: internal address of -1 is 1->2 and internal address of 0.9999 is 1Topics from One-Dimensional Dynamics by Karen M. Brucks,Henk Bruin. page 265 exercise 14.2.12 of hyperbolic component is defined as a internal address of it's center In an internal address, the numbers (period) must be increasing by definition. The internal address is describing a kneading sequence by increasing periods.Combinatorics, external rays, and twisted polynomials. by Wolf Jung These correspond to hyperbolic components in M, where the kneading sequence is changing. Example: AABA∗ is obtained by changing A → AAB → AABA∗ , so the internal address is 1-3-5. Conversely, the internal address 1-3-5 gives A → AAB → AABA∗ . angled Angled internal address is an extension of internal address. The angled internal address of the end of a finite chain of child bulbs would be: Examples: describes period 6 component which is a satelite of period 3 component. Mandelbrot Artists by Claude Heiland-Allen Elements period of hyperbolic componnet angle of internal ray One can see the address as: sequence of hyperbolic components path inside Mandelbrot set Path inside Mandelbrot set: start with center of period 1 ( nucleus) internal ray with angle n/m root point n/m ( bond) internal angle center with given period ... Problems Infinite sequences: islands infinite sequence of bifurcations Angle Types of angle thumbnail|right|Principal branch or complex number argument external angle internal angle plain angleparameter plane dynamic plane where: is a multiplier map is a Boettcher function external The external angle is an angle of point of set's exterior the boundary. It is: the same on all points on the external ray. It is important for proving connectedness of the Mandelbrot set. a proper fraction an approximation of directional derivative internal The internal anglemuency - internal angle (the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2016.) is an angle of point of component's interior it is a rational number and proper fraction measured in turns (see multiplier map) it is the same for all point on the internal ray in a contact point (root point) it agrees with the rotation number root point has internal angle 0 (inside child component) "The internal angles start at 0, at the cusp, and increase counterclockwise. " Robert Munafointernal angle from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017 Internal angle of the wake root point angles of the wake = angles of parameter rays that land on the root point angles of dynamic rays that land on the alpha fixed point angles of dynamic rays that land on the critical point and critical value angles of principal Misiurewicz point See also binary search for internal angle . From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022.     Mu-ency index plain The plain angle is an angle of complex point = its argumentargument of complex number bearing angle in CSS By convention, when an angle denotes a direction in CSS, it is typically interpreted as a bearing angle, where:w3.org docs: CSS and angle-value 0deg is "up" or "north" on the screen and larger angles are more clockwise (so 90deg is "right" or "east") Units turns degrees radians Number types Angle (for example, external angle in turns) can be used in different number types Examples: the external arguments of the rays landing at z = −0.15255 + 1.03294i are:A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set M. Romera, G. Pastor, A. B. Orue, A. Martin, M.-F. Danca, and F. Montoya where: Bifurcation Numerical Bifurcation Analysis of Maps MatContMatcont - is a Matlab software project for the numerical continuation and bifurcation study of continuous and discrete parameterized dynamical systems. Leaders of the project are Willy Govaerts (Gent,B) and Yuri A. Kuznetsov (Utrecht,NL). Coordinate Coordinate: Fatou coordinate for every repelling and attracting petal (linearization of function near parabolic fixed point) Boettcher Koenigs "The coordinates are the current location, measured on the x-y-z axis. The gradient is a direction to move from our current location" Sadid Hasanquora: What-is-gradient? Curves Types: topology: closed versus open simple versus not simple ( complex) infinite, finite at one end ( ray), finite at both ends ( segment) self-intersections, crossing, singularities other properities: invariant critical Points of the curve: regular singular: A point on the curve at which the curve behaves in an extraordinary manner is called a singular point. Points of inflexion Multiple points( n-tuple points):statistics how to: double-points A point on the curve through which more than one branches of curve double : "A double point is a point on a curve where two branches of the curve intersect; in other words, it’s a point traced twice when a curve is traversed." Triple point: A point on the curve through which three branches of curve pass Descriptiongeometry by Dr. Carol JVF Burns plane curve = it lies in a plane. closed = it starts and ends at the same place. simple = it never crosses itself. only regular points See Fractals curves closed Closed curves are curves whose ends are joined. Closed curves do not have end points. Simple Closed Curve: A connected curve that does not cross itself and ends at the same point where it begins. It divides the plane into exactly two regions (Jordan curve theorem). Examples of simple closed curves are ellipse, circle and polygons.What is a Curve ? Complex Closed Curve (not simple = non-simple) It divides the plane into more than two regions. Example: Lemniscates. "non-self-intersecting continuous closed curve in plane" = "image of a continuous injective function from the circle to the plane" Circle Inner circle Unit circle Unit circle is a boundary of unit diskUnit circle in Wikipedia where coordinates of point of unit circle in exponential form are: Critical curves Diagrams of critical polynomials are called critical curves.The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653 These curves create skeleton of bifurcation diagram. Hao Bailin Bailin Hao Elementary Symbolic Dynamics and Chaos in Dissipative Systems World Scientific 1989 http://power.itp.ac.cn/~hao/ 9971-5-0682-3 (the dark linesM. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint) dendrit a locally connected branched curve "Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic."LAMINATIONAL MODELS FOR SOME SPACES OF POLYNOMIALS OF ARBITRARY DEGREE by ALEXANDER BLOKH, LEX OVERSTEEGEN, ROSS PTACEK, AND VLADLEN TIMORIN "a dendrite is a locally connected continuum that does not contain Jordan curves." Models_for_spaces_of_dendritic_polynomials by ALEXANDER BLOKH, LEX OVERSTEEGEN, ROSS PTACEK,AND VLADLEN TIMORIN "a locally connected continuum without subsets homeomorphic to a circle" connected with no interior locally connected, uniquely arcwise connected, compact metric space See also: Misiurewicz point on the parameter plane Dendrite Modeling: Modeling dendrites, including trees, lightning, river systems, and all manner of branching structures, has been frequently undertaken in computer graphics. We propose a new dendritic modeling framework using path planning as the basic operationMODELING DENDRITIC SHAPES Using Path Planning by Ling Xu, David Mould Procedural Branching Textureprocedural-branching-texture in blender by LordoftheFleas Escape lines Escape line = boundary of escape time's level sets "If the escape radius is equal to 2 the contour lines have a contact point (c= -2) and cannot be considered as equipotential lines" Escape lines versus equipotential lines in the Mnadelbrot set by M. Romera, Pastor G, D. de la Guía, Montoya geodesic In geometry, a geodesic is a curve representing in some sense the shortest path (arc) between two points in a surfacePseudosphere Geodesics by Tim Hutton Integral integral curve is a parameterized curve, whose tangent vectors agree with the vectors from this vector field. In physics, integral curves for an electric field or magnetic field are known as field lines. Invariant Types: topological shift invariants examples: curve is invariant for the map f (evolution function) if images of every point from the curve stay on that curve The Computation of Invariant Circles of Maps Article in Physica D Nonlinear Phenomena 16(2):243-251 · June 1985 DOI: 10.1016/0167-2789(85)90061-2 1st I.G. KevrekidisA Newton-Raphson method for numerically constructing invariant curves Marty, WolfgangNumerical Approximation of Rough Invariant Curves of Planar Maps Article in SIAM Journal on Scientific Computing 25(1) · September 2003 DOI: 10.1137/S106482750241373X K. D. Edoh and Jens Lorenz curve is invariant for a system of ordinary differential equationsSIAM J. Sci. and Stat. Comput., 8(6), 951–962. (12 pages) A New Algorithm for the Numerical Approximation of an Invariant Curve Published online: 14 July 2006 Keywords invariant manifold, polygonal approximation AMS Subject Headings 65L99, 65H10, 34C40 Publication Data ISSN (print): 0196-5204 ISSN (online): 2168-3417 Publisher: Society for Industrial and Applied Mathematics M. van Veldhuizen "Quasi-invariant curves are used in the study of hedgehog dynamics" RICARDO PEREZ-MARCOON QUASI-INVARIANT CURVES by RICARDO PEREZ-MARCO Examples: field lines external ray internal ray Isocurves Isocurve = level curve = curve which consist of points which have the same value (level) of parameter / variable Equipotential lines Equipotential lines = Isocurves of complex potential "If the escape radius is greater than 2 the contour lines are equipotential lines" Escape lines versus equipotential lines in the Mnadelbrot set by M. Romera, Pastor G, D. de la Guía, Montoya Examples desmos examples: isovalues by Fabrice Jordan curve thumb|200px|Illustration of the Jordan curve theorem. The Jordan curve (drawn in black) divides the plane into an "inside" region (light blue) and an "outside" region (pink). Jordan curve = a simple closed curve that divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior pointsWikipedia: Jordan curve theorem Lamination Lamination of the unit disk is a closed collection of chords in the unit disc, which can intersect only in an endpoint of each on the boundary circleModeling Julia Sets with Laminations: An Alternative Definition by Debra MimbsLaminations of the unit disk with irrational rotation gaps by John C. Mayer It is a model of Mandelbrot or Julia set. A lamination, L, is a union of leaves and the unit circle which satisfies:Rational maps represented by both rabbit and aeroplane matings Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Freddie R. Exall July 2010 leaves do not cross (although they may share endpoints) and L is a closed set. "The pattern of rays landing together can be described by a lamination of the disk. As θ is varied, the diameter defined by θ/2 and (θ +1)/2 is moving and disconnecting or reconnecting chords. " Wolf Jung Core entropy and biaccessibility of quadratic polynomials by Wolf Jung Leaf Chords = leaves = arcs A leaf on the unit disc is a path connecting two points on the unit circle.Rational maps represented by both rabbit and aeroplane matings Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Freddie R. Exall July 2010 "In Thurston’s fundamental preprint, the two characteristic rays and their common landing point are the “minor leaf” of a “lamination”"Rational Parameter Rays of the Mandelbrot Set by Dierk Schleicher Level curve LCM = Level Curve Method = method for drawing level curves Examples: equipotential line (the same potential) external ray (the same external angle) boundary of level set (see Level Set Method = LSM) Open curve Curve which is not closed. Examples: line, ray. Path Path in geometry is a curve Ray Rays are: invariant curves dynamic or parameter external, internal or extended Extended "We prolong an external ray R θ supporting a Fatou component U (ω) up to its center ω through an internal ray and call the resulting set the extended ray E θ with argument θ." Alfredo PoirierCritical portraits for postcritically finite polynomials by Alfredo Poirier External ray The closure of an external ray is called a closed ray. If ray lands, then the closure of the ray is the union of the external ray and its landing point.NON-ACCESSIBLE CRITICAL POINTS OF CERTAIN RATIONAL FUNCTIONS WITH CREMER POINTS by Lia Petracovici "A ray R is said to land or converge, if the accumulation set is a singleton subset of J. The conjecture that the Mandelbrot set is locally connected is equivalent to the continuous landing of all external rays."Convergence of external rays in parameter spaces of symmetric polynomials by Ahmad Zireh. Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 6, 291 - 296 where: is a closure of = the bar is taken to mean the closure rather than the complex conjugate MLC = Mandelbrot Local connectivity Conjecture: M is locally connectedA survey on MLC, Rigidity and related topics by Anna Miriam Benini singelton set is a set with exactly one element "If the MLC were proved true, the theorem of Caratheodory would give us an extension of the Riemann map to , giving a conformal equivalence of M with D. Given the fractal nature of M, this would be a very surprising result.Local Connectivity of the Mandelbrot Set. by Matt Koster December 4, 2019 A dynamic periodic ray pair is called characteristic if it separates the critical value from all rays and for all k ≥ 1 (except of course from those on the ray pair itself). Every cycle of periodic ray pairs has a unique characteristic ray pair with angles in the union Symbolic dynamics of quadratic polynomials. Preprint (2002) page 96 For non-periodic rays, we allow a characteristic ray pair to contain the critical value: a ray pair is characteristic if consists of two components so that contains all rays Rc(2kϑ) and Rc(2kϑ0) for k ≥ 1 and contains the critical value Internal ray Definition: "The internal rays are the preimages of the radial segments under the coordinate with componenet center corresponding to 0." Alfredo PoirierCritical portraits for postcritically finite polynomials by Alfredo Poirier The internal rays of U are the images of radial lines under the Riemann maps.Graph Replacement Systems for Julia Sets of Quadratic Polynomials by Yuan J. Liu Internal rays are: dynamic (on dynamic plane, inside filled Julia set) |parameter (on parameter plane, inside Mandelbrot set) usuning multiplier map dynamic For a parameter c with superattracting orbit: for every Fatou component of filled julia setwikipedia: Filled_Julia_set there is: a unique periodic or pre-periodic point of the super-attracting orbit a Riemann map that maps:Rational Parameter Rays of The Multibrot Sets by Dominik Eberlein, Sabyasachi Mukherjee, Dierk Schleicher component to unit disc: and point to the origin: The point is called the center of component . For any angle the pre-image of the radial segment of the unit disc is called an internal ray of component with well-defined landing point. where: is The quotient group R/Z See also: Hubbard tree intertwined The internal rays are the curves that connects endpoints of external rays to the origin (the only pole) by winding in the specific way through the Julia set. Unlike the external rays the internal rays allways cross other internal rays, usually at multiple points, hence they are interwinedRobert L. Devaney. "Intertwined internal rays in Julia sets of rational maps." Fundamenta Mathematicae 206.1 (2009): 139-159. <http://eudml.org/doc/283146>. parameter Escape route Escape route is a path inside Mandelbrot set. Escape route 1/2 Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set by Anne M. Burns. Mathematics Magazine Vol. 75, No. 2 (Apr., 2002), pp. 104-116 is part of the real slice of the mandelbrot set) part of the real line x=0 Steps: start from center of period 1 go along internal ray 1/2 to root point of period 2 component go along internal ray 0 to the center of period 2 component go along internal ray 1/2 to root point of period 4 component ... Spider A spider S is a collection of disjoint simple curves called legs Iterated Monodromy Groups of Quadratic Polynomials, I Laurent Bartholdi, Volodymyr V. Nekrashevych (extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infinity GROWTH OF GROUPS DEFINED BY AUTOMATA: ASHLEY S. DOUGHERTY, LYDIA R. KINDELIN, AARON M. REAVES, ANDREW J. WALKER, AND NATHANIEL F. ZAKAHI See: spider algorithm spider program Spine In the case of complex_quadratic_polynomial the spine of the filled Julia set is defined as arc between -fixed point and , with such properties: spine lies inside .Douglas C. Ravenel: External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester https://web.archive.org/web/20120208022215/http://www.math.rochester.edu/u/faculty/doug/oldcourses/215s98/lecture10.html 2012-02-08 This makes sense when is connected and full John Milnor: Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004) spine is invariant under 180 degree rotation, spine is a finite topological tree, Critical point always belongs to the spine.Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case -fixed point is a landing point of external ray of angle zero , is landing point of external ray . Algorithms for constructing the spine: detailed version is described by A. DouadyA. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Georgia, USA, 1986. Simplified version of algorithm: connect and within by an arc, when has empty interior then arc is unique, otherwise take the shortest way that contains .K M. Brucks, H Bruin: Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257 Curve : divides dynamical plane into two components. Computing external angle for c from centers of hyperbolic components and Misiurewicz points: The spine of K is the arc from beta to minus beta. Mark 0 each time C is above the spine and 1 each time it is below. You obtain the expansion in base 2 of the external argument theta of z by C. This simply comes from the two following facts: * 0 < theta < 1/2 if access to z is above the spine, 1/2 < theta < 1 if it is below * function f doubles the external arguments with respect to K, as well as the potential, since Riemman map (Booettcher map) conjugates f to . Note that if c and z are real, the tree reduces to the segment [beta',beta] of the real line, and the sequence of 0 and 1 obtained is just the kneading sequence studied by Milnor and Thurston (except for convention: they use 1 and -1). This sequence appears now as the binary expansion of a number which has a geometrical interpretation. " A. Douady Relation between spine and major leaf of the lamination Vein "A vein in the Mandelbrot set is a continuous, injective arc inside in the Mandelbrot set" "The principal vein is the vein joining to the main cardioid" (Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems. A dissertation by Giulio Tiozzo) Discriminant In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them. Distance thumb|Distance Function See also: metric The applications of non-euclidean distance | Metric Spaces by Zach Star Algorithm Distance Estimation Method DEM/J DEM/M SDF = Signed Distance Function distance fields EDT Euclidean Distance Transform SEDT = squared Euclidean distance transform. Algorithms generating distance fields from boolean fields:distance fields by Philip RideoutDistance Transforms of Sampled Functions by Pedro Felipe Felzenszwalbdsp.stackexchange question: fastest-algorithm-for-distance-transform Marching Parabolas, a linear-time CPU-amenable algorithm. Min Erosion, a simple-to-implement GPU-amenable algorithm. Dynamics symbolicSymbolic Dynamics of Quadratic Polynomials by H. Bruin and D. SchleicherSymbolic Dynamics and Rotation Numbers J. J. P. Veerman Phys. 13A, 1986, 543-576. Symbolic Dynamics of Order-Preserving Orbits J. J. P. Veerman Phys. 29D, 1987, 191-201. complex Walter Bergweiler: A gallery of complex dynamics pictures.Around the boundary of complex dynamics by Roland K. W. Roeder Arithmetic combinatorial local/global discrete/continous parabolic/hyperbolic/eliptic Examples: discrete local complex parabolic dynamics + evolution of dynamics along escape route 0 ( parabolic implosion); Im(c) = 0 parameter c location of c Julia set interior type of critical orbit dynamics critical point fixed points stability of alfa c = 0 center, interior connected = Circle Julia set exist superattracting attracted to alfa fixed point fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling r = 0 0<c<1/4 internal ray 0, interior connected exist attracting attracted to alfa fixed point alfa is attracting, beta is repelling 0 < r < 1.0 c = 1/4 cusp, boundary connected = cauliflower exist parabolic attracted to alfa fixed point alfa fixed point equal to beta fixed point, both are parabolic r = 1 c>1/4 external ray 0, exterior disconnected = imploded cauliflowerdisappears repelling repelling to infinity both finite fixed points are repelling r > 1 symbolic "Symbolic dynamics encodes: Freely downloadable book Elementary Symbolic Dynamics and Chaos in Dissipative Systems by Bailin HAO, World Scientific, 1989, by kind permission of the publisher. a dynamical system by a shift map on a space of sequences over finite alphabet using Markov partition of the space the points of space by their itineraries with respect to the partition " (Volodymyr Nekrashevych - Symbolic dynamics and self-similar groups) entropy image entropy Image entropy by Dave O'Brien equation differential differential equations exact analytic solutions. approximated solution use perturbation theory to approximate the solutions Field Field is a region in space where each and every point is associated with a value. The field types according to the value type: scalar field Distance field – Some mapping , where for any given input the output is the distance to the nearest surface (where the field value is 0).fractal-rendering from cglearn vector field, for example gradient field Function types: by application map = iterated map mappings = transformation of the plane by function type polynomial Derivative Derivative of Iterated function (map)mathoverflow question: whats-a-natural-candidate-for-an-analytic-function-that-interpolates-the-tower/43003Faa di Bruno and derivatives of an iterated function ON MAY 20, 2017 BY DCHOYLE of the function f with respect to (wrt) variable following the derivativeA Cheritat wiki: Mandelbrot_set - Following_the_derivative angular Angular derivative Shapiro, J.H. (1993). The Angular Derivative. In: Composition Operators. Universitext: Tracts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0887-7_5 The Schwarzian Derivative The Schwarzian Derivative MAT335H1F Lecture Notes by Burbulla (Chapter 11, 12 and 13) Schwarzian-Derivative-Poster Schwarzian derivatives of rational functions by Alex EremenkoWhat is ... Schwarzian Derivative? (Notices of AMS Jan 2009), Wirtinger derivatives gradient the gradient is the generalization of the derivative for the multivariable functionsbetterexplained: vector-calculus-understanding-the-gradientkhan academy: the-gradient definitions: (field): Gradient field is the vector field with gradient vector (function): The gradient of a scalar-valued multivariable function is a vector-valued function denoted (vector): The gradient of the function f at the point (x,y) is defined as the unique vector (result of gradient function) representing the maximum rate of increase of a scalar function (length of the vector) and the direction of this maximal rate (angle of the vector). Such vector is given by the partial derivatives with respect to each of the independent variablesGradient-Based Optimization by Jason Hicken, Prof. Juan Alonso, and Prof. Charbel Farhat (operator): Del or nabla is an gradient operator = a vector differential operator Notations: See also Gradient Descent Algorithmgradient-descent-algorithm-and-its-variants by Imad DabburaThe chaotic nature of faster gradient descent methods by Kees van den Doel and Uri Ascher Gradient Ascent Algorithm image gradient Jacobian The Jacobian is the generalization of the gradient for vector-valued functions of several variables multiplier The multiplier of a fixed point α is the derivative A′(α) calculated in any local chart around αConformal Geometry and Dynamics of Quadratic Polynomials, vol I-II Mikhail Lyubich, page 61 Germ Germ Germ in wikipedia of the function f in the neighborhood of point z is a set of the functions g which are indistinguishable in that neighborhood See: parabolic germ the linearization of a germ Linearization of germs: regular dependence on the multiplier by Carlo Carminati, Stefano Marmi map differences between map and the function math.stackexchange question: is-there-any-difference-between-mapping-and-function Iterated function = mapIterated function (map) in wikipedia an evolution functionevolution function of the discrete nonlinear dynamical systemthe discrete nonlinear dynamical system is called map , examples: rational maps Newton maps complex quadratic map: exponential maps trigonometric maps landing map: " A theorem of Caratheodory states that if is a full compact and locally connected set, then external rays land and the landing map is continuous."math.stackexchange question: why-is-local-connectivity-important-for-polynomial-julia-sets types or names Brjuno Brjuno function Links: Quelle est la régularité de la fonction de Brjuno ? harmonic An harmonic or spherical function is a: "set of orthogonal functions all of whose curvatures are changing at the same rate."riemann-for-anti-dummies by the LaRouche Youth Movement in Canada "harmonic functions relate two sets of different curves such that the rate of change of their respective curvatures is always equal. " and they are orthogonal "One set of curves of the harmonic function expressed the pathways of minimal change in the potential for action, while the other, orthogonal curves expressed the pathways of maximum change in the potential for action." "a pair of harmonic conjugate functions, u and v. They satisfy the Cauchy-Riemann equations. Geometrically, this implies that the contour lines of u and v intersect at right angles"chebfun docs Geometric examples: " A set of concentric circles and radial lines comprises an harmonic function because both the circles and the radial lines intersect orthogonally and both have constant curvature." "a set of orthogonal ellipses and hyperbolas." How to find harmonic conjugate function ? HarmonicFunction by (c) 2011 John H. Mathews, Russell W. Howell meromorphic meromorphic maps: Those with NO FINITE, NON-ATTRACTING FIXED POINTSConnectivity of Julia sets of Newton maps: A unified approach by K. Baranski N. Fagella X. Jarque B. Karpinska Polynomial Critical Critical polynomial: so These polynomials are used for finding: centers of period n Mandelbrot set components. Centers are roots of n-th critical polynomials (points where critical curve Qn croses x axis) Misiurewicz points post-critically finite a post-critically finite polynomial = all critical points have finite orbit Resurgent "resurgent functions display at each of their singular points a behaviour closely related to their behaviour at the origin. Loosely speaking, these functions resurrect, or surge up - in a slightly different guise, as it were - at their singularities" J. Écalle, 1980A Beginners’ Guide to Resurgence and Trans-series in Quantum Theories Gerald DunneA Primer on Resurgent Transseries and Their Asymptotics by Inês Aniceto, Gökçe Başar, Ricardo SchiappaUniversality of Resurgence in Quantization Theories - video transformation In mathematics, a transformation is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f : X → X.Olexandr Ganyushkin Volodymyr Mazorchuk Classical Finite Transformation Semigroups: An Introduction https://archive.org/details/classicalfinitet00gany_719 limited 2008 Springer Science & Business Media 978-1-84800-281-4 1Pierre A. Grillet Semigroups: An Introduction to the Structure Theory https://books.google.com/books?id=yM544W1N2UUC&pg=PA2 1995 CRC Press 978-0-8247-9662-4 2Wilkinson, Leland & Graham The Grammar of Graphics Springer 2005 978-0-387-24544-7 29 https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA29 2nd Examples include: linear transformations of vector spaces geometric transformations projective transformations affine transformations rotations reflections translationshttps://www.mathsisfun.com/geometry/transformations.html Transformations www.mathsisfun.com 2019-12-13https://www.basic-mathematics.com/transformations-in-math.html Types of Transformations in Math Basic-mathematics.com 2019-12-13 coordinate transformations There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described by coordinate transformations, which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x, y) and polar coordinates (r, θ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x = r cosθ and y = r sinθ. With every bijection from the space to itself two coordinate transformations can be associated: Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation) Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation) For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more. Yoccoz’s function glitches thumb|Interior of the Cauliflower Julia set. .The black structure around fixed point and it's preimages is a numerical error (glitch) }} Definition: Incorrect (noisy) parts of rendersdinkydauset at deviantar: Perturbation-for-the-Mandelbrot-set-450766847 using perturbation technique pixels which dynamics differ significantly from the dynamics of the reference pixelmath.stackexchange question: selecting-reference-orbit-for-fractal-rendering-with-perturbation-theory"These can be detected and corrected by using a more appropriate reference."math.stackexchange question: coloring-the-mandelbrot-set-using-iterated-points? Examples: glitches in perturbation method graf Dessin d'enfant Wikipedia article: Dessin d'enfant :commons:Category:Dessins d'enfants (mathematics) See also: Polynomials Invertible in k-Radicals by Y. Burda, A. Khovanskii Tree tree is a simply connected graph See also: fractalforums.org: functional-graph-of-modular-arithmetic Oleg Ivrii (Tel Aviv University), "Shapes of trees" Farey tree Farey tree = Farey sequence as a tree Hubbard tree a simplified, combinatorial model of the Julia set (MARY WILKERSON) "Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane." Dessins d’enfants and Hubbard trees by Kevin M. Pilgrim " Hubbard trees are invariant trees connecting the points of the critical orbits of post-critically finite polynomials. Douady and Hubbard showed in the Orsay Notes that they encode all combinatorial properties of the Julia sets. For quadratic polynomials, one can describe the dynamics as a subshift on two symbols, and itinerary of the critical value is called the kneading sequence." Henk Bruin and Dierk SchleicherAdmissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials by Henk Bruin, Dierk Schleicher the Hubbard tree is the convex hull of the critical orbits within the filled Julia set, i.e., the complement of the basion of infinity Rooted tree rooted tree of preimages: where a vertex is connected by an edge with . Iteration Iteration Magnitude magnitude of the point (complex number in 2D case) = it's distance from the originwikipedia: Magnitude in mathematics radius is the absolute value of complex number (compare to arguments or angle) Map description types The map f is hyperbolic if every critical orbit converges to a periodic orbit.Hyperbolic Components by John Milnor Complex quadratic map Forms c form: z^2+c quadratic mapComplex quadratic map in wikipedia math notation: Maxima CAS function: f(z,c):=z*z+c; (%i1) z:zx+zy*%i; (%o1) %i*zy+zx (%i2) c:cx+cy*%i; (%o2) %i*cy+cx (%i3) f:z^2+c; (%o3) (%i*zy+zx)^2+%i*cy+cx (%i4) realpart(f); (%o4) -zy^2+zx^2+cx (%i5) imagpart(f); (%o5) 2*zx*zy+cy Iterated quadratic map math notation ... or with subscripts: Maxima CAS function: fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p-1, z, c),c); zp:fn(p, z, c); lambda form: z^2+m*z More description Maxima CAS code (here m not lambda is used): (%i2) z:zx+zy*%i; (%o2) %i*zy+zx (%i3) m:mx+my*%i; (%o3) %i*my+mx (%i4) f:m*z+z^2; (%o4) (%i*zy+zx)^2+(%i*my+mx)*(%i*zy+zx) (%i5) realpart(f); (%o5) -zy^2-my*zy+zx^2+mx*zx (%i6) imagpart(f); (%o6) 2*zx*zy+mx*zy+my*zx Switching between forms Start from: internal angle internal radius r Multiplier of fixed point: When one wants change from lambda to c:Michael Yampolsky, Saeed Zakeri: Mating Siegel quadratic polynomials. or from c to lambda: Example values: r c fixed point alfa fixed point 1/1 1.0 0.25 0.5 1.0 0 1/2 1.0 -0.75 -0.5 -1.0 0 1/3 1.0 0.64951905283833*i-0.125 0.43301270189222*i-0.25 0.86602540378444*i-0.5 0 1/4 1.0 0.5*i+0.25 0.5*i i 0 1/5 1.0 0.32858194507446*i+0.35676274578121 0.47552825814758*i+0.15450849718747 0.95105651629515*i+0.30901699437495 0 1/6 1.0 0.21650635094611*i+0.375 0.43301270189222*i+0.25 0.86602540378444*i+0.5 0 1/7 1.0 0.14718376318856*i+0.36737513441845 0.39091574123401*i+0.31174490092937 0.78183148246803*i+0.62348980185873 0 1/8 1.0 0.10355339059327*i+0.35355339059327 0.35355339059327*i+0.35355339059327 0.70710678118655*i+0.70710678118655 0 1/9 1.0 0.075191866590218*i+0.33961017714276 0.32139380484327*i+0.38302222155949 0.64278760968654*i+0.76604444311898 0 1/10 1.0 0.056128497072448*i+0.32725424859374 0.29389262614624*i+0.40450849718747 0.58778525229247*i+0.80901699437495 One can easily compute parameter c as a point c inside main cardioid of Mandelbrot set: of period 1 hyperbolic component (main cardioid) for given internal angle (rotation number) t using this c / cpp code by Wolf JungMandel: software for real and complex dynamics by Wolf Jung double InternalAngleInTurns; double InternalRadius; double t = InternalAngleInTurns *2*M_PI; // from turns to radians double R2 = InternalRadius * InternalRadius; double Cx, Cy; /* C = Cx+Cy*i */ // main cardioid Cx = (cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4; Cy = (sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4; or this Maxima CAS code: /* conformal map from circle to cardioid (boundary of period 1 component of Mandelbrot set */ F(w):=w/2-w*w/4; /* circle D={w:abs(w)=1 } where w=l(t,r) t is angle in turns ; 1 turn = 360 degree = 2*Pi radians r is a radius */ ToCircle(t,r):=r*%e^(%i*t*2*%pi); GiveC(angle,radius):= ( [w], /* point of unit circle w:l(internalAngle,internalRadius); */ w:ToCircle(angle,radius), /* point of circle */ float(rectform(F(w))) /* point on boundary of period 1 component of Mandelbrot set */ )$ compile(all)$ /* ---------- global constants & var ---------------------------*/ Numerator :1; DenominatorMax :10; InternalRadius:1; /* --------- main -------------- */ for Denominator:1 thru DenominatorMax step 1 do ( InternalAngle: Numerator/Denominator, c: GiveC(InternalAngle,InternalRadius), display(Denominator), display(c), /* compute fixed point */ alfa:float(rectform((1-sqrt(1-4*c))/2)), /* alfa fixed point */ display(alfa) )$ Circle map Circle map three-cool-facts-about-rotations-of-the-circle by David Richeson irrational rotationirrational-rotations-of-the-circle-and-benfords-law by David Richeson Caratheodory semiconjugacy "The map is called the Caratheodory semiconjugacy, with the associated identity in the degree 2 case. This identity allows us to easily track forward iteration of external rays and their landing points in by doubling the angle of their associated external rays modulo 1." Mary WilkersonSubdivision rule constructions on critically preperiodic quadratic matings by Mary Wilkerson where is the real numbers modulo the integers group ( quotient group )Quotient group which is isomorphic to the circle groupCircle group in wikipedia the group of complex numbers of absolute value 1 under multiplication or correspondingly, the group of rotations in 2D about the origin, that is, the special orthogonal group a dyadic rational number An isomorphism is given by (see Euler's identity). Doubling map Angle doubling map Feigenbaum map "the Feigenbaum map F is a solution of Cvitanovic-Feigenbaum equation"The measure of the Feigenbaum Julia set by Artem Dudko and Scott Sutherland Feigenbaum quadratic polynomials (= Feigenbaum point of parameter plane) , i.e., infinitely renormalizable polynomials of bounded type.MLC at Feigenbaum points by Dzmitry Dudko, Mikhail Lyubich First return map definition Poincaré map video: Intro to Poincare map (Poincaré), the first return map. This map helps us determine the stability of a limit cycle using the eigenvalues (Floquet multipliers) associated with the map. "In contrast to a phase portrait, the return map is a discrete description of the underlying dynamics. .... A return map (plot) is generated by plotting one return value of the time series against the previous one "General principles of chaotic dynamics by P.B. Persson, C.D. Wagner "If x is a periodic point of period p for f and U is a neighborhood of x, the composition maps U to another neighborhood V of x. This locally defined map is the return map for x." (W P Thurston: On the geometry and dynamics of Iterated rational maps) "The first return map S → S is the map defined by sending each x0 ∈ S to the point of S where the orbit of x0 under the system first returns to S." Continuous time and discrete time dynamical systems by Shaun Bullett "way to obtain a discrete time system from a continuous time system, called the method of Poincar´e sections Poincar´e sections take us from: continuous time dynamical systems on (n + 1)-dimensional spaces to discrete time dynamical systems on n-dimensional spaces"Continuous time and discrete time dynamical systems by Shaun Bullett postcritically finite postcritically finite: maps whose critical orbits are all periodic or preperiodicEXPONENTIAL THURSTON MAPS AND LIMITS OF QUADRATIC DIFFERENTIALS by JOHN HUBBARD, DIERK SCHLEICHER, AND MITSUHIRO SHISHIKURA " In the theory of iterated rational maps, the easiest maps to understand are postcritically finite: maps whose critical orbits are all periodic or preperiodic. These maps are also the most important maps for understanding the combinatorial structure of parameter spaces of rational maps. " A postcritically finite quadratic polynomial fc(z) = z^2+c may be:The Thurston Algorithm for quadratic matings by Wolf Jung periodic of satellite type periodic of primitive type critically preperiodic (Misiurewicz type) Examples are given by: the Basilica Q(z) = z^2 − 1 the Kokopelli P(z) = z^2 + i (dendrite) Critically preperiodic polynomials the critical point of fc is strictly preperiodic parameter c is from Thurston-Misiurewicz points–values on the boundary of the Mandelbrot set = Misiurewicz point Julia set is dendrite Multiplier map thumb|right|Mandelbrot set - multiplier map Multiplier map associated with hyperbolic component gives an explicit uniformization of hyperbolic component by the unit disk : it is (d-1) to one function. Where d is a degree of iterated function In other words it maps hyperbolic component H to unit disk D. It maps point c from parameter plane to point b from reference plane: where: c is a point in the parameter plane b is a point in the reference plane. It is also internal coordinate is a multiplier map Multiplier map is a conformal isomorphism.Conformal Geometry and Dynamics of Quadratic Polynomials Mikhail Lyubich It can be computed using: Newton method: internal_coordinate by Claude Heiland-Allen m-interior function First derivative wrt z Approximation Mapping a polar grid onto the Mandelbrot Set on iteration at a time.by Maths town Quadratic like maps quadratic like maps is nothing but complexification of the concept of unimodal mapYouTube: Mikhail Lyubich: Story of the Feigenbaum point. Centre International de Rencontres Mathématiques Riemann map Riemann mapping theoremwikipedia: Riemann mapping theorem says that every simply connected subset U of the complex number plane can be mapped to the open unit disk D where: D is a unit disk f is Riemann map (function). It is 1to-1 function U is subset of complex plane Examples (approximations of Riemann mapping): multiplier map on the parameter plane binary decomposition Böttcher coordinates on the parameter plane the Riemann map for the complement of the Mandelbrot set on dynamic planeA THOMPSON GROUP FOR THE BASILICA by JAMES BELK AND BRADLEY FORREST for the Fatou component containing a superattracting fixed point for a rational mapmath stackexchange question: explicit-riemann-mappings a Riemann map for the complement of the filled Julia set of a quadratic polynomial with connected Julia: "The Riemann map for the central component for the Basilica was drawn in essentially the same way, except that instead of starting with points on a big circle, I started with sample points on a circle of small radius (e.g. 0.00001) around the origin." Jim Belk zeros of qn algorithm function: explicit formula (only in simple cases) numerical approximation (in most of the cases)mathoverflow question: complex-function-for-mapping-a-circle-to-a-superellipse Zipper https://code.google.com/archive/p/zipper/ https://sites.math.washington.edu/~marshall/zipper.html " Thurston and others have done some beautiful work involving approximating arbitrary Riemann maps using circle packings. See Circle Packing: A Mathematical Tale by Stephenson." " To some extent, constructing a Riemann map is simply a matter of constructing a harmonic function on a given domain (as well as the associated harmonic conjugate), subject to certain boundary conditions. The solution to such problems is a huge topic of research in the study of PDE's, although the connection with Riemann maps is rarely mentioned." Jim Belkmath.stackexchange question: explicit-riemann-mappings PDE's approach to construct a Riemann map explicitly on a given domain D First, translate the domain so that it contains the origin. Next, use a numerical method to construct a harmonic function F satisfying for all , and let Then and is harmonic so: R is the radial component (i.e. modulus) of a Riemann map on D. The angular component can now be determined by the fact that its level curves are perpendicular to the level curves of R, and have equal angular spacing near the origin." "Using the Riemann mapping BM we can define the parameter external rays and equipotentials as the preimages of the straight rays going to ∞ and round circles centered at 0. This gives us two orthogonal foliations in the complement of the Mandelbrot set." Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle by Mikhail Lyubich See Commons: Category:Riemann mapping A Riemann map on the central componentA THOMPSON GROUP FOR THE BASILICA by JAMES BELK AND BRADLEY FORREST Some internal rays of the BasilicaGraph Replacement Systems for Julia Sets of Quadratic Polynomials by Yuan J. Liu The Bottcher Map B gives rise to internal angles in each bubbleA Thompson-Like Group for the Bubble Bath Julia Set by Jasper Weinrich-Burdref Rotation map "If a is rational, then every point is periodic. If a is irrational, then every point has a dense orbit." David Richesonthree-cool-facts-about-rotations-of-the-circle by David Richeson rational Rotation map describes counterclockwise rotation of point thru turns on the unit circle: It is used for computing: itinerary irrational ROTATE IN A CIRCLE WITH A ROTATION by Sylvie Ruette (in fr.) Wikipedia: Irrational rotation Shift map names: bit shift map (because it shifts the bit) = if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero. 2x mod 1 map (because it is math description of its action) Shift map (one-sided binary left shift) acts on one-sided infinite sequence of binary numbers by It just drops first digit of the sequence. If we treat sequence as a binary fraction: then shift map = the dyadic transformation = dyadic map = bit shift map= 2x  mod 1 map = Bernoulli map = doubling map = sawtooth map and "shifting N places left is the same as multiplying by 2 to the power N (written as 2N)"binary_shift_left (operator <<) In Haskell: shift k = genericTake q . genericDrop k . cycle -- shift map See also: How to compute external angles of principal Misiurewicz points of wakes? On quotients of the shift associated with dendrite Julia sets of quadratic polynomials by Christopher Penrose Published 1990 subsection-sequence_space by Mark McClure Dehn twist Dehn twistDehn_twist in wikipedia Number complex number numerical value: x+y*i vector from origin to point (x,y) point (x,y) od 2D Cartesion plain constant Fegenbaum constant first (delta)Feigenbaum constants second (alpha) How to compute: Keith Briggs: How to calculate octave program by Anton Hendricson python program by cdlane Rosettacode An efficient method for the computation of the Feigenbaum constants to high precision by Andrea Molteni (Submitted on 7 Feb 2016) degree It hase many meanings:Degree in Wikipedia (disambiguation page) unit of the angle degree of a function polynomial rational functionfractalforums.org: definitions-of-degree-of-rational-function Multiplier The multiplier of periodic z-point:Multiplier at wikipediaInternal angles and multipliers from Fractal Geometry Yale University Michael Frame, Benoit Mandelbrot (1924-2010), and Nial Neger September 3, 2017 is a complex number "The value of is the same at any point in the orbit of a: it is called the multiplier of the cycle."A Cheritat wiki-draw: Mandelbrot_set#Following_the_derivative The multiplier is invariant under conjugacyscholarpedia: Siegel disks Linearization Linearizability depends on the multiplier Math notation: Maxima CAS function for computing multiplier of periodic cycle: m(p):=diff(fn(p,z,c),z,1); where p is a period. It takes period as an input, not z point. period 1 2 3 It is used to: compute stability index of periodic orbit (periodic point) = (where r is a n internal radius) multiplier map "The multiplier of a fixed point gives information about its stability (the behaviour of nearby orbits)" Periodic cycles and singular values of entire transcendental functions by Anna Miriam Benini and Nuria Fagella See also: multiplier map Buff, Xavier. “VIRTUALLY REPELLING FIXED POINTS.” Publicacions Matemàtiques, vol. 47, no. 1, Universitat Autònoma de Barcelona, 2003, pp. 195–209, http://www.jstor.org/stable/43736773. Rotation number The rotation numberWikipedia: Rotation numberrotation number From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2016scholarpedia: Rotation_theory The Fractal Geometry of the Mandelbrot Set II. How to Count and How to Add Robert L. Devaney An Introduction to Rotation Theory Prize winner, DSWeb Student Competition, 2007 By Christian Kue of the disk (component) attached to the main cardioid of the Mandelbrot set is a proper, positive rational number p/q in lowest terms where: q is a period of attached disk (child period) = the period of the attractive cycles of the Julia sets in the attached disk p describes fc action on the cycle: fc turns clockwise around z0 jumping, in each iteration, p points of the cycle Complex systems simulation Curso 2012-2013 by Antonio Giraldo and María Asunción Sastre Features: in a contact point (root point) it agrees with the internal angle the rotation numbers are ordered clockwise along the boundary of the componant " For parameters c in the p/q-limb, the filled Julia set Kc has q components at the fixed point αc . These are permuted cyclically by the quadratic polynomial fc(z), going p steps counterclockwise " Wolf Jung Winding number of the map (iterated function)Weisstein, Eric W. "Map Winding Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MapWindingNumber.htmlWikipedia: Rotation number "the winding number of the dynamic ray at angle a around the critical value, which is defined as follows: denoting the point on the dynamic a-ray at potential t greater or equal to zero by zt and decreasing t from +infinity to 0, the winding number is the total change of arg(zt - c) (divided by 2*Pi so as to count in full turns). Provided that the critical value is not on the dynamic ray or at its landing point, the winding number is well-defined and finite and depends continuously on the parameter. " DIERK SCHLEICHER RATIONAL PARAMETER RAYS OF THE MANDELBROT SET by Dierk Schleicher "the winding number of the dynamic ray at angle ϑ around the critical value, which is defined as follows: denoting the point on the dynamic ϑ-ray at potential t ≥ 0 by zt and decreasing t from +∞ to 0, the winding number is the total change of arg(zt − c) (divided by 2π so as to count in full turns). Provided that the critical value is not on the dynamic ray or at its landing point, the winding number is well-defined and finite and depends continuously on the parameter. When the parameter c moves in a small circle around c0 and if the winding number is defined all the time, then it must change by an integer corresponding to the multiplicity of c as a root of z(c) − c. However, when the parameter returns back to where it started, the winding number must be restored to what it was before. This requires a discontinuity of the winding number, so there are parameters arbitrarily close to c0 for which the critical value is on the dynamic ray at angle ϑ, and c0 is a limit point of the parameter ray at angle ϑ. Since this parameter ray lands, it lands at c0." of the curve https://plus.maths.org/content/winding-numbers-topography-and-topology-iiWinding-Number by empet the winding number of a curve is the number of complete rotations, in the counterclockwise sense, of the curve around the point(0, 0).Finding the number of roots of a polynomial in a plane region using the winding number by Juan Luis García Zapataa, Juan Carlos Díaz Martín w(γ, x) = number of times curve γ winds round point x. The winding number is signed: + for counterclockwise, − for clockwise.MATH 145: SUPPLEMENTARY NOTES by VIN DE SILVA Computing winding number of the curve (which is not crossing the origin) using: numerical integration computational geometry The discrete winding number = winding number of polygon approximating curve Orbit Orbit is a sequence of pointsWikipedia: Orbit (dynamics) phase space trajectories of dynamical systems The orbit of periodic point is finite and it is called a cycle. Backward Critical Critical orbit is forward orbit of a critical point. Forward Homoclinic / heteroclinic Inverse Inverse = Backward periodic skipped set containing first n iterations of initial point without initial point and its k iterations number of elements = n - k It is used in the average colorings truncated set containing initial point and first n iterations of initial point number of elements = n+1 Parameter Parameter point of the parameter plane " is renormalizable if restriction of some of its iterate gives a polinomial-like map of the same or lower degree. " Ouadratic-like maps and Renormalization by Nuria Fagella parameter of the function Period Period of point under the iterarted function f is the smallest positive integer value p for which this equality holds is the periodPeiod From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015. of the orbit.scholarpedia: Periodic Orbit for a Map is a point of periodic orbit (limit cycle) . More is here Plane Planes Wikipedia: Complex quadratic polynomial - Planes Douady’s principle: “sow in dynamical plane and reap in parameter space”. 2-sphere In topology: two-dimensional sphere = 2-sphere = the two-dimensional surface of a three-dimensional ballN-sphere in wikipedia Geometrically, the set of extended complex numbers is referred to as the Riemann sphere or extended complex plane. partition Examples: Markow partition Yoccoz puzzle critical portrait lamination critical portrait partition A critical portrait naturally induces partitions: Df , If , and Pf of the closed unit disk D, the unit circle T, and the plane C, respectively; Kneading partition of the dynamic plane In case of critically preperiodic polynomials the partition of the dynamic plane used in the definition of the kneading sequence. Partition is formed by the dynamic rays at angles: t/2 (t + 1)/2 which land together at the critical point. Angle t is angle which lands on the critical value: How to find angle of the dynamic external ray that land on the critical value z = c ? Spine partition of the dynamic plane Curve : where: R is an dynamic external ray S is the spine of Julia set the angles 0 and 1/2 are landing at the fixed point and at its preimage divides dynamical plane into two components. crossing/noncrossing noncrossing: "A partition of a (finite) set is just a subdivision of the set into disjoint subsets. If the set is represented as points on a line (or around the edge of a disc), we can represent the partition with lines connecting the dots. The lines usually have lots of crossings. When the partition diagram has no crossing lines, it is called a non-crossing partition. ... They have a lot of beautiful algebraic structure, and are related to lots of old enumeration problems. More recently (and importantly), they turn out to be a crucial tool in understanding how the eigenvalues of large random matrices behave." Todd Kemp (UCSD)UC San Diego : MATH 196/296: Student Colloquium Key words: Enumerative combinatorics types slit plane = plane with the slit deletedmath.stackexchange question: entire-function-with-image-contained-in-slit-plane-is-constant: Let S be the "slit plane" chessboard or checkerboards types in case of discrete dynamical system Dynamic plane or phase space z-plane for fc(z)= z^2 + c z-plane for fm(z)= z^2 + m*z Parameter plane See:Alternate Parameter Planes by David E. Joyce Types of the parameter plane: c-plane (standard plane) exponential plane (map) mu-ency: exponential map by R MunafoExponential mapping and OpenMP by Claude Heiland-Allen flatten' the cardiod (unroll) Linas Vepstas : Self Similar?the flattened cardioid of a Mandelbrot by Tom Rathborne = "A region along the cardioid is continuously blown up and stretched out, so that the respective segment of the cardioid becomes a line segment. .." (Figure 4.22 on pages 204-205 of The Science Of Fractal Images)Stretching cusps by Claude Heiland-Allen transformations Twisted Mandelbrot Sets by Eric C. Hill Points Band-merging the band-merging points are Misiurewicz pointsdoubling bifurcations on complex plane by E Demidov Biaccessible If there exist two distinct external rays landing at point we say that it is a biaccessible point.On biaccessible points in the Julia set of the family z(a+z^{d}) by Mitsuhiko Imada We call p biaccessible if it is accessible through at least two distinct external raysOn biaccessible points in the Julia set of a Cremer quadratic polynomial by Dierk Schleicher, Saeed Zakeri blowup point blowup point = parameter for which the critical orbits map to ∞, so the Julia set is the entire sphere Campbell, J.T., Collins, J.T. Blowup Points and Baby Mandelbrot Sets for a Family of Singularly Perturbed Rational Maps. Qual. Theory Dyn. Syst. 16, 31–52 (2017). https://doi.org/10.1007/s12346-015-0169-5 branched A point in the complex plane is branched, if it is in the Julia set and is the landing point of more than two rays.Criterion for rays landing together by Jinsong Zeng Buried " a point of the Julia set is buried if it is not in the boundary of any Fatou component." Topological Variety of Buried Points by Clinton P. Curry, Logan C. Hoehn, and John C. Mayer polynomials do not have buried points some rational Julia sets have (Residual Julia Set = Buried Points) Center Nucleus or center of hyperbolic component A center of a hyperbolic component H is a parameter (or point of parameter plane) such that the corresponding periodic orbit has multiplier= 0." Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper it has a superstable periodic orbit Synonyms: Nucleus of a Mu-Atom Nucleus - From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015. How to find center/s ? Center of Siegel Disc Center of Siegel disc is an irrationally indifferent periodic point. Mane's theorem: "... appart from its center, a Siegel disk cannot contain any periodic point, critical point, nor any iterated preimage of a critical or periodic point. On the other hand it can contain an iterated image of a critical point." Siegel disks by Xavier Buff and Arnaud Ch ́ritat e Univ. Toulouse Roma, April 2009 Critical A critical pointWikipedia: Critical point (mathematics) of is a point in the dynamical plane such that the derivative vanishes ( is equal to zero): A critical value is an image of critical point complex quadratic polynomial For the complex quadratic polynomial in the c form implies we see that the only (finite) critical point of is the point . is an initial point for Mandelbrot set iteration.Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations Cut right|thumb|The "neck" of this eight-like figure is a cut-point. right|thumb|Cut points in the San Marco Basilica Julia set. Biaccessible points = landing points for 2 external rays Cut point k of set S is a point for which set S-k is dissconected (consist of 2 or more sets).Cut point in wikipedia This name is used in a topology. Examples: root points of Mandelbrot set Misiurewicz points of boundary of Mandelbrot set cut points of Julia sets (in case of Siegel disc critical point is a cut point) These points are landing points of 2 or more external rays. Point which is a landing point of 2 external rays is called biaccessible Cut ray is a ray which converges to landing point of another ray.On local connectivity for the Julia set of rational maps: Newton’s famous example By P. Roesch Cut rays can be used to construct puzzles. Cut angle is an angle of cut ray. fixed names fixed point invariant = The number of fixed points of a dynamical system is invariant under many mathematical operations. fixpoint Periodic point when period = 1 steady state of dynamical system stable behaviour equilibrium point = fixed point of DE :w:Hyperbolic equilibrium point p of f, such that (Df)p has no eigenvalue with :w:absolute value 1. In this case, Λ = {p} In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas. This fails to hold in general. Strogatz notes that "hyperbolic is an unfortunate name—it sounds like it should mean 'saddle point'—but it has become standard."Strogatz Steven Nonlinear Dynamics and Chaos 2001 Westview Press 0-7382-0453-6 registration https://archive.org/details/nonlineardynamic00stro Several properties hold about a neighborhood of a hyperbolic point, notablyOtt Edward Chaos in Dynamical Systems https://archive.org/details/chaosindynamical0000otte registration 1994 Cambridge University Press 0-521-43799-7 Feigenbaum The Feigenbaum Pointmuency: feigenbaum point is a: point c of parameter plane is the limit of the period doubling cascade of bifurcations = the limit of the sequence of real period doubling parameters the accumulation point of the period-doubling cascade in the real-valued x^2+c mapping an infinitely renormalizable parameter of bounded type boundary point between chaotic (-2 < c < MF) and periodic region (MF< c < 1/4)On Periodic and Chaotic Regions in the Mandelbrot Set by G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya Generalized Feigenbaum points are: the limit of the period-q cascade of bifurcations landing points of parameter ray or rays with irrational angles Examples: -.1528+1.0397i) The Mandelbrot set is conjectured to be self- similar around generalized Feigenbaum pointsfractal-faq : section 6 when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each timePeriod doubling and Feigenbaum's scaling be E Demidov {| class="wikitable" |- ! n ! Period = 2^n ! Bifurcation parameter = cn ! Ratio |- | 1 || 2 || -0.75 || N/A |- | 2 || 4 || -1.25 || N/A |- | 3 || 8 || -1.3680989 || 4.2337 |- | 4 || 16 || -1.3940462 || 4.5515 |- | 5 || 32 || -1.3996312 || 4.6458 |- | 6 || 64 || -1.4008287 || 4.6639 |- | 7 || 128 || -1.4010853 || 4.6682 |- | 8 || 256 || -1.4011402 || 4.6689 |- |9 ||512 ||-1.401151982029 || |- |10 ||1024 ||-1.401154502237 || |- |infinity || || -1.4011551890 ... || |} Bifurcation parameter is a root point of period = 2^n component. This series converges to the Feigenbaum point c = −1.401155 The ratio in the last column converges to the first Feigenbaum constant. " a "Feigenbaum point" (an infinitely renormalizable parameter of bounded type, such as the famous Feigenbaum value which is the limit of the period-2 cascade of bifurcations), then Milnor's hairiness conjecture, proved by Lyubich, states that rescalings of the Mandelbrot set converge to the entire complex plane. So there is certainly a lot of thickness near such a point, although again this may not be what you are looking for. It may also prove computationally intensive to produce accurate pictures near such points, because the usual algorithms will end up doing the maximum number of iterations for almost all points in the picture." Lasse Rempe-Gillenmathoverflow question: is-there-a-way-to-find-regions-of-depth-in-the-mandelbrot-set-other-than-simply?rq=1 Fibonacci Fibonacci pointFractalforums: fibonacci-and-the-mandelbrot-set Parameter scaling for the Fibonacci point by Leroy WenstromThe Fibonacci unimodal map by Mikhail Lyubich, John W. Milnor germ Catastrophe theory analyzes degenerate critical points of the potential function — points where not just the first derivative, but one or more higher derivatives of the potential function are also zero. These are called the germs of the catastrophe geometries. The degeneracy of these critical points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters. In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local has some meaning.The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain. infinity The point at infinity [w:Point at infinity|Point at infinity in wikipedia]" is a superattracting fixed point, but more importantly its immediate basin of attraction - that is, the component of the basin containing the fixed point itself - is completely invariant (invariant under forward and backwards iteration). This is the case for all polynomials (of degree at least two), and is one of the reasons that studying polynomials is easier than studying general rational maps (where e.g. the Julia set - where the dynamics is chaotic - may in fact be the whole Riemann sphere). The basin of infinity supports foliations into "external rays" and "equipotentials", and this allows one to study the Julia set. This idea was introduced by Douady and Hubbard, and is the basis of the famous "Yoccoz puzzle"." Lasse Rempe-GillenMathoverflow question: Attractive Basins and Loops in Julia Sets thumb|Mandelbrot set at Fibonacci point Misiurewicz Misiurewicz pointWikipedia: Misiurewicz point = " parameters where the critical orbit is pre-periodic. Myrberg-Feigenbaum MF = the Myrberg-Feigenbaum point is the different name for the Feigenbaum Point. node branch point of the shrub type of the Misiurewicz point Parabolic point parabolic points: this occurs when two singular points coalesce in a double singular point (parabolic point)The bifurcation diagram of cubic polynomial vector fields on CP1 by Christiane Rousseau "the characteristic parabolic point (i.e. the parabolic periodic point on the boundary of the critical value Fatou component) of fc"A rigidity result for some parabolic germs by Luna Lomonaco, Sabyasachi Mukherjee Periodic Point z has period p under f if: In other words point is periodic See also: fixed point stability of periodic point attracting repelling indifferent multiplier of periodic cycle Pinching "Pinching points are found as the common landing points of external rays, with exactly one ray landing between two consecutive branches. They are used to cut M or K into well-defined components, and to build topological models for these sets in a combinatorial way. " (definition from Wolf Jung program Mandel) other names pinch points cut points See for examples: period 2 = Mandel, demo 2 page 3. period 3 = Mandel, demo 2 page 5 http://www.mndynamics.com/indexp.html| program Mandel by Wolf Jung, demo 2 page 3 Pool "A point in the dendrite is called a pool if it is the landing point for two external rays, both of whose angles are of the form for some k, n ∈ N, where k ≡ 1 mod 6. ... central pool ... it is geometrically the center of the dendrite; a one half rotation around this point maps the dendrite to itself." Thompson-Like Groups for Dendrite Julia Sets by Will Smith post-critical A post-critical point is a point where is a critical point.GROWTH OF GROUPS DEFINED BY AUTOMATA : ASHLEY S. DOUGHERTY, LYDIA R. KINDELIN, AARON M. REAVES, ANDREW J. WALKER, AND NATHANIEL F. ZAKAHI See also: post-critical set precritical precritical points, i.e., the preimages of the critical point reference point Reference point of the image: its orbit (reference orbit) is computed with arbitrary precision and saved orbits of the other points of the image (no-reference points) are computed from reference orbit using standard precision (with hardware floating point numbers) = faster than using arbitrary precision renormalizable point of the parameter plane " is renormalizable if restriction of some of its iterate gives a polinomial-like map of the same or lower degree. " Ouadratic-like maps and Renormalization by Nuria Fagella infinitely renormalizable " a "Feigenbaum point" (an infinitely renormalizable parameter of bounded type, such as the famous Feigenbaum value which is the limit of the period-2 cascade of bifurcations), then Milnor's hairiness conjecture, proved by Lyubich, states that rescalings of the Mandelbrot set converge to the entire complex plane. So there is certainly a lot of thickness near such a point, although again this may not be what you are looking for. It may also prove computationally intensive to produce accurate pictures near such points, because the usual algorithms will end up doing the maximum number of iterations for almost all points in the picture." Lasse Rempe-Gillenmathoverflow question: is-there-a-way-to-find-regions-of-depth-in-the-mandelbrot-set-other-than-simply?rq=1 IMMEDIATE RENORMALIZATION " A cubic polynomial P with a non-repelling fixed point b is said to be immediately renormalizable if there exists a (connected) quadratic-like invariant filled Julia set K∗ such that b ∈ K∗ . In that case exactly one critical point of P does not belong to K∗." Immediate renormalization of complex polynomials by Alexander Blokh, Lex Oversteegen, Vladlen Timorin repelling Virtually repelling virtually repelling fixed pointsBuff, Xavier. "Virtually repelling fixed point.." Publicacions Matemàtiques 47.1 (2003): 195-209. <http://eudml.org/doc/41482>. root or bond The root point of the hyperbolic component of the Mandelbrot set: A point where two mu-atoms meet has a rotational number 0 it is a biaccessible point (landing point of 2 external rays) Names: bond Bond the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018. singular the singular points of a dynamical system In complex analysis there are four classes of singularities: Isolated singularities: Suppose the function f is not defined at a, although it does have values defined on U \ {a}. The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U \ {a}. The function g is a continuous replacement for the function f. The point a is a pole or non-essential singularity of f if there exists a holomorphic function g defined on U with g(a) nonzero, and a natural number n such that f(z) = g(z) / (z − a)n for all z in U \ {a}. The least such number n is called the order of the pole. The derivative at a non-essential singularity itself has a non-essential singularity, with n increased by 1 (except if n is 0 so that the singularity is removable). The point a is an essential singularity of f if it is neither a removable singularity nor a pole. The point a is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree. Branch points are generally the result of a multi-valued function, such as or being defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, however, it must connect two different branch points (like and for ) which are fixed in place. tip from Mu-Ency: "the point in a primary filament that has the simplest external angle; this is the point that you get by appending FS[(1/2B1)] an infinite number of times to the primary filament's name." This is also the "limit" of the ... series. Misurewicz point triple "A point in the dendrite is called a triple point if its removal separates the dendrite into three connected components. Such a point is the landing point for three external rays, whose angles all have of the form for some k, n ∈ N, where k is congruent to 1, 2 or 4, mod 7." Will Smith in Thompson-Like Groups for Dendrite Julia Sets wandering A point is called wandering if its forward orbit under the iteration of f is infinite.Criterion for rays landing together by Jinsong Zeng There is no wandering branched point for any quadratic polynomial. However, this is not true in general. Blokh and Oversteegen constructed cubic polynomials whose Julia sets contain wandering branched points;A. Blokh and L. Oversteegen, Wandering gaps for weakly hyperbolic polynomials. Complex Dynamics: Families and Friends. Ed. D.Schleicher A.K.Peters, Wellesley, MA, 2008,139-168. Portrait orbit portrait types There are two types of orbit portraits: primitive and satellite.Wikipedia: Orbit portrait If is the valence of an orbit portrait and is the recurrent ray period, then these two types may be characterized as follows: Primitive orbit portraits have and . Every ray in the portrait is mapped to itself by . Each is a pair of angles, each in a distinct orbit of the doubling map. In this case, is the base point of a baby Mandelbrot set in parameter space. Satellite (non-primitive) orbit portraits have . In this case, all of the angles make up a single orbit under the doubling map. Additionally, is the base point of a parabolic bifurcation in parameter space. Critical Critical orbit portrait = portrait of the critical orbit ... for the polynomial we may note the critical orbit portrait: for this map, or we may double the angles of external rays and record the locations of landing points in order to observe the same behavior." THURSTON’S ALGORITHM AND RATIONAL MAPS FROM QUADRATIC POLYNOMIAL MATINGS by Mary Wilkerson critical portrait: orbit portrait of critical point z = 0 = portrait of forward orbit of critical point a collection of subsets of the unit circle paritition of the unit circle and the dynamic plane. The partition is formed by the dynamic rays at angles and , which land together at the critical point. The ray for angle is landing at the critical value collection of angles of rays landing on the critical point Examples: for critical portrait is (1/8, 7/12) for critical portrait is (1/12, 7/12) Precision Precision of: data type used for computation. Measured in bits (width of significant (fraction) = number of binary digits) or in decimal digits input values result (number of significant figures) See: Numerical Precision: " Precision is the number of digits in a number. Scale is the number of digits to the right of the decimal point in a number. For example, the number 123.45 has a precision of 5 and a scale of 2."stackoverflow question: how-do-i-interpret-precision-and-scale-of-a-number-in-a-database error taking-the-error-out-of-the-error-function by Fredrik Johansson Principle Douady’s principle Douady’s principle: “sow in dynamical plane and reap in parameter space”. Problem small divisor problem Types One-Dimensional Small Divisor ProblemsAn Introduction To Small Divisors by S. Marmi (On Holomorphic Germs and Circle Diffeomorphisms) linearization problem in complex dimension one dynamical systems: "Given a fixed point of a differentiable map, seen as a discrete dynamical system, the linearization problem is the question whether or not the map is locally conjugated to its linear approximation at the fixed point. Since the dynamics of linear maps on finite dimensional real and complex vector spaces is completely understood, the dynamics of a map on a finite dimensional phase space near a linearizable fixed point is tractable."scholarpedia: Siegel disks Linearization Where it can be found: stability in mechanics, particularly in celestial mechanics relations between the growth of the entries in the continued fraction expansion of t and the behaviour of f around z=0 under iteration. See: Linearization in scholarpedia Processes or transformations and phenomenona Aliasing and antialiasing aliasingserious_statistics_aliasing by GuestJim Conjugation Topological conjugacy two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy also known as topological equivalenceArnold V. I. Geometric Methods in the Theory of Ordinary Differential Equations (Springer, 2020) is important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially. To illustrate this directly: suppose that and are iterated functions, and there exists a homeomorphism such that so that and are topologically conjugate. Then one must have and so the iterated systems are topologically conjugate as well. Here, denotes function composition. right|150px Commutative square diagram a collection of maps square diagram that commutes = all map compositions starting from the same set A and ending with the same set D give the same result Examples The logistic map and the tent map are topologically conjugate.Alligood, K. T., Sauer, T., and Yorke, J.A. Chaos: An Introduction to Dynamical Systems 1997 Springer 0-387-94677-2 114–124 The logistic map of unit height and the Bernoulli map are topologically conjugate.November 2010 For certain values in the parameter space, the Hénon map when restricted to its Julia set is topologically conjugate or semi-conjugate to the shift map on the space of two-sided sequences in two symbols.Devaney R. Nitecki Z. 1979 Shift automorphisms in the Hénon mapping https://projecteuclid.org/euclid.cmp/1103905161 Comm. Math. Phys. 67 2 137–146 2 September 2016 10.1007/bf01221362 1979CMaPh..67..137D 121479458 Contraction and dilatation the contraction z → z/2 the dilatation z → 2z. convolution In the digital image processingDigital_image_processing in wikipedia: image convolution Convolution is used to extract certain features from an input image, like edge Image convolutions by dimensions of the kernel array: 1D LIC 2D Gaussian blur (Gaussian smoothing) Sobel filter See also feature detection (Feature extraction) edge detection Ridge detection Motion detection Blob detection differentiation Method of computing the derivative of a mathematical function types: symbolic differentiation Automatic Differentiation (AD)autodiff.org numeric differentiation Wikipedia: Numerical_differentiationNumerical Recipies In C : 5.7 Numerical Derivativesstackoverflow question: is-there-any-standard-way-to-calculate-the-numerical-gradient = the method of finite differencesWikipedia: Finite difference Discretizations discretizationWikipedia: discretization and its reverse mathinsight: from_discrete_to_continuous_dynamical_systems discretize/homogenize in the DDG (Discrete Differential Geometry) Discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts.J.M. Hyman, M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Computers & Mathematics with Applications, Volume 33, Issue 4, 1997, Pages 81-104, ISSN 0898-1221, https://doi.org/10.1016/S0898-1221(97)00009-6. (https://www.sciencedirect.com/science/article/pii/S0898122197000096) Abstract: This is the first in series of papers creating a discrete analog of vector analysis on logically rectangular, nonorthogonal, nonsmooth grids. We introduce notations for 2-D logically rectangular grids, describe both cell-valued and nodal discretizations for scalar functions, and construct the natural discretizations of vector fields, using the vector components normal and tangential to the cell boundaries. We then define natural discrete analogs of the divergence, gradient, and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates, such as length of elements of coordinate lines, areas of elements of coordinate surfaces, and elementary volumes. We introduce the discrete volume integral of scalar functions, the discrete surface integral, and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects. These theorems include the following: the discrete analog of relationship div A→ = 0 if and only if A→ = curl B→; curl A→ = 0 if and only if A→ = grad ϕ; if A→ = grad ϕ, then the line integral does not depend on path; and if the line integral of a vector function is equal to zero for any closed path, then this vector is the gradient of a scalar function. Last, we define the discrete operators DIV, GRAD, and CURL in terms of primitive differencing operators (based on forward and backward differences) and primitive metric operators (related to multiplications of discrete functions by length of edges, areas of surfaces, and volumes of 3-D cells). These formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints. Keywords: Finite-difference; Logically-rectangular grids; Discrete vector analysis Examples: Cartesian coordinate system ( regular grid ) of continous space distorsion distorsionfilterforge filters: distorsion of the plane = plane transformation distortion of Mandelbrot set island Implosion and explosion thumbnail|right|Explosion (above) and implosion (below) Implosion is: the process of sudden change of quality fuatures of the object, like collapsing (or being squeezed in) the opposite of explosion Example: parabolic implosion in complex dynamics () when filled Julia for complex quadratic polynomial set looses all its interior (when c goes from 0 along internal ray 0 thru parabolic point c=1/4 and along external ray 0 = when c goes from interior, crosses the boundary to the exterior of Mandelbrot set) Airplane primitive parabolic implosion by Wolf Jung " We can see that looks somewhat like from the "outside", but on the "inside" there are curlicues; pairs of them are vaguely reminiscent of "butterflies". As t→0, these butterflies persist and remain uniformly large. We think of t as representing time, which decreases to 0. The fact that they suddenly disappear for t=0 is the phenomenon called "implosion". Or, if we think of time starting at t=0, then the instantaneous appearance of large "butterflies" for t>0 may be thought of as "explosion". " the Julia set implodes when under small perturbations (epsilon) near parabolic parameter (like c = 1/4)Parabolic Implosion by Han Peters, L. Vivas Published 2020 Semi-parabolic implosion in SemiParabolicImplosion by Eric Bedford Explosion is a: sudden change of quality features of the object in an extreme manner, the opposite of implosion Example: in exponential dynamics when λ> 1/e, the Julia set of is the entire plane.CANTOR BOUQUETS, EXPLOSIONS, AND KNASTER CONTINUA: DYNAMICS OF COMPLEX EXPONENTIALS by Robert L. Devaney Publicacions Matematiques, Vol 43 (1999), 27–54. integrating integrating along some vector field means finding a solution curve. Example: finding extrrernal ray using Runge-Kutta method for numerical integrationfractalforums.com: smooth-external-angle-of-mandelbrot-set Linearization changing from non-linear to linear " ... turn the perturbated linear map into the exactly linear map (it linearizes )" Jean-Christophe YoccozJean-Christophe Yoccoz Small Divisors: Number Theory in Dynamical Systems, page 46 linearization in english wikipedia Linearization in scholarpedia "System is linearizable at the origin if and only if there exists a change of coordinates which linearizes the system, that is, all the coefficients of the normal form vanish." Integrability and Linearizability of the Lotka^Volterra System with a Saddle Point with Rational Hyperbolicity Ratio by Simon Gravel and Pierre Thibault thumb|right|Linearization with inverse function Systèmes dynamiques linéaires 1 : introduction, systèmes dynamiques linéaires à temps discret by Emmanuel Risler Examples: Parabolic Linearization Linearisation Theorems Dynamics of f near a fixed or periodic pointShaun Bullett: Holomorphic Dynamics and Hyperbolic Geometry (LTCC Feb-March 2013) notes In the neighbourhood of a fixed point, which we take to be 0, (Taylor series with big O notation), where is the multiplier at the fixed point. We say that f is linearisable if there is a neighbourhood U on which f is conjugate to (by a complex analytic conjugacy). Examples: Koenigs’ Linearization Theorem 1884 Boettcher 1904 Mating Mating On The Notions of Mating by Carsten Lunde Petersen, Daniel Meyer Moebius Transformation Moebius transformation Monodromy TypesBook: Linear Differential Equations in the Complex Domain by Yoshishige Haraoka the local monodromy, which describes the change of the fundamental system of solutions caused by the analytic continuation along a loop encircling a regular singular point. the (global) monodromy, which describes the changes caused by global analytic continuations Normalization Normalize normalize = transformation to the modelNormal Forms, Bifurcations and Finiteness Problems in Differential Equations by Yulij Ilyashenko, Christiane Rousseau, Gert Sabidussi " normalize this vector so it has modulus one " A Cheritat move fixed point to the origin (z = 0) mapping the range of variable to standard range [0.0, 1.0] [0,255], like rgb values converting closed curve to unit circle converting closed curves to concentric circles with center at the origintowardsdatascience: gradient-descent-algorithm-and-its-variants See also: uniformization renormalization Parametrization Parametrization is the process of finding parametric equations of a curvemath.stackexchange question: what-is-parameterization Perturbation Perturbation technque for fast rendering the deep zoom images of the Mandelbrot setPerturbation for the Mandelbrot set by DinkydauSet, Apr 28, 2014, 3:46:13 PM perturbation of parabolic point PARABOLIC IMPLOSION A MINI-COURSE ARNAUD CHERITAT use perturbation theory to approximate the solutions of the differential equationsscholarpedia article: Perturbation theory (dynamical systems) perturbation of point x: where epsilon is absolute value of approximation errorwikipedia: Approximation error adding some small value ( epsilon denoted by Greek letter ) to the constant value to see how function changes near hard to analyze values Renormalization logistic map renormalizationLogistic map: Renormalizationestimate in wikipedia renormalization of the point Demo 5: renormalization from program mandel Mitsuhiro Shishikura: Renormalization in complex dynamics "to any quadratic map f we can associate a canonical sequence of periods p1 < p2 <... for which f is renormalizable. Depending on whether the sequence is: empty finite infinite the map f is called respectively: non-renormalizable at most finitely renormalizable infinitely renormalizable" Baby Mandelbrot sets, Renormalization and MLC Mikhail Lyubich "Sectorial renormalizations are useful in the nonlinearizable situation. " Ricardo Pérez-MarcoON QUASI-INVARIANT CURVES by Ricardo Pérez-Marco The self-similarity is a result of something called "renormalization" (which as far as I know is not related to the concept with the same name in quantum field theory). Jim Belkmathoverflow question: fine-structure-of-bifurcation-diagram-of-logistic-family Examples: Near parabolic renormalization for unicritical holomorphic mapsNear Parabolic Renormalization for Unicritical Holomorphic Maps by Arnaud Chéritat. Arnold Mathematical Journal https://doi.org/10.1007/s40598-020-00172-6 how theories change as we move to more or less detailed descriptions is known as renormalization. by Simon DeDeo Separation "the double fixed point 0 of usually splits into two fixed points. ... These points separate at some speed" ( PARABOLIC IMPLOSION. A MINI-COURSE by ARNAUD CHERITAT ) Surgery surgery in differential topology wikipedioa: Surgery theory regluing Regluing of rational functions by Vladlen Timorin Links: http://mathoverflow.net/questions/tagged/surgery-theory Tuning definition M-set and Julia setTuning From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2014. external angles examples of external angles tuningOnline tuning in the Mandelbrot set by Claude Heiland-Allen subwake island "tuning is a procedure to replace the bounded superattracting Fatou components with the copies of a filled Julia set of another polynomial and respect some combinatorial properties. Douady-Hubbard proved any quadratic polynomial which has a periodic critical point can be tuned with any quadratic polynomials in the Mandelbrot set M." YIMIN WANGPrimitive tuning for non-hyperbolic polynomials by Yimin Wang 2021 Types PRIMITIVE TUNING Uniformization Uniformization of Hyperbolic Components of Mandelbrot set to the unit disc = multiplier map basin of superattractive fixed point - Bottcher map (The Bottcher uniformization theorem) Vectorisation property or feature behavior local behavior is the behavior of a complex analytic function near some point (fixed, periodic) = Local theory of periodic orbits = local dynamics global behavior chaos Universal routes to chaotic behavior (routes into chaos, deterministic chaos) period doubling route to chaos the Pomeau-Manneville scenario the Ruelle-Takens-Newhouse scenario, Density High Density Nodes in the Chaotic Region of 1D Discrete Maps by George Livadiotis January 2018Entropy 20(1):24 DOI: 10.3390/e20010024, LicenseCC BY 4.0 density of the image Dense imageWikipedia: dense setmathoverflow question: is-there-an-almost-dense-set-of-quadratic-polynomials-which-is-not-in-the-inte/254533#254533fractalforums: dense-image downsaling with gamma correctionA Cheritat wiki: see image showing gamma-correct downscale of dense part of Mandelbropt set path finding fractal forums: pathfinding-in-the-mandelbrot-set/ supersampling: "ots of detail but fractal fades away as you get more accurate, as n increases in nxn supersampling" TGlad Hyperbolic/parabolic/elliptic The meaning of the terms "elliptic, hyperbolic, parabolic" in different disciplines in mathematicsquora: Where-is-the-best-summary-on-the-meaning-of-the-terms-elliptic-hyperbolic-parabolic-as-used-in-different-disciplines-in-mathematics PDE (Linear Second Order PDE’s in two Independent Variables): Moebius transformations: moebius-transformation by Claude Heiland-Allen2015-02-06_interpolating_moebius_transformations by Claude Heiland-Allen Discrete local complex dynamics Conic section Quadratic form Probability distributions. Invariant sth is invariant with respect to the transformation = non modified, steady Topological methods for the analysis of dynamical systems Invariants type metric invariants dynamical invariants, topological invariants. dynamical Dynamical invariants = invariants of the dynamical system periodic points fixed point invariant curve periodic ray external: fixed curves near fixed point internal Dynamical Invariants Derived from Recurrence PlotsN. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329, 2007. Orientation A compass rose: Notice that the convention for measuring angles is different to the convention we used in the unit circle definition of the trigonometric functions. Firstly 0o is North, rather than the x axis. Secondly the direction in which angles increase is clockwise rather than counter-clockwise. Unit circle : the direction in which angles increase is counter-clockwise angle zero is the x axis direction Cartesian coordinate systemmath.stackexchange question: direction-of-x-axis-with-respect-to-y-axis-in-cartesian-coordinate-system smooth smooth = changing without visible (noticeable) edges use: smooth gradient similar: continuous compare: discrete Stability stability of quasiperiodic motion under small perturbation. In the celestial mechanics dynamics of 3 bodies around sun is described by the system of differential equations. In such case it "becomes fantastically complicated and remains largely mysterious even today." See KAM = Kolmogorov–Arnold–Moser theorem and small divisor problem stability of the fixed point under small perturbation there is equivalence (for |f′(0)| ≤ 1) of stability (a topological notion) and linearizability (an analytical notion) Compare with: shadowing lemma Sensitive dependence on initial conditions - Butterfly effect Radius Radius of complex number The absolute value or modulus or magnitude or radius of a complex number Conformal radius Conformal radius of Siegel Disk Wikipedia: Conformal radiusscholarpedia: Quadratic Siegel disks Escape radius (ER) Escape radius (ER) or bailout value is a radius of circle centered at origin (z=0). This set is used as a target set in the bailout test (escape time method = ETM) Minimal Minimal Escape Radius should be grater or equal to 2: Better estimation is:Julia Sets of Complex Polynomials and Their Implementation on the Computer by Christoph Martin Strohfractalforums: bounding circle of julia sets by knighty crossing How to choose parameters for which level curves cross critical point (and its preimages)? Choose escape radius equal to n=th iteration of critical value. // find such ER for LSM/J that level curves croses critical point and it's preimages double GiveER(int i_Max){ complex double z= 0.0; // critical point int i; ; // critical point escapes very fast here. Higher valus gives infinity for (i=0; i< i_Max; ++i ){ z=z*z +c; } return cabs(z); } Another way: choose the parameter c such that it is on an escape line, then the critical value will be on an escape line as well. Inner radius Inner radius of Siegel Disc radius of inner circle, where inner circle with center at fixed point is the biggest circle inside Siegel Disc. minimal distance between center of Siel Disc and critical orbit Internal radius Internal radius is a: absolute value of multiplier See also: the N-2 ruleThe Mandelbrot Set and Julia Sets Combinatorics in the Mandelbrot Set - The 1/n2 rule, and deviations from it Sequences A sequence is an ordered list of objects (or events).Wikipedia: Sequence A series is the sum of the terms of a sequence of numbers.Wikipedia: series Some times these names are not used as in above definitions. Itinerary is an itinerary of point x under the map f relative to the paritirtion. It is a right-infinite sequence of zeros and ones Structure of Inverse Limit Spaces of Tent Maps with Nonrecurrent Critical Points by Brian Raines and Sonja Stimac where Examples: internal angle and wake For rotation map and invariant interval (circle): one can compute : and split interval into 2 subintervals (lower circle partition): then compute s according to it's relation with critical point: Itinerary can be convertedBifurcation structures in maps of Henon type by Kai T Hansen and Predrag Cvitanovic to point itinerary with respect to a critical portrait kneading sequence "the kneading sequence of an external angle ϑ (here ϑ = 1/6) is defined as the itinerary of the orbit of ϑ under angle doubling, where the itinerary is taken with respect to the partition formed by the angles ϑ/2, and (ϑ + 1)/2 "Hausdorff dimension of biaccessible angles of quadratic Julia sets and of the Mandelbrot set by Henk Bruin and Dierk Schleicher The itinerary ν = ν1ν2ν3 . . . of the critical value is called the kneading sequence.ADMISSIBILITY OF KNEADING SEQUENCES AND STRUCTURE OF HUBBARD TREES FOR QUADRATIC POLYNOMIALS by HENK BRUIN AND DIERK SCHLEICHER One can start from the critical point but neglect the initial symbol. Such sequence is computed with the Hubbard tree See also: kneading theory Milnor–Thurston_kneading_theory in wikipedia combinatorial dynamics Thue–Morse sequence Thue–Morse sequence how to compute itrosetta code: Thue-Morse Orbit Orbit can be: forward = sequence of points backward (inverse) tree in case of multivalued function sequence Series A series is the sum of the terms of a sequence of numbers.Wikipedia: series Some times these names are not used as in above definitions. Taylor Taylor series and Mandelbrot setquora: Why-is-the-Mandelbrot-set-a-fractal? Answer by Anders Kaseorg The Existence and Uniqueness of the Taylor Series of Iterated Functions The Existence and Uniqueness of the Taylor Series of Iterated Functions by Daniel Geisler Set Attracting set Informal definition: "an attracting set for a dynamical system is a closed subset A of its phase space such that for "many" choices of initial point the system will evolve towards A ." John W Milnorscholarpedia.org: Attractor Continuum definitionWikipedia: Continuum in set theory Band chaotic band 1 800 border:none; middle Chaotic bands from B0 to B10 Mandelbrot set - real slice chotic band 0.png Mandelbrot set - real slice chotic band 1.png Mandelbrot_set_-_real_slice_chotic_band_2.png Mandelbrot_set_-_real_slice_chotic_band_3.png Mandelbrot_set_-_real_slice_chotic_band_4.png Mandelbrot_set_-_real_slice_chotic_band_5.png Mandelbrot_set_-_real_slice_chotic_band_6.png Mandelbrot_set_-_real_slice_chotic_band_7.png Mandelbrot_set_-_real_slice_chotic_band_8.png Mandelbrot_set_-_real_slice_chotic_band_9.png Mandelbrot_set_-_real_slice_chotic_band_10.png period- chaotic band Harmonic structure of one-dimensional quadratic maps by G. Pastor, M. Romera, and F. Montoya is between Misiurewicz points (primary separators) and it's biggest midget has period contains Sharkovsky subsequence: sequence of islands for periods: for k = 1, 2, ..... (in the increasing order = increasing from left to right). These are first appearance of hyperbolic components with such period in Sharkowsky ordering is on n-place in Sharkowsky ordering Dwell bands "Dwell bands are regions where the integer iteration count is constant, when the iteration count decreases (increases) by 1 then you have passed a dwell band going outwards (inwards). " fractalforums: binary-decomposition-and-external-angles by Claude Other names: level sets of integer escape time Basin Basin can consist of one component, like basin of infinity of attraction definitions: An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor The collection of all points whose iterates under f converge to the attractor Alicia Cordero, Fazlollah Soleymani, Juan R. Torregrosa, Stanford Shateyi, "Basins of Attraction for Various Steffensen-Type Methods", Journal of Applied Mathematics, vol. 2014, Article ID 539707, 17 pages, 2014. https://doi.org/10.1155/2014/539707 immediate basin of attraction the component of the basin containing the periodic point itself Examples basin of infinity (whole basin = one component) Component connected component (blob) in the image Components of parameter plane Names: mu-atommu-atom From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018.   ball bud bulb decoration: "A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid."A priori bounds for some infinitely renormalizable quadratics: II. Decorations by Jeremy Kahn, Mikhail Lyubich lake lakelet.Mu-atom From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013. filament from Mu-Ency: "Any contiguous subset of the Mandelbrot Set which consists of the infinitely convoluted and branching structures that connect the island mu-molecules to each other." Some colloquial names for filaments: antenna main antenna spike spoke. "A filament consists of a) minibrots and b) limit points of sequences of those minibrots. The latter include Misiurewicz points (rational external angles, one for filament termini and two or more for interior points such as multi-armed spiral centers) and other points (with irrational external angles). My intuition says if you zoom to a succession of smaller minis along a filament, if this is done in a pattern for infinitely long you tend to a Misiurewicz point, and if it's done randomly for infinitely long you tend to an irrational point. But I have no proof of this. Other noninterior points on filaments mostly belong to individual minibrots: cardioid cusps (two rational external angles, odd denominator) and minibrot-filament branch tips (Misiurewicz points, two rational external angles, even denominator). There is one last point: the exact base of the filament where it attaches to something (minibrot or main set). This point has irrational external angles. The Feigenbaum point at the base of the spike is one of these." pauldelbrotfractalforums.org: how-thick-are-the-filaments Islands Names: mini Mandelbrot set 'baby'-Mandelbrot set island mu-molecules = embedded copy of the Mandelbrot SetIsland Mu-Molecule by Robert P. Munafo, 2012 Aug 18. Bug Island Mandelbrotie Midget List of islands: http://mrob.com/pub/mu-data/largest-islands.txt http://mrob.com/pub/muency/largestislands.html http://www.math.cornell.edu/~rperez/Documents/maximals.pdf http://fraktal.republika.pl/mset_external_ray_mini.html http://mathr.co.uk/mandelbrot/feature-database.csv.bz2 (a database of all islands up to period 16, found by tracing external rays): period, islandhood, angled internal address, lower external angle numerator, denominator, upper numerator, denominator, orientation, size, centre realpart, imagpart features of island period symbolic sequence angled internal address lower and upper external angle of rays landing on it's root center ( root orientation size distortion tip (Misiurewicz point, c value period and preperiod lower and upper external angle of rays landing on it Primitive and satellite "Hyperbolic components come in two kinds, primitive and satellite, depending on the local properties of their roots." Internal addresses in the Mandelbrot set and Galois groups of polynomials by Dierk Schleicher, page 31 primitive =non-satellite = island the root of component is not on the boundary of another component = "it was born from another hyperbolic component by the period increasing bifurcation"notes by Lyubich ones that have a cusp likes the main cardioid, when the little Julia sets are disjoint Satellite copies of the Mandelbrot set by Luna Lomonaco satellite ones that don't have a cuspmathoverflow: precise-location-of-the-mandelbrot-bulb-attachment-to-the-main-cardioid it's root is on the boundary of another hyperbolic component ON BIACCESSIBLE POINTS OF THE MANDELBROT SET by SAEED ZAKERI when the little Julia sets touch at their β-fixed point primare Child (Descendant) and the parent (ancestor) ancestor of hyperbolic component descendant of hyperbolic component = child Child From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013. Hyperbolic component of Mandelbrot set right|thumb|Boundaries of hyperbolic components of Mandelbrot set Domain is an open connected subset of a complex plane. "A hyperbolic component H of Mandelbrot set is a maximal domain (of parameter plane) on which has an attracting periodic orbit. A center of a H is a parameter (or point of parameter plane) such that the corresponding periodic orbit has multiplier= 0." Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake Internal addresses in the Mandelbrot set and irreducibility of polynomials by Dierk Schleicher features of hyperbolic component period islandhood (shape = cardiod or circle) angled internal address lower and upper external angle of rays landing on it's root center ( root orientation size Abbreviations: LAHCs = the last appearance HCs placed in the chaotic region Limb The part of the Mandelbrot set contained in the wake together with the root is called the limb of the Mandelbrot set originated at H (hyperbolic component of the Mandelbrot set)Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle by Mikhail Lyubich right|thumbnail|13/34 limb and wake on the left image p/q-limb is a part of Mandelbrot set contained inside p/q-wake For every rational number , where p and q are relatively prime, a hyperbolic component of period q bifurcates from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the p/q-limb. Computer experiments suggest that the diameter of the limb tends to zero like . The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like . A period-q limb will have q − 1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas. In an attempt to demonstrate that the thickness of the p/q-limb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for the series to converge for z = ( being the location thereof). As the series doesn't converge for the exact value of z = , the number of iterations required increases with a small ε. It turns out that multiplying the value of ε with the number of iterations required yields an approximation of π that becomes better for smaller ε. For example, for ε = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928.Gary William Flake, The Computational Beauty of Nature, 1998. p. 125. 978-0-262-56127-3. Types:Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle by Mikhail Lyubich The limbs attached to the main cardioid are called primary. Let H be a hyperbolic component attached to the main cardioid. The limbs attached to such a component are called secondary We refer to a truncated limb if we remove from it a neighborhood of its root As n tends to infinity the limbs converge to a limiting elephant. See demo 2 page 10 from program Mandel by Wolf Jung molecule The main molecule is the union of all hyperbolic components attached to the main cardioid through a chain of finitely many components.Counting Hyperbolic Components in the Main Molecule by Schinella D'Souza island mu-molecule = island mu-unit the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018 shrub "what emerges from Myrrberg-Feigenbaum point is what we denominate a shrub due to its shape" M Romera filament, chaotic part of the p/q limb: "The chaotic region is made up of an infinity of hyperbolic components mounted on an infinity of shrub branches in each one of the infinity shrubs of the family."Romera, Miguel & Pastor, Gerardo & Alvarez, Gonzalo & Vitini, Fausto. (2003). Shrubs in the Mandelbrot Set Ordering. International Journal of Bifurcation and Chaos. 13. 10.1142/S0218127403007941. Examples main antenna is a shrub of family representative of a branch is the smallest period hyperboloic componenet in the branch spokes "Colloquial term for a filament, specifically one of the "arms" radiating from a branch point." - from Mu-Ency Wake thumbnail|right|Wakes of Mandelbrot Set to Period 10 p/q-wake is the region of parameter plane enclosed by two external rays landing on the same root point on the boundary of main cardioid (period 1 hyperbolic component). Angles of the external rays that land on the root point one can find by: Combinatorial algorithm = Devaney's method book program by Claude Heiland-Allen wake function from program Mandel by Wolf Jung p/q-Subwake of W is a wake of a p/q-satellite component of W thumb|right|Wake 1/3 (bounded by 2 external rays) and internal ray 1/3 wake is named after: rotation number p/q (as above) angles of external rays landing in it's root point: "If two M-rays land at the same point we denote by wake the component of which does not contain 0."Local properties of the Mandelbrot set at parabolic points by Tan Lei Components of dynamical plane Fatou set components components of interior of Julia sets In case of Siegel disc critical orbit is a boundary of component containing Siegel Disc. For a quadratic polynomial with a parabolic orbit, the unique Fatou componentwikipedia: Classification_of_Fatou_components containing the critical value will be called the characteristic Fatou component; (Dierk Schleicher in Rational Parameter Rays of the Mandelbrot Set) "for rational maps (iterating maps of the form f(x)=p(x)/q(x) where p,q are polynomials) result in 1, 2 or infinitely many components."math.stackexchange question: quasiconformal-automorphism-groups-of-julia-sets See also: interior and exterior of filled Julia set for polynomials immediate basin of attraction Domain Domain in mathematical analysis it is an open connected set Jordan domain "A Jordan domainWikipedia: Carathéodory's theorem (conformal mapping) J is the homeomorphic image of a closed disk in E2. The image of the boundary circle is a Jordan curve, which by the Jordan Curve Theorem separates the plane into two open domains, one bounded, the other not, such that the curve is the boundary of each." The intrinsic geometry of a Jordan domain by Richard L. Bishop Examples: Petal of the Leau-Fatou flower Canonical domain One of the simply-connected Riemann surfacesIncorporating topography into crustal-scale wave-equation imaging through conformal mapping by Jeffrey Shraggec and Paul Sava characterized by rectangular grid Flower Lea-Fatu flower Interval a partition of an interval into subintervals Markov partitionmath.stackexchange question: definition-of-markov-partition Invariant sth is invariant if it does't change under transformation "A subset S of the domain Ω is an invariant set for the system (7.1) if the orbit through a point of S remains in S for all t ∈ R. If the orbit remains in S for t > 0, then S will be said to be positively invariant. Related definitions of sets that are negatively invariant, or locally invariant, can easily be given" Norman Lebovitz: Textbook for Mathematics 27300 Examples: invariant set invariant point = fixed point invariant cycle = periodic point invariant curve invariant circle petal = invariant planar set Julia set Feigenbaum Julia set Julia set for Feigenbaum parameter c Successive zooms lead to a Julia set which grows more and more hairs. (Similarly, the Mandelbrot set gains more decorations while limiting on the Feigenbaum point.) This leads to the natural question: Does the Julia set of the Feigenbaum quadratic polynomial have positive or zero measure? If zero, is its Hausdorff dimension less than 2?On the Measure of the Feigenbaum Julia Set Scott Sutherland Stony Brook University Joint work with Artem Dudko, IMPAN Bedlewo, Poland 22 March 2018 Level set a level set of a real-valued function fWikipedia: Level set (see also dwell band) Level set methods (LSM) on the dynamic plane in case of: dynamic plane integer escape time target set: exterior of the circle (used in the escaping test) attracting case On the dynamic plane level set is defined: Boundaries of level sets (lemniscates) are On the parameter plane where is Escape Radius, bailout value, radius of circle which is used to measure if orbit of is bounded; it is integer number are complex numbers (points of 2-D planes) is point of dynamical plane (z-plane) is point of parameter plane (c-plane) critical point of Then: ... is a circle, is an Cassini oval, is a pear curveWeisstein, Eric W. "Pear Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PearCurve.html Mandelbrot lemniscate at 2DCurves by Jan Wassenaar. These curves tend to boundary of Mandelbrot set as n goes to infinity. If ER < 2 they are inside Mandelbrot set Polynomial_lemniscate. If ER = 2 curves meet together (have common point) c = −2. Thus they can't be equipotential lines. If ER ≥ 2 they are outside of Mandelbrot set. They can also be drawn using Level Curves Method. If ER >> 2 they approximate equipotential lines (level curves of real potential, see CPM/M). parabolic case Where: d is a diameter of circle through 2 points: and radius r is half of diameter: is n*p iteration of critical point fixed point of p iteration of f function p is a period of the cycle Locus Cantor The Cantor locus is the unique hyperbolic component, in the moduli space of quadratic rational maps rat2, consisting of maps with totally disconnected Julia sets Limits of Polynomial-like Quadratic Rational Maps III: The Cantor Locus by Eva Uhre Connectedness In one-dimensional complex dynamics, the connectedness locus is a subset of the parameter space of rational functions, which consists of those parameters for which the corresponding Julia set is connected. the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. Shift The shift locus of complex polynomials of degree d ≥ 2 is a collection of polynomials that every critical point escapes to infinity under iterations of itself. The reason we call it a shift polynomial is the following theorem. The most famous and the simplest one is the exterior of Mandelbrot set, C −M, which is the shift locus of quadratic polynomials S2.Monodromy through bifurcation locus of the Mandelbrot set by Hyungryul Baik, Juhun Baik 2023 See also monodromy Planar set a non-separating planar set is a set whose complement in the plane is connected.A. Blokh, X. Buff, A. Cheritat, L. Oversteegen The solar Julia sets of basic quadratic Cremer polynomials, Ergodic Theory and Dynamical Systems, 30 (2010), #1, 51-65 postsingular "The postsingular set P(f) of a meromorphic function f is the closure of the union of forward iterates of the singular set S(f):"Prescribing the Postsingular Dynamics of Meromorphic Functions by Christopher J. Bishop, Kirill Lazebnik post-critical the iterates of the critical set "For a rational map of the Riemann sphere f, the post-critical set PC(f) is defined as closure of orbits of all critical points of f. It is proved by Lyubich [Ly83b] that the post-critical set of a rational map is the measure theoretic attractor of points in the Julia set of that map. That is, for every neighborhood of the post-critical set, orbit of almost every point in the Julia set eventually stays in that neighborhood" Dynamics of complex unicritical polynomials. A Dissertation Presented by Davoud Cheraghi "The postcritical set P(f) of a rational map f is the smallest forward invariant subset of that contains the critical values of f."On the postcritical set of a rational map by Laura G. DeMarco, Sarah C. Koch and Curtis T. McMullen "The analysis of the post-critical set plays a central role in the dynamics of rational maps, mainly because of the following two properties: the set of attracting cycles is always finite for rational maps f every attracting cycle attracts the orbit of a critical point of f."Dynamics of hyperbolic correspondences by Carlos Siqueira region Shell-Thron regionHyperoperations Wiki: Shell-Thron_region Sepal Sepal Singular set "The singular set S(f) of a meromorphic function f : C → Cˆ is the collection of values w at which one can not define all branches of the inverse f −1 in any neighborhood of w. If f is rational, then S(f) coincides with the collection of critical values of f. If f is transcendental meromorphic, f −1 may also fail to be defined in a neighborhood of an asymptotic value" Prescribing the Postsingular Dynamics of Meromorphic Functions by Christopher J. Bishop, Kirill Lazebnik Target set Target set trap for forward orbit it is a set which captures any orbit tending to fixed / periodic point Trap Trap is another name of the target set Test Bailout test or escaping test right|thumb|Two sets after bailout test: escaping white and non-escaping black right|thumb|Distance to fixed point for various types of dynamics It is used to check if point z on dynamical plane is escaping to infinity or not.Fractus doc by Richard Rosenman It allows to find 2 sets: escaping points (it should be also the whole basing of attraction to infinity)Wikipedia: Escaping set not escaping points (it should be the complement of basing of attraction to infinity) In practice for given IterationMax and Escape Radius: some pixels from set of not escaping points may contain points that escape after more iterations then IterationMax (increase IterMax) some pixels from escaping set may contain points from thin filaments not choosed by maping from integer to world (use DEM) If is in the target set then is escaping to infinity (bailouts) after n forward iterations (steps).fractint doc: bailout The output of test can be: boolean (yes/no) integer: integer number (value of the last iteration) Types of bailout test: in Fractalzoom other description kf - p-norm with weights Criterion criterion = an algorithm which will always give an answer Attraction test Theorem The Douady-Hubbard landing theorem for periodic external rays of polynomial dynamics: "for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray." A landing theorem for entire functions with bounded post-singular sets by Anna Miriam Benini, Lasse Rempe-Gillen Let f be a polynomial whose postcritical set P(f) is bounded. Then every periodic ray of f lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point of f is the landing point of at least one periodic dynamic ray, and at most finitely may dynamic rays, all of which are periodic with the same period. The Douady-Hubbard Strumienianin Theorem says that each polynomial-like map g with connected Julia set is hybrid to a unique polynomial up to an affine conjugacy. To determine the straightening uniquely, it is convenient to introduce an external marking for gPrimitive tuning for non-hyperbolic polynomials by Yimin Wang References