Fractals/Iterations in the complex plane/def cqp


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See also

Address edit

 "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)" Claude Heiland-Allen[1]


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

Internal edit

  • 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


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 edit

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:



  •   describes period 6 component which is a satelite of period 3 component.
  • Mandelbrot Artists by Claude Heiland-Allen


  • 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
  • ...

Problems edit

Infinite sequences:

  • islands
  • infinite sequence of bifurcations

Angle edit

Types of angle edit

Principal branch or complex number argument
external angle internal angle plain angle
parameter plane      
dynamic plane    


external edit

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

internal edit

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]


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

plain edit

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

bearing angle in CSS edit

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")

Units edit

  • turns
  • degrees
  • radians

Number types edit

Angle (for example, external angle in turns) can be used in different number types


the external arguments of the rays landing at z = −0.15255 + 1.03294i are:[13]




Bifurcation edit

  • Numerical Bifurcation Analysis of Maps

Coordinate edit


   "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]

Curves edit


  • 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


  • plane curve = it lies in a plane.
  • closed = it starts and ends at the same place.
  • simple = it never crosses itself. only regular points


closed edit

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"

Circle edit

Inner circle edit

Unit circle edit

Unit circle   is a boundary of unit disk[19]


where coordinates of   point of unit circle in exponential form are:


Critical curves edit

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

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

dendrit edit

  • 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 lines edit

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]

geodesic edit

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

Integral edit

  • 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 edit


  • topological
  • shift invariants


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


  • field lines
    • external ray
    • internal ray

Isocurves edit

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

Equipotential lines edit

Equipotential lines = Isocurves of complex potential

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


Jordan curve edit

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 points[35]

Lamination edit

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]

Leaf edit

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]

Level curve edit

LCM = Level Curve Method = method for drawing level curves


  • equipotential line (the same potential)
  • external ray (the same external angle)
  • boundary of level set (see Level Set Method = LSM)

Open curve edit

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

Path edit

  • Path in geometry is a curve

Ray edit

Rays are:

  • invariant curves
  • dynamic or parameter
  • external, internal or extended

Extended edit

"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]

External ray edit

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   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]


  •   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 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   to  , 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   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  [47]

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 edit


  • "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 edit

For a parameter c with superattracting orbit: for every Fatou component   of filled julia set[50]   there is:

  • a unique periodic or pre-periodic point   of the super-attracting orbit
  • a Riemann map that maps:[51]

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.


See also:

intertwined edit

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]

parameter edit

Escape route edit

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


  • 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 edit

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]


Spine edit

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  .[56] This makes sense when   is connected and full [57]
  • spine is invariant under 180 degree rotation,
  • spine is a finite topological tree,
  • Critical point   always belongs to the spine.[58]
  •  -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. Douady[59]
  • 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  .[60]

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 edit

"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 edit

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 edit

Distance Function

See also:

  • metric [61]
  • Algorithm
  • 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.

Dynamics edit

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


  • 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 cauliflower disappears repelling repelling to infinity both finite fixed points are repelling r > 1

symbolic edit

"Symbolic dynamics encodes: [70]

entropy edit

equation edit

differential edit

differential equations

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

Field edit

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).[72]
  • vector field, for example gradient field

Function edit


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

Derivative edit

angular edit

Angular derivative [76]

The Schwarzian Derivative edit

The Schwarzian Derivative [77] [78][79][80]

Wirtinger derivatives edit

gradient edit

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


  • (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 variables[83]
  • (operator): Del or nabla is an gradient operator = a vector differential operator




See also

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

Jacobian edit

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

multiplier edit

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

Germ edit

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



map edit

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

is called map  , examples:

  • rational maps
  • 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."[93]

types or names edit

Brjuno edit

  • Brjuno function


harmonic edit

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 edit

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

Polynomial edit

Critical edit

Critical polynomial:






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 edit

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

Resurgent edit

"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]

transformation edit

In mathematics, a transformation is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f : XX.[101][102][103]

Examples include:

  • linear transformations of vector spaces
  • geometric transformations
    • projective transformations
    • affine transformations

coordinate transformations edit

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 (xy) and polar coordinates (rθ) 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 edit

glitches edit

Interior of the Cauliflower Julia set. .The black structure around fixed point and it's preimages is a numerical error (glitch) }}


  • 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]


graf edit

Dessin d'enfant edit

See also:

Tree edit

  • tree is a simply connected graph

See also:

Farey tree edit

Farey tree = Farey sequence as a tree

Hubbard tree edit

  • 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 edit

rooted tree of preimages:


where a vertex   is connected by an edge with  .

Iteration edit


Magnitude edit

  • 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)

Map edit


types edit

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

Complex quadratic map edit

Forms edit

c form: z^2+c edit

quadratic map[113]

  • math notation:  
  • Maxima CAS function:
(%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 edit

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 edit

Start from:

  • internal angle  
  • internal radius r

Multiplier of fixed point:


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


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 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 */

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 

 /* 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 */


/* ---------- global constants & var ---------------------------*/
Numerator :1;
DenominatorMax :10;

/* --------- main -------------- */
for Denominator:1 thru DenominatorMax step 1 do
 InternalAngle: Numerator/Denominator,
 c: GiveC(InternalAngle,InternalRadius),
  /* compute fixed point */
 alfa:float(rectform((1-sqrt(1-4*c))/2)), /* alfa fixed point */

Circle map edit

Circle map [116]

Caratheodory semiconjugacy edit

"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 Wilkerson[118]


  •   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  
  • a dyadic rational number  

An isomorphism is given by   (see Euler's identity).

Doubling map edit

Angle doubling map


Feigenbaum map edit

First return map edit

"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   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 edit

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)

Critically preperiodic polynomials edit

  • 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 edit

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:



  • 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.[129]

It can be computed using:


Quadratic like maps edit

quadratic like maps is nothing but complexification of the concept of unimodal map[130]

Riemann map edit

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



  • 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 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


  • explicit formula (only in simple cases)
  • numerical approximation (in most of the cases)[134]
    • Zipper
    • " 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

for all  , and let



  • and   is harmonic


  • 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]


Rotation map edit

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

rational edit

Rotation map   describes counterclockwise rotation of point   thru   turns on the unit circle:


It is used for computing:

irrational edit

Shift map edit


  • 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)"[141] (operator <<)

In Haskell:

 shift k = genericTake q . genericDrop k . cycle  -- shift map

See also:

Dehn twist edit

Dehn twist[142]

Number edit

complex number edit

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

constant edit

Fegenbaum constant edit

  • first (delta)[143]
  • second (alpha)

How to compute:

degree edit

It hase many meanings:[144]

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

Multiplier edit

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

  • 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."[148]
  • The multiplier is invariant under conjugacy[149]
  • Linearizability depends on the multiplier

Math notation:


Maxima CAS function for computing multiplier of periodic cycle:


where p is a period. It takes period as an input, not z point.


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)" [150]

See also:

Rotation number edit

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]


  • 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 edit

  • 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 edit

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.

Backward edit

Critical edit

Critical orbit is forward orbit of a critical point.

Forward edit

Homoclinic / heteroclinic edit

Inverse edit

Inverse = Backward

periodic edit

skipped edit

  • 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 edit

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


Parameter edit


Period edit

Period of point   under the iterarted function f is the smallest positive integer value p for which this equality


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

  is a point of periodic orbit (limit cycle)  .

More is here

Plane edit

Planes [168]

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

2-sphere edit

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.

partition edit


critical portrait partition edit

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 edit

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 edit

Curve  :



  • 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 edit

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

types edit

  • slit plane = plane with the slit deleted[171]: Let S be the "slit plane"  
  • chessboard or checkerboards

types in case of discrete dynamical system edit

Dynamic plane or phase space edit

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

Parameter plane edit


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]

Points edit

Band-merging edit

the band-merging points are Misiurewicz points[179]

Biaccessible edit

  • 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 edit

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

branched edit

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.[183]

Buried edit

" 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)

Center edit

Nucleus or center of hyperbolic component edit

A center of a hyperbolic component H is a parameter   (or point of parameter plane) such that

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


  • Nucleus of a Mu-Atom [186]

How to find center/s ?

Center of Siegel Disc edit

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]

Critical edit

A critical point[188] 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 edit

For the complex quadratic polynomial in the c form




we see that the only (finite) critical point of   is the point  .

  is an initial point for Mandelbrot set iteration.[189]

Cut edit

The "neck" of this eight-like figure is a cut-point.
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).[190] This name is used in a topology.


  • 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.

fixed edit


  • 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]

Feigenbaum edit

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]


Generalized Feigenbaum points are:

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


  • -.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  
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 edit

Fibonacci point[199] [200][201]

germ edit

  • 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 edit

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]

Mandelbrot set at Fibonacci point

Misiurewicz edit

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

Myrberg-Feigenbaum edit

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

node edit

Parabolic point edit

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]

Periodic edit

Point z has period p under f if:


In other words point is periodic

See also:

Pinching edit

"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 [207]

Pool edit

"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." [208]

post-critical edit

A post-critical point is a point


where   is a critical point.[209]

See also:

precritical edit

precritical points, i.e., the preimages of the critical point

reference point edit

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 then using arbitrary precision

renormalizable edit

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. " [210]

infinitely renormalizable edit

" 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[211]


" 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∗." [212]

repelling edit

Virtually repelling edit

virtually repelling fixed points[213]

root or bond edit

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)


singular edit

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) / (za)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 edit

  • 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 edit

"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 edit

A point is called wandering if its forward orbit under the iteration of f is infinite.[215]

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;[216]

Portrait edit

orbit portrait edit

types edit

There are two types of orbit portraits: primitive and satellite.[217] 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 edit

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." [218]

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  


  • for   critical portrait is (1/8, 7/12)
  • for   critical portrait is (1/12, 7/12)

Precision edit

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)


  • 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."[219]
  • error [220]

Principle edit

Douady’s principle edit

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

Problem edit

small divisor problem edit


  • One-Dimensional Small Divisor Problems[221] (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."[222]

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.


Processes or transformations and phenomenona edit

Aliasing and antialiasing edit

Conjugation edit

Topological conjugacy edit

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 equivalence[224] 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.


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


  • The logistic map and the tent map are topologically conjugate.[225]
  • The logistic map of unit height and the Bernoulli map are topologically conjugate.[citation needed]
  • 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.[226]

Contraction and dilatation edit

  • the contraction z → z/2
  • the dilatation z → 2z.

convolution edit

In the digital image processing[227]: 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 edit

Method of computing the derivative of a mathematical function


  • symbolic differentiation
  • Automatic Differentiation (AD)[228]
  • numeric differentiation [229][230][231] = the method of finite differences[232]

Discretizations edit

  • discretization[233] and its reverse [234]
  • discretize/homogenize in the DDG (Discrete Differential Geometry)

Discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts.[235]


  • Cartesian coordinate system ( regular grid ) of continous space

distorsion edit

Implosion and explosion edit

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


  • 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)[237]
    • " 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)[238]
  • Semi-parabolic implosion in  [239]

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.[240]

integrating edit

  • integrating along some vector field means finding a solution curve. Example: finding extrrernal ray using Runge-Kutta method for numerical integration[241]

Linearization edit

  • changing from non-linear to linear
  • " ... turn the perturbated linear map   into the exactly linear map   (it linearizes  )" Jean-Christophe Yoccoz[242]
  • 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." [243]
Linearization with inverse function


  • Parabolic Linearization

Linearisation Theorems edit

Dynamics of f near a fixed or periodic point[244]

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).


  • Koenigs’ Linearization Theorem 1884
  • Boettcher 1904

Mating edit

Mating [245]

Moebius Transformation edit

Monodromy edit


  • 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 edit


  • normalize = transformation to the model[247]
  • " 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 origin[248]

See also:

  • uniformization
  • renormalization

Parametrization edit

  • Parametrization is the process of finding parametric equations of a curve[249]

Perturbation edit

  • Perturbation technque for fast rendering the deep zoom images of the Mandelbrot set[250]
  • perturbation of parabolic point [251]
  • use perturbation theory to approximate the solutions of the differential equations[252]
  • perturbation of point x:   where epsilon is absolute value of approximation error[253]
  • adding some small value ( epsilon denoted by Greek letter  ) to the constant value to see how function changes near hard to analyze values

Renormalization edit

"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" [255]

"Sectorial renormalizations are useful in the nonlinearizable situation. " Ricardo Pérez-Marco[256]

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 Belk[257]


Separation edit

  • "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 edit

  • surgery in differential topology [259]
  • regluing [260]


Tuning edit

  • definition
    • M-set and Julia set[261]
    • external angles

examples of external angles tuning[262]

"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 WANG[263]



Uniformization edit

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 edit

property or feature edit

behavior edit

  • 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 edit

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 edit

density of the image edit

Dense image[264][265][266]

  • downsaling with gamma correction[267]
  • path finding[268]
  • supersampling: "ots of detail but fractal fades away as you get more accurate, as n increases in nxn supersampling" TGlad

Hyperbolic/parabolic/eliptic edit

The meaning of the terms "elliptic, hyperbolic, parabolic" in different disciplines in mathematics[269]

Invariant edit

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 edit

Dynamical invariants = invariants of the dynamical system

Dynamical Invariants Derived from Recurrence Plots[272]

Orientation edit

  • 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 system[273]

smooth edit

smooth = changing without visible (noticeable) edges


  • smooth gradient


  • continuous


  • discrete

Stability edit

  • 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:

Radius edit

Radius of complex number edit

The absolute value or modulus or magnitude or radius of a complex number

Conformal radius edit

Conformal radius of Siegel Disk [274][275]

Escape radius (ER) edit

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 edit

Minimal Escape Radius should be grater or equal to 2:


Better estimation is:[276][277]


crossing edit

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 edit

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 edit

Internal radius is a:

  • absolute value of multiplier  

See also: the N-2 rule[278]

Sequences edit

A sequence is an ordered list of objects (or events).[279]

A series is the sum of the terms of a sequence of numbers.[280] Some times these names are not used as in above definitions.

Itinerary edit

  is an itinerary of point x under the map f relative to the paritirtion.

It is a right-infinite sequence of zeros and ones [281]




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 converted[282] to point  


itinerary with respect to a critical portrait edit

kneading sequence edit

  • "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 "[283]
  • The itinerary ν = ν1ν2ν3 . . . of the critical value is called the kneading sequence.[284] One can start from the critical point but neglect the initial symbol. Such sequence is computed with the Hubbard tree

See also:

Thue–Morse sequence edit

Thue–Morse sequence

Orbit edit

Orbit can be:

  • forward = sequence of points
  • backward (inverse)
    • tree in case of multivalued function
    • sequence

Series edit

A series is the sum of the terms of a sequence of numbers.[286] Some times these names are not used as in above definitions.

Taylor edit

  • Taylor series and Mandelbrot set[287]
  • The Existence and Uniqueness of the Taylor Series of Iterated Functions [288]

Set edit

Attracting set edit

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 Milnor[289]

Continuum edit


Band edit

chaotic band edit

Chaotic bands from B0 to B10

period-  chaotic band  [291]

  • 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 edit

"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). " [292] Other names:

  • level sets of integer escape time

Basin edit

Basin can consist of

  • one component, like basin of infinity

of attraction edit


  • 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 [293]

immediate basin of attraction edit

the component of the basin containing the periodic point itself


  • basin of infinity (whole basin = one component)


Component edit

connected component (blob) in the image edit

Components of parameter plane edit


  • mu-atom[294]
  • 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."[295]
  • lake
  • lakelet.[296]

filament edit

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." pauldelbrot[297]

Islands edit


  • mini Mandelbrot set
  • 'baby'-Mandelbrot set
  • island mu-molecules = embedded copy of the Mandelbrot Set[298]
  • Bug
  • Island
  • Mandelbrotie
  • Midget

List of islands:

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 edit

"Hyperbolic components come in two kinds, primitive and satellite, depending on the local properties of their roots." [299]

  • 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"[300]
    • ones that have a cusp likes the main cardioid, when the little Julia sets are disjoint [301]
  • satellite
    • ones that don't have a cusp[302]
    • it's root is on the boundary of another hyperbolic component [303]
    • when the little Julia sets touch at their β-fixed point

primare edit

Child (Descendant) and the parent (ancestor) edit

  • ancestor of hyperbolic component
  • descendant of hyperbolic component = child [304]

Hyperbolic component of Mandelbrot set edit

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." [305]

A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake [306]

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


  • LAHCs = the last appearance HCs placed in the chaotic region

Limb edit

  • 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)[307]
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.[308]


  • 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 edit

  • The main molecule is the union of all hyperbolic components attached to the main cardioid through a chain of finitely many components.[310]
  • island mu-molecule = island mu-unit [311]

shrub edit

  • "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."[312]


  • main antenna is a shrub of   family

representative of a branch is the smallest period hyperboloic componenet in the branch

spokes edit

"Colloquial term for a filament, specifically one of the "arms" radiating from a branch point." - from Mu-Ency

Wake edit

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:

p/q-Subwake of W is a wake of a p/q-satellite component of W

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."[313]

Components of dynamical plane edit

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 component[314] 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."[315]

See also:

  • interior and exterior of filled Julia set for polynomials
  • immediate basin of attraction

Domain edit

Domain in mathematical analysis it is an open connected set

Jordan domain edit

"A Jordan domain[316] 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." [317]


Canonical domain edit

  • One of the simply-connected Riemann surfaces[318]
  • characterized by rectangular grid

Flower edit

Lea-Fatu flower

Interval edit

a partition of an interval into subintervals

Invariant edit

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" [320]


  • invariant set
  • invariant point = fixed point
  • invariant cycle = periodic point
  • invariant curve
    • invariant circle
  • petal = invariant planar set

Julia set edit

Feigenbaum Julia set edit

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?[321]

Level set edit

in case of:

attracting case edit

On the dynamic plane level set is defined:


Boundaries of level sets (lemniscates) are


On the parameter plane



  •   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  






  is a circle,

  is an Cassini oval,

  is a pear curve[323][324].

These curves tend to boundary of Mandelbrot set as n goes to infinity.

  • If ER < 2 they are inside Mandelbrot set[325].
  • 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 edit



  • 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 edit

Cantor edit

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 [326]

Connectedness edit

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 edit

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.[327]

See also

  • monodromy

Planar set edit

a non-separating planar set is a set whose complement in the plane is connected.[328]

postsingular edit

"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):"[329]


post-critical edit

  • 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" [330]
  • "The postcritical set P(f) of a rational map f is the smallest forward invariant subset of that contains the critical values of f."[331]
  • "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."[332]

region edit

Sepal edit


Singular set edit

"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" [334]

Target set edit

Target set

  • trap for forward orbit
  • it is a set which captures any orbit tending to fixed / periodic point

Trap edit

Trap is another name of the target set

Test edit

Bailout test or escaping test edit

Two sets after bailout test: escaping white and non-escaping black
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.[335] It allows to find 2 sets:

  • escaping points (it should be also the whole basing of attraction to infinity)[336]
  • 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).[337]

The output of test can be:

  • boolean (yes/no)
  • integer: integer number (value of the last iteration)

Types of bailout test:

Criterion edit

criterion = an algorithm which will always give an answer

Attraction test edit

Theorem edit

  • 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." [338]
    • 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 g[339]

References edit

  1. haskell package: ruff-0.2by Claude Heiland-Allen
  2. On the Locus of Crossed Renormalization (Problems on complex dynamical systems) by Riedl, Johannes; Schleicher, Dierk
  3. Trees of visible components in the Mandelbrot set by Virpi K a u k o
  4. Rational Maps with Clustering and the Mating of Polynomials by Thomas Joseph Sharland
  5. analytical naming system From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020.  
  6. math.stackexchange question: naming-bulbs-on-the-mandelbrot-set
  7. Topics from One-Dimensional Dynamics by Karen M. Brucks,Henk Bruin. page 265 exercise 14.2.12
  8. Combinatorics, external rays, and twisted polynomials. by Wolf Jung
  9. muency - internal angle (the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2016.)
  10. internal angle from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017
  11. argument of complex number
  12. docs: CSS and angle-value
  13. 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
  14. Matcont - 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).
  15. quora: What-is-gradient?
  16. statistics how to: double-points
  17. geometry by Dr. Carol JVF Burns
  18. What is a Curve  ?
  19. Unit circle in Wikipedia
  20. 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
  21. Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971-5-0682-3. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
  22. M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint
  24. Models_for_spaces_of_dendritic_polynomials by ALEXANDER BLOKH, LEX OVERSTEEGEN, ROSS PTACEK,AND VLADLEN TIMORIN
  25. MODELING DENDRITIC SHAPES Using Path Planning by Ling Xu, David Mould
  26. procedural-branching-texture in blender by LordoftheFleas
  27. Escape lines versus equipotential lines in the Mnadelbrot set by M. Romera, Pastor G, D. de la Guía, Montoya
  28. Pseudosphere Geodesics by Tim Hutton
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  39. Core entropy and biaccessibility of quadratic polynomials by Wolf Jung
  40. 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
  41. Rational Parameter Rays of the Mandelbrot Set by Dierk Schleicher
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  62. distance fields by Philip Rideout
  63. Distance Transforms of Sampled Functions by Pedro Felipe Felzenszwalb
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  81. betterexplained: vector-calculus-understanding-the-gradient
  82. khan academy: the-gradient
  83. Gradient-Based Optimization by Jason Hicken, Prof. Juan Alonso, and Prof. Charbel Farhat
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  85. The chaotic nature of faster gradient descent methods by Kees van den Doel and Uri Ascher
  86. Conformal Geometry and Dynamics of Quadratic Polynomials, vol I-II Mikhail Lyubich, page 61
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  108. math.stackexchange question: coloring-the-mandelbrot-set-using-iterated-points?
  109. Dessins d’enfants and Hubbard trees by Kevin M. Pilgrim
  110. Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials by Henk Bruin, Dierk Schleicher
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  116. three-cool-facts-about-rotations-of-the-circle by David Richeson
  117. irrational-rotations-of-the-circle-and-benfords-law by David Richeson
  118. Subdivision rule constructions on critically preperiodic quadratic matings by Mary Wilkerson
  119. Quotient group
  120. Circle group in wikipedia
  121. The measure of the Feigenbaum Julia set by Artem Dudko and Scott Sutherland
  122. MLC at Feigenbaum points by Dzmitry Dudko, Mikhail Lyubich
  123. Poincaré map
  124. General principles of chaotic dynamics by P.B. Persson, C.D. Wagner
  125. Continuous time and discrete time dynamical systems by Shaun Bullett
  126. Continuous time and discrete time dynamical systems by Shaun Bullett
  128. The Thurston Algorithm for quadratic matings by Wolf Jung
  129. Conformal Geometry and Dynamics of Quadratic Polynomials Mikhail Lyubich
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  134. mathoverflow question: complex-function-for-mapping-a-circle-to-a-superellipse
  135. math.stackexchange question: explicit-riemann-mappings
  136. Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle by Mikhail Lyubich
  138. Graph Replacement Systems for Julia Sets of Quadratic Polynomials by Yuan J. Liu
  139. A Thompson-Like Group for the Bubble Bath Julia Set by Jasper Weinrich-Burdref
  140. three-cool-facts-about-rotations-of-the-circle by David Richeson
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  150. Periodic cycles and singular values of entire transcendental functions by Anna Miriam Benini and Nuria Fagella
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  154. The Fractal Geometry of the Mandelbrot Set II. How to Count and How to Add Robert L. Devaney
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  164. Wikipedia: Orbit (dynamics)
  165. Ouadratic-like maps and Renormalization by Nuria Fagella
  166. Peiod From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
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  173. mu-ency: exponential map by R Munafo
  174. Exponential mapping and OpenMP by Claude Heiland-Allen
  175. Linas Vepstas : Self Similar?
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  177. Stretching cusps by Claude Heiland-Allen
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  180. On biaccessible points in the Julia set of the family z(a+z^{d}) by Mitsuhiko Imada
  181. On biaccessible points in the Julia set of a Cremer quadratic polynomial by Dierk Schleicher, Saeed Zakeri
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  184. Topological Variety of Buried Points by Clinton P. Curry, Logan C. Hoehn, and John C. Mayer
  185. Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
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  187. Siegel disks by Xavier Buff and Arnaud Ch ́ritat e Univ. Toulouse Roma, April 2009
  188. Wikipedia: Critical point (mathematics)
  189. Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
  190. Cut point in wikipedia
  191. On local connectivity for the Julia set of rational maps: Newton’s famous example By P. Roesch
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  195. On Periodic and Chaotic Regions in the Mandelbrot Set by G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya
  196. fractal-faq : section 6
  197. Period doubling and Feigenbaum's scaling be E Demidov
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  199. Fractalforums: fibonacci-and-the-mandelbrot-set
  200. Parameter scaling for the Fibonacci point by Leroy Wenstrom
  201. The Fibonacci unimodal map by Mikhail Lyubich, John W. Milnor
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  203. Mathoverflow question: Attractive Basins and Loops in Julia Sets
  204. Wikipedia: Misiurewicz point
  205. The bifurcation diagram of cubic polynomial vector fields on CP1 by Christiane Rousseau
  206. A rigidity result for some parabolic germs by Luna Lomonaco, Sabyasachi Mukherjee
  207. program Mandel by Wolf Jung, demo 2 page 3
  208. Thompson-Like Groups for Dendrite Julia Sets by Will Smith
  210. Ouadratic-like maps and Renormalization by Nuria Fagella
  211. mathoverflow question: is-there-a-way-to-find-regions-of-depth-in-the-mandelbrot-set-other-than-simply?rq=1
  212. Immediate renormalization of complex polynomials by Alexander Blokh, Lex Oversteegen, Vladlen Timorin
  213. Buff, Xavier. "Virtually repelling fixed point.." Publicacions Matemàtiques 47.1 (2003): 195-209. <>.