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

 "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-Allen[1]

• 
  Parameter plane with internal rays (green) used for creating internal addresses
• 
  ${\displaystyle 1\xrightarrow {4/5} 5\xrightarrow {1/17} 85}$
• 
  ${\displaystyle 1{\xrightarrow {1/3}}3{\xrightarrow {1/7}}21{\xrightarrow {1/13}}273}$
• 
  ${\displaystyle 1\xrightarrow {1/3} 3\xrightarrow {1/3} 9}$
• 
  ${\displaystyle 1\xrightarrow {1/2} 2\xrightarrow {1/3} 6}$

types

• finite / infinite
• accessible/non-accessible
• on the parameter plane / on th edynamic plane
• simple/ angled
• for Crossed Renormalizations^[2]

• 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 A^[3]
• Internal addresses describe the combinatorial structure of the Mandelbrot set.^[4] It is one of the Analytical Naming Systems^[5][6]
• the ancestral route of a hyperbolic component is the ordered sequence of all its ancestors

 ${\displaystyle 1\quad {\xrightarrow {\quad }}\ 3\quad {\xrightarrow {\quad }}\ 6}$

Internal address:

• is not constant within hyperbolic component. Example: internal address of -1 is 1->2 and internal address of 0.9999 is 1^[7]
• 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.^[8] 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 internal address is an extension of internal address. The angled internal address of the end of a finite chain of child bulbs ${\displaystyle p_{j}/q_{j},j\in 1,2,\ldots ,k}$ would be:

${\displaystyle 1\xrightarrow {p_{1}/q_{1}} q_{1}\xrightarrow {p_{2}/q_{2}} q_{1}q_{2}\ldots \xrightarrow {p_{k}/q_{k}} \prod _{j=1}^{k}q_{j}}$

Examples:

• 
  ${\displaystyle 1\quad \xrightarrow {4/5} \ 5\quad \xrightarrow {1/17} \ 85\quad }$

• ${\displaystyle 1\quad \xrightarrow {1/3} \ 3\quad \xrightarrow {1/2} \ 6\quad }$ 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 adress 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
• ...

Infinite sequences:

• islands
• infinite sequence of bifurcations

	external angle	internal angle	plain angle
parameter plane	${\displaystyle arg(\Phi _{M}(c))\,}$	${\displaystyle arg(\rho _{n}(c))\,}$	${\displaystyle arg(c)\,}$
dynamic plane	${\displaystyle arg(\Phi _{c}(z))\,}$		${\displaystyle arg(z)\,}$

where:

• ${\displaystyle \rho _{n}(c)}$ is a multiplier map
• ${\displaystyle \Phi (c)\,}$ is a Boettcher function

The external angle is a 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

The internal angle^[9] 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 Munafo^[10]

${\displaystyle \alpha ={\frac {p}{q}}\in \mathbb {Q} }$

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

The plain angle is an angle of complex point = its argument^[11]

By convention, when an angle denotes a direction in CSS, it is typically interpreted as a bearing angle, where:^[12]

• 0deg is "up" or "north" on the screen
• and larger angles are more clockwise (so 90deg is "right" or "east")

• turns
• degrees
• radians

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:^[13]

${\displaystyle (\theta _{20}^{-},\theta _{20}^{+})=(0.{\overline {00110011001100110100}},0.{\overline {00110011001101000011}})}$

where:

${\displaystyle \theta _{20}^{-}=0.{\overline {00110011001100110100}}_{2}=0.{\overline {20000095367522590181913549340772}}_{10}={\frac {209716}{1048575}}={\frac {209716}{2^{20}-1}}}$

• Numerical Bifurcation Analysis of Maps
  • MatCont^[14]

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 Hasan[15]

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):^[16] 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

Description^[17]

• 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 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.^[18]
• 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"

Unit circle ${\displaystyle \partial D\,}$ is a boundary of unit disk^[19]

${\displaystyle \partial D=\left\{w:abs(w)=1\right\}}$

where coordinates of ${\displaystyle w\,}$ point of unit circle in exponential form are:

${\displaystyle w=e^{i*t}\,}$

Diagrams of critical polynomials are called critical curves.^[20]

These curves create skeleton of bifurcation diagram.^[21] (the dark lines^[22])

• a locally connected branched curve
• "Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic."^[23]
• "a dendrite is a locally connected continuum that does not contain Jordan curves." ^[24]
• "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 operation^[25]
• Procedural Branching Texture^[26]

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" ^[27]

In geometry, a geodesic is a curve representing in some sense the shortest path (arc) between two points in a surface^[28]

• 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.

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^[29][30][31]
• curve is invariant for a system of ordinary differential equations^[32]

"Quasi-invariant curves are used in the study of hedgehog dynamics" RICARDO PEREZ-MARCO^[33]

Examples:

• field lines
  • external ray
  • internal ray

Isocurve = level curve = curve which consist of points which have the same value (level) of parameter / variable

Equipotential lines = Isocurves of complex potential

"If the escape radius is greater than 2 the contour lines are equipotential lines" ^[34]

Examples

• desmos examples: isovalues by Fabrice

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 points^[35]

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 circle^[36][37]

It is a model of Mandelbrot or Julia set.

A lamination, L, is a union of leaves and the unit circle which satisfies:^[38]

• 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 ^[39]

Chords = leaves = arcs

A leaf on the unit disc is a path connecting two points on the unit circle.^[40]

"In Thurston’s fundamental preprint, the two characteristic rays and their common landing point are the “minor leaf” of a “lamination”"^[41]

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)

Curve which is not closed. Examples: line, ray.

• Path in geometry is a curve

Rays are:

• invariant curves
• dynamic or parameter
• external, internal or 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 Poirier^[42]

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.^[43]

  "A ray R is said to land or converge, if the accumulation set ${\displaystyle {\bar {R}}-R}$ 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."[44]

where:

• ${\displaystyle {\bar {R}}}$ is a closure of ${\displaystyle R}$ = the bar is taken to mean the closure rather than the complex conjugate
• MLC = Mandelbrot Local connectivity Conjecture: M is locally connected^[45]
• 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 ${\displaystyle \Phi :D\to Int(M)}$ to ${\displaystyle S1}$, giving a conformal equivalence of M with D. Given the fractal nature of M, this would be a very surprising result.[46]

A dynamic periodic ray pair ${\displaystyle P_{c}(\vartheta ,\vartheta ')}$ is called characteristic if it separates the critical value from all rays ${\displaystyle R_{c}(2^{k}\vartheta )}$ and ${\displaystyle R_{c}(2^{k}\vartheta ')}$ for all k ≥ 1 (except of course from those on the ray pair ${\displaystyle P_{c}(\vartheta ,\vartheta ')}$ itself).

Every cycle of periodic ray pairs has a unique characteristic ray pair with angles in the union ${\displaystyle \bigcup _{k\geq 0}\{2^{k}\vartheta ,2^{k}\vartheta '\}}$^[47]

For non-periodic rays, we allow a characteristic ray pair to contain the critical value: a ray pair ${\displaystyle P_{c}(\vartheta ,\vartheta ')}$ is characteristic if ${\displaystyle \mathbb {C} \setminus P_{c}(\vartheta ,\vartheta ')}$ consists of two components ${\displaystyle X0,X1}$ so that

• ${\displaystyle {\overline {X}}}$ contains all rays Rc(2kϑ) and Rc(2kϑ0) for k ≥ 1
• and ${\displaystyle {\overline {X}}1}$ contains the critical value

Definition:

• "The internal rays are the preimages of the radial segments under the coordinate with componenet center corresponding to 0." Alfredo Poirier^[48]
• The internal rays of U are the images of radial lines under the Riemann maps.^[49]

Internal rays are:

• dynamic (on dynamic plane, inside filled Julia set)
• |parameter (on parameter plane, inside Mandelbrot set) usuning multiplier map

• 
  Dynamic internal (blue segment) and external (red ray) rays

For a parameter c with superattracting orbit: for every Fatou component ${\displaystyle {\mathit {U}}}$ of filled julia set^[50] ${\displaystyle K_{c}}$ there is:

• a unique periodic or pre-periodic point ${\displaystyle z_{\mathit {U}}}$ of the super-attracting orbit
• a Riemann map that maps:^[51]

component to unit disc:

${\displaystyle \varphi _{\mathit {U}}:{\mathit {U}}\to \mathbb {D} }$

and point ${\displaystyle z_{\mathit {U}}}$ to the origin:

${\displaystyle \varphi _{\mathit {U}}(z_{\mathit {U}})=0}$

The point ${\displaystyle z_{\mathit {U}}}$ is called the center of component ${\displaystyle {\mathit {U}}}$.

For any angle ${\displaystyle \vartheta \in \mathbb {R} /\mathbb {Z} }$ the pre-image of the radial segment of the unit disc

${\displaystyle \varphi _{\mathit {U}}^{-1}(r^{2\pi \vartheta }):r\in [0,1]}$

is called an internal ray of component ${\displaystyle {\mathit {U}}}$ with well-defined landing point.

where:

• ${\displaystyle \mathbb {R} /\mathbb {Z} }$ is The quotient group R/Z

See also:

• Hubbard tree

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 interwined^[52]

• 
• 
  Uniformization of the interior of Mandelbrot set components using multiplier map: internal rays, internal coordinate
• 
  Method of computing internal ray using explicit equation
• 
  Wakes near the period 3 island in the Mandelbrot set, internal angles and rays (green) and external angles and rays (red)

Escape route is a path inside Mandelbrot set.

Escape route 1/2 ^[53]

• 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
• ...

• escape routes
• 
  1/2
• 
  1/3

A spider S is a collection of disjoint simple curves called legs ^[54] (extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infinity ^[55]

See:

• spider algorithm
• spider program

• 
  Rabbit Julia set with spine
• 
  Basilica Julia set with spine

In the case of complex_quadratic_polynomial ${\displaystyle f_{c}(z)=z^{2}+c}$ the spine ${\displaystyle S_{c}\,}$ of the filled Julia set ${\displaystyle \ K\,}$ is defined as arc between ${\displaystyle \beta \,}$-fixed point and ${\displaystyle -\beta \,}$,

${\displaystyle S_{c}=\left[-\beta ,\beta \right]\,}$

with such properties:

• spine lies inside ${\displaystyle \ K\,}$.^[56] This makes sense when ${\displaystyle K\,}$ is connected and full ^[57]
• spine is invariant under 180 degree rotation,
• spine is a finite topological tree,
• Critical point ${\displaystyle z_{cr}=0\,}$ always belongs to the spine.^[58]
• ${\displaystyle \beta \,}$-fixed point is a landing point of external ray of angle zero ${\displaystyle {\mathcal {R}}_{0}^{K}}$,
• ${\displaystyle -\beta \,}$ is landing point of external ray ${\displaystyle {\mathcal {R}}_{1/2}^{K}}$.

Algorithms for constructing the spine:

• detailed version is described by A. Douady^[59]
• Simplified version of algorithm:
  • connect ${\displaystyle -\beta \,}$ and ${\displaystyle \beta \,}$ within ${\displaystyle K\,}$ by an arc,
  • when ${\displaystyle K\,}$ has empty interior then arc is unique,
  • otherwise take the shortest way that contains ${\displaystyle 0}$.^[60]

Curve ${\displaystyle R\,}$:

${\displaystyle R\ {\overset {\underset {\mathrm {def} }{}}{=}}\ R_{1/2}\ \cup \ S_{c}\ \cup \ R_{0}\,}$

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 ${\displaystyle z\to z^{2}}$.
 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

"A vein in the Mandelbrot set is a continuous, injective arc inside in the Mandelbrot set"

"The principal vein ${\displaystyle v_{p/q}}$ is the vein joining ${\displaystyle c_{p/q}}$ to the main cardioid" (Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems. A dissertation by Giulio Tiozzo)

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.

See also:

• metric ^[61]
• 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:^[62][63][64]
    • Marching Parabolas, a linear-time CPU-amenable algorithm.
    • Min Erosion, a simple-to-implement GPU-amenable algorithm.

• symbolic^[65][66][67]
• complex ^[68][69]
• Arithmetic
• combinatorial
• local/global
• discrete/continous
• parabolic/hyperbolic/eliptic

Examples:

• discrete local complex parabolic dynamics

"Symbolic dynamics encodes: ^[70]

• a dynamical system ${\displaystyle f:X\to X}$ by a shift map on a space of sequences over finite alphabet using Markov partition of the space ${\displaystyle X}$
• the points of space ${\displaystyle X}$ by their itineraries with respect to the partition " (Volodymyr Nekrashevych - Symbolic dynamics and self-similar groups)

• image entropy ^[71]

differential equations

• exact analytic solutions.
• approximated solution
  • use perturbation theory to approximate the solutions

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 ${\displaystyle R^{n}\to R}$, where for any given input the output is the distance to the nearest surface (where the field value is 0).^[72]
• vector field, for example gradient field

types:

• by application
  • map = iterated map
  • mappings = transformation of the plane
• by function type
  • polynomial

• Derivative of Iterated function (map)^[73][74]
• of the function f with respect to (wrt) variable
• following the derivative^[75]

Angular derivative ^[76]

The Schwarzian Derivative ^{[77] [78][79][80]}

the gradient is the generalization of the derivative for the multivariable functions^[81][82]

definitions:

• (field): Gradient field is the vector field with gradient vector
• (function): The gradient of a scalar-valued multivariable function ${\displaystyle f(x,y)}$ is a vector-valued function denoted ${\displaystyle \nabla f}$
• (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 variables^[83]
• (operator): Del or nabla is an gradient operator = a vector differential operator

• 
  Gradient 2D field with gradient vectors
• 
  one gradient vector (red)

Notations:

${\displaystyle \nabla f(x,y)=[{\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}}]={\begin{bmatrix}{\frac {\partial f}{\partial x}}\\{\frac {\partial f}{\partial y}}\end{bmatrix}}={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} }$

${\displaystyle \operatorname {grad} f=\nabla f}$

See also

• Gradient Descent Algorithm^[84][85]
• Gradient Ascent Algorithm
• image gradient

The Jacobian is the generalization of the gradient for vector-valued functions of several variables

The multiplier of a fixed point α is the derivative A′(α) calculated in any local chart around α[86]

Germ ^[87] of the function f in the neighborhood of point z is a set of the functions g which are indistinguishable in that neighborhood

${\displaystyle [f]_{z}=\{g:g\sim _{z}f\}.}$

See:

• parabolic germ
• the linearization of a germ^[88]

• differences between map and the function ^[89]
• Iterated function = map^[90]
• an evolution function^[91] of the discrete nonlinear dynamical system^[92]

${\displaystyle z_{n+1}=f(z_{n})\,}$

is called map ${\displaystyle f}$, examples:

• rational maps
  • Newton maps
  • complex quadratic map: ${\displaystyle f_{c}:z\to z^{2}+c.\,}$
• exponential maps
• trigonometric maps
• landing map: " A theorem of Caratheodory states that if ${\displaystyle K\subset \mathbb {C} }$ is a full compact and locally connected set, then external rays land and the landing map ${\displaystyle \ell :\mathbb {R} /\mathbb {Z} \to \partial K}$ is continuous."^[93]

• Brjuno function

Links:

• Quelle est la régularité de la fonction de Brjuno ?

An harmonic or spherical function is a:

• "set of orthogonal functions all of whose curvatures are changing at the same rate."^[94]
• "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"^[95]

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 ? ^[96]

meromorphic maps: Those with NO FINITE, NON-ATTRACTING FIXED POINTS^[97]

Critical polynomial:

${\displaystyle Q_{n}=f_{c}^{n}(z_{cr})=f_{c}^{n}(0)\,}$

so

${\displaystyle Q_{1}=f_{c}^{1}(0)=c\,}$

${\displaystyle Q_{2}=f_{c}^{2}(0)=c^{2}+c\,}$

${\displaystyle Q_{3}=f_{c}^{3}(0)=(c^{2}+c)^{2}+c\,}$

These polynomials are used for finding:

• centers of period n Mandelbrot set components. Centers are roots of n-th critical polynomials ${\displaystyle centers=\{c:f_{c}^{n}(z_{cr})=0\}\,}$ (points where critical curve Qn croses x axis)
• Misiurewicz points ${\displaystyle M_{n,k}=\{c:f_{c}^{k}(z_{cr})=f_{c}^{k+n}(z_{cr})\}\,}$

a post-critically finite polynomial = all critical points have finite orbit

"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, 1980^{[98][99][100]}

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.^{[101][102][103]}

Examples include:

• linear transformations of vector spaces
• geometric transformations
  • projective transformations
  • affine transformations
    • rotations
    • reflections
    • translations^[104][105]

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.

Definition:

• Incorrect (noisy) parts of renders^[106] using perturbation technique
• pixels which dynamics differ significantly from the dynamics of the reference pixel^[107]"These can be detected and corrected by using a more appropriate reference."^[108]

Examples:

• glitches in perturbation method

• Wikipedia article: Dessin d'enfant
• commons:Category:Dessins d'enfants (mathematics)

See also:

• Polynomials Invertible in k-Radicals by Y. Burda, A. Khovanskii

• 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 sequence as a 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." ^[109]
• " 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 Schleicher^[110]
• 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 of preimages:

${\displaystyle T_{f}:=\bigsqcup _{n\geq 0}f^{-n}(t),}$

where a vertex ${\displaystyle z\in f^{-n}(t)}$ is connected by an edge with ${\displaystyle f(z)\in f^{-(n-1)}(t)}$.

Iteration

• magnitude of the point (complex number in 2D case) = it's distance from the origin^[111]
• radius is the absolute value of complex number (compare to arguments or angle)

description

• The map f is hyperbolic if every critical orbit converges to a periodic orbit.^[112]

quadratic map^[113]

• math notation: ${\displaystyle f_{c}(z)=z^{2}+c\,}$
• 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

${\displaystyle \ f_{c}^{(0)}(z)=z=z_{0}}$

${\displaystyle \ f_{c}^{(1)}(z)=f_{c}(z)=z_{1}}$

...

${\displaystyle \ f_{c}^{(p)}(z)=f_{c}(f_{c}^{(p-1)}(z))}$

or with subscripts:

${\displaystyle \ z_{p}=f_{c}^{(p)}(z_{0})}$

• 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);

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

Start from:

• internal angle ${\displaystyle \theta ={\frac {p}{q}}}$
• internal radius r

Multiplier of fixed point:

${\displaystyle \lambda =re^{2\pi \theta i}}$

When one wants change from lambda to c:^[114]

${\displaystyle c=c(\lambda )={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right)={\frac {\lambda }{2}}-{\frac {\lambda ^{2}}{4}}}$

or from c to lambda:

${\displaystyle \lambda =\lambda (c)=1\pm {\sqrt {1-4c}}}$

Example values:

${\displaystyle \theta }$	r	c	fixed point alfa ${\displaystyle z_{c}}$	${\displaystyle \lambda }$	fixed point ${\displaystyle z_{\lambda }}$
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:

${\displaystyle c=c_{x}+c_{y}*i}$

of period 1 hyperbolic component (main cardioid) for given internal angle (rotation number) t using this c / cpp code by Wolf Jung^[115]

 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 ^[116]

• irrational rotation^[117]

"The map ${\displaystyle \gamma :\mathbb {R} /\mathbb {Z} \to J_{c}}$ is called the Caratheodory semiconjugacy, with the associated identity

${\displaystyle \gamma (2*t)=f_{c}(\gamma (t))}$ 

in the degree 2 case. This identity allows us to easily track forward iteration of external rays and their landing points in ${\displaystyle J_{c}}$ by doubling the angle of their associated external rays modulo 1." Mary Wilkerson^[118]

where

• ${\displaystyle \mathbb {R} /\mathbb {Z} }$ is the real numbers modulo the integers group ( quotient group )^[119] which is isomorphic to the circle group^[120]
  • 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 ${\displaystyle {\mbox{SO}}(2)}$
• a dyadic rational number ${\displaystyle \theta \in S^{1}=\mathbb {R} /\mathbb {Z} }$

An isomorphism is given by ${\displaystyle f(a+\mathbb {Z} )=\exp(2\pi ia)}$ (see Euler's identity).

Angle doubling map

${\displaystyle T(x)=(2x){\bmod {1}}}$

• "the Feigenbaum map F is a solution of Cvitanovic-Feigenbaum equation"^[121]
• Feigenbaum quadratic polynomials (= Feigenbaum point of parameter plane) , i.e., infinitely renormalizable polynomials of bounded type.^[122]

• definition ^[123]
• 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 "^[124]

"If x is a periodic point of period p for f and U is a neighborhood of x, the composition ${\displaystyle f^{\circ p}\,}$ 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." ^[125]

"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"^[126]

postcritically finite: maps whose critical orbits are all periodic or preperiodic^[127]

  " 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:^[128]

• 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)

• 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 ${\displaystyle \lambda }$ associated with hyperbolic component ${\displaystyle \mathrm {H} }$

• gives an explicit uniformization of hyperbolic component ${\displaystyle \mathrm {H} }$ by the unit disk ${\displaystyle \mathbb {D} }$:
• it is (d-1) to one function. Where d is a degree of iterated function

${\displaystyle \lambda _{p}:\mathrm {H} \to \mathbb {D} }$

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:

${\displaystyle \lambda _{p}(c)=b}$

where:

• c is a point in the parameter plane
• b is a point in the reference plane. It is also internal coordinate
• ${\displaystyle \lambda }$ is a multiplier map

Multiplier map is a conformal isomorphism.^[129]

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 is nothing but complexification of the concept of unimodal map^[130]

Riemann mapping theorem^[131] says that every simply connected subset U of the complex number plane can be mapped to the open unit disk D

${\displaystyle f:U\to D}$

where:

• D is a unit disk ${\displaystyle D=\{z\in \mathbf {C} :|z|<1\}.}$
• f is Riemann map (function). It is 1to-1 function
• U is subset of complex plane

• 
  Multiplier map
• 
  BDM
• 
  MBDM
• 
  DLD
• 
  SAC
• 
  zeros of qn algorithm

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 plane^[132]
    • for the Fatou component containing a superattracting fixed point for a rational map^[133]
    • 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)^[134]
  • 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 Belk^[135]

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

 ${\displaystyle F(z)\;=\;-\log |z|}$

for all ${\displaystyle z\in \partial D}$, and let

 ${\displaystyle R(z)=|z|e^{F(z)}}$

Then

• ${\displaystyle R(0)=0}$
• ${\displaystyle R|_{\partial D}\equiv 1}$
• and ${\displaystyle \log R}$ 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." [136]

See

• Commons: Category:Riemann mapping
• A Riemann map on the central component^[137]
• Some internal rays of the Basilica^[138]
• The Bottcher Map B gives rise to internal angles in each bubble^[139]

     "If a is rational, then every point is periodic. If a is irrational, then every point has a dense orbit." David Richeson[140]

Rotation map ${\displaystyle R}$ describes counterclockwise rotation of point ${\displaystyle \theta }$ thru ${\displaystyle {\frac {p}{q}}}$ turns on the unit circle:

 ${\displaystyle R_{\frac {p}{q}}(\theta )=\theta +{\frac {p}{q}}}$

It is used for computing:

• itinerary

• rotation maps
• 
• 

• ROTATE IN A CIRCLE WITH A ROTATION by Sylvie Ruette (in fr.)
• Wikipedia: Irrational rotation

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

 ${\displaystyle \sigma (b_{1},b_{2},b_{3},\ldots )=(b_{2},b_{3},b_{4},\ldots )}$

It just drops first digit of the sequence.

  ${\displaystyle \sigma ^{2}(S)=\sigma (\sigma (S))}$

  ${\displaystyle \sigma ^{k}(b_{1}b_{2}\ldots )=b_{k+1}b_{k+2}\ldots }$

If we treat sequence as a binary fraction:

 ${\displaystyle x=0.b_{1},b_{2},b_{3},\ldots }$

then shift map = the dyadic transformation = dyadic map = bit shift map= 2x  mod 1 map = Bernoulli map = doubling map = sawtooth map

 ${\displaystyle \sigma (x)=2x{\bmod {1}}}$

and "shifting N places left is the same as multiplying by 2 to the power N (written as 2N)"^[141] (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^[142]

• numerical value: x+y*i
• vector from origin to point (x,y)
• point (x,y) od 2D Cartesion plain

• first (delta)^[143]
• 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)

It hase many meanings:^[144]

• unit of the angle
• degree of a function
  • polynomial
  • rational function^[145]

The multiplier of periodic z-point:^[146][147]

• is a complex number
• "The value of ${\displaystyle (f^{p})^{\prime }}$ is the same at any point in the orbit of a: it is called the multiplier of the cycle."^[148]
• The multiplier is invariant under conjugacy^[149]
• Linearizability depends on the multiplier

Math notation:

${\displaystyle \lambda _{c}(z)={\frac {df_{c}^{(p)}(z)}{dz}}\,}$

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	${\displaystyle f^{p}(z)\,}$	${\displaystyle \lambda _{c}(z)\,}$
1	${\displaystyle z^{2}+c\,}$	${\displaystyle 2z\,}$
2	${\displaystyle z^{4}+2cz^{2}+c^{2}+c}$	${\displaystyle 4z^{3}+4cz}$
3	${\displaystyle z^{8}+4cz^{6}+6c^{2}z^{4}+2cz^{4}+4c^{3}z^{2}+4c^{2}z^{2}+c^{4}+2c^{3}+c^{2}+c}$	${\displaystyle 8z^{7}+24cz^{5}+24c^{2}z^{3}+8cz^{3}+8c^{3}z+8c^{2}z}$

It is used to:

• compute stability index of periodic orbit (periodic point) = ${\displaystyle |\lambda |=r}$ (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)" ^[150]

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.

The rotation number^{[151][152][153][154][155]} 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 ^[156]

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

• of the map (iterated function)^[157][158]
  • "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 ^[159]
  • "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 ^[160][161]
  • the winding number of a curve is the number of complete rotations, in the counterclockwise sense, of the curve around the point(0, 0).^[162]
  • w(γ, x) = number of times curve γ winds round point x. The winding number is signed: + for counterclockwise, − for clockwise.^[163]

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 is a sequence of points^[164]

• phase space trajectories of dynamical systems
• The orbit of periodic point is finite and it is called a cycle.

Critical orbit is forward orbit of a critical point.

• 
  homoclinic orbit
• 
  Heteroclinic orbit

Inverse = Backward

• set containing first n iterations of initial point without initial point and its k iterations
• number of elements = n - k

${\displaystyle {\mathcal {O}}_{n,k}(z)=\{z_{n-k},z_{n-k+1},\ldots z_{n}\}}$

It is used in the average colorings

• set containing initial point and first n iterations of initial point
• number of elements = n+1

${\displaystyle {\mathcal {O}}_{n}(z)=\{z,f(z),\ldots f^{n}(z)\}=\{z_{0},z_{1},\ldots z_{n}\}}$

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. " ^[165]
• parameter of the function

Period of point ${\displaystyle z_{0}}$ under the iterarted function f is the smallest positive integer value p for which this equality

${\displaystyle f^{p}(z_{0})=z_{0}}$

holds is the period^[166] of the orbit.^[167]

${\displaystyle z_{0}}$ is a point of periodic orbit (limit cycle) ${\displaystyle \{z_{0},\dots ,z_{p-1}\}}$.

More is here

Planes ^[168]

Douady’s principle: “sow in dynamical plane and reap in parameter space”.

In topology: two-dimensional sphere = 2-sphere = the two-dimensional surface of a three-dimensional ball^[169]

Geometrically, the set of extended complex numbers is referred to as the Riemann sphere or extended complex plane.

Examples:

• Markow partition
• Yoccoz puzzle
• critical portrait
• lamination

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;

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:

${\displaystyle z=c}$

• 
  t = 1/4 preperiod = 2 period = 1
• 
  t = 1/6 preperiod = 1 and period = 2
• 
  t = 9/56 preperiod = 3 and period = 3

How to find angle of the dynamic external ray that land on the critical value z = c ?

Curve ${\displaystyle R\,}$:

${\displaystyle R\ {\overset {\underset {\mathrm {def} }{}}{=}}\ R_{1/2}\ \cup \ S_{c}\ \cup \ R_{0}\,}$

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 ${\displaystyle \beta _{c}}$ and at its preimage ${\displaystyle -\beta _{c}}$

divides dynamical plane into two components.

• 
  Plane paritition in case of Rabbit Julia set
• 
  Plane paritition in case of Basilica Julia set

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)^[170]

Key words:

• Enumerative combinatorics

• slit plane = plane with the slit deleted^[171]: Let S be the "slit plane" ${\displaystyle S=\mathbb {C} -\{t\in \mathbb {R} :t\leq 0\}}$
• chessboard or checkerboards

• z-plane for fc(z)= z^2 + c
• z-plane for fm(z)= z^2 + m*z

See:^[172]

Types of the parameter plane:

• c-plane (standard plane)
• exponential plane (map) ^[173][174]
• flatten' the cardiod (unroll) ^[175][176] = "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)^[177]
• transformations ^[178]

• 
  c-plane
• 
  inverted c plane = 1/c plane
• 
  unrolled

• 
  lambda plane

the band-merging points are Misiurewicz points^[179]

• If there exist two distinct external rays landing at point we say that it is a biaccessible point.^[180]
• We call p biaccessible if it is accessible through at least two distinct external rays^[181]

blowup point = parameter for which the critical orbits map to ∞, so the Julia set is the entire sphere ^[182]

A point in the complex plane ${\displaystyle \mathbb {C} }$ is branched, if

• it is in the Julia set
• and is the landing point of more than two rays.^[183]

" a point of the Julia set is buried if it is not in the boundary of any Fatou component." ^[184]

polynomials do not have buried points

some rational Julia sets have (Residual Julia Set = Buried Points)

A center of a hyperbolic component H is a parameter ${\displaystyle c_{0}\in H\,}$ (or point of parameter plane) such that

• the corresponding periodic orbit has multiplier= 0." ^[185]
• it has a superstable periodic orbit

Synonyms:

• Nucleus of a Mu-Atom ^[186]

How to find center/s ?

Center of Siegel disc is a 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." ^[187]

A critical point^[188] of ${\displaystyle f_{c}\,}$ is a point ${\displaystyle z_{cr}\,}$ in the dynamical plane such that the derivative vanishes ( is equal to zero):

${\displaystyle f_{c}'(z_{cr})=0.\,}$

A critical value is an image of critical point

For the complex quadratic polynomial in the c form

${\displaystyle f_{c}'(z)={\frac {d}{dz}}f_{c}(z)=2z}$

implies

${\displaystyle z_{cr}=0\,}$

we see that the only (finite) critical point of ${\displaystyle f_{c}\,}$ is the point ${\displaystyle z_{cr}=0\,}$.

${\displaystyle z_{0}}$ is an initial point for Mandelbrot set iteration.^[189]

Cut point k of set S is a point for which set S-k is dissconected (consist of 2 or more sets).^[190] 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.^[191] Cut rays can be used to construct puzzles.

Cut angle is an angle of cut ray.

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."^[192] Several properties hold about a neighborhood of a hyperbolic point, notably^[193]

• 
  types of fixed points
• 
  1a) stable fixpoint 1b) instable fixpoint, stable limit cycle 1c) phase space dynamics. Subcritical Hopf bifurcation: 2a) stable fixpoint, unstable limit cycle 2b) instable fixpoint 2c) phase space dynamics. \omega determines the angular dynamics and therefore the direction of winding for the trajectories.

• 
  Zoom toward Feigenbaum point
• 
  The Feigenbaum point (red arrow) is limit of the bifurcation point
• 
  Julia set for the Feigenbaum point

The Feigenbaum Point^[194] 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)^[195]

${\displaystyle MF^{(n)}({\tfrac {p}{q}})=c}$

Generalized Feigenbaum points are:

• the limit of the period-q cascade of bifurcations
• landing points of parameter ray or rays with irrational angles

Examples:

• ${\displaystyle MF^{(0)}=MF^{(1)}({\tfrac {1}{2}})=c=-1.401155}$
• -.1528+1.0397i)

The Mandelbrot set is conjectured to be self- similar around generalized Feigenbaum points^[196] when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time^[197]

n	Period = 2^n	Bifurcation parameter = cn	Ratio ${\displaystyle ={\dfrac {c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}}}$
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-Gillen^[198]

Fibonacci point^{[199] [200][201]}

• 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.

The point at infinity ^[202]" 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-Gillen^[203]

Misiurewicz point^[204] = " parameters where the critical orbit is pre-periodic.

MF = the Myrberg-Feigenbaum point is the different name for the Feigenbaum Point.

• branch point of the shrub
• type of the Misiurewicz point

parabolic points: this occurs when two singular points coalesce in a double singular point (parabolic point)^[205]

"the characteristic parabolic point (i.e. the parabolic periodic point on the boundary of the critical value Fatou component) of fc"^[206]

Point z has period p under f if:

${\displaystyle z:\ f^{p}(z)=z}$

In other words point is periodic

See also:

• fixed point
• stability of periodic point
  • attracting
  • repelling
  • indifferent
• multiplier of periodic cycle