Fractals/Iterations in the complex plane/misiurewicz
Misiurewicz point is the parameter c ( point oc parameter plane) where the critical orbit is preperiodic.
ProperitiesEdit
 "Around Misiurewicz points (which are only countable but dense), the Mandelbrot set looks locally like the corresponding Julia set, in particular it looks locally very different for each Misiurewicz point." Pablo Shmerkin^{[1]}
 The Mandelbrot is asymptotically selfsimilar about preperiodic Misiurewicz points.
 the set of all Misiurewicz points is dense on the boundary of the Mandelbrot set^{[2]}
Relation between external angles and Misurewicz pointEdit
The dynamics of the external angle under angle doubling is not the same as the dynamics of the landing point under iteration:^{[3]} the preperiod and period can differ between the external angle and the landing Misiurewicz point
Examples:
 the external angle 0.1(0) has preperiod 1 and period 1, and it lands on the point c = −2 = ( preperiod 2 and period 1)
 external angle .001001(010010100) ray is landing on the point c = 0.026593792304386393+0.8095285579867694i. The external angle have preperiod 6 and period 9, but landing point has preperiod 7 and period 3
notationEdit
Misiurewicz point (parameter, polynomial, map) can be marked by:^{[4]}
 preperiod and period
 the parameter coordinate c ∈ M
 the angle of the external ray
angleEdit
"Critically preperiodic polynomials are typically parameterized by the angle θ of the external ray landing at the critical value rather than by the critical value." MARY WILKERSON^{[5]}
The preperiodic angle (decimal fraction with even denominator ) of the external ray that lands:
 at z = c in the Julia set J(f) on the dynamic plane
 at in Mandelbrot set M on the parameter plane
so
Examples:
 Ray for angle lands on the point from the parameter plane. It is tip of the main antenna ( end of 1/2 limb). On the the dynamic plane it gives the Line Julia set
 Ray for angle lands on the point from the parameter plane. It is the first tip of wake 1/3. On the the dynamic plane it gives the dendrite Julia set
 Ray for angle lands on the point from the parameter plane. It is last tip of wake 1/3. On the the dynamic plane it gives the classic Dendrite Julia set
partition of the dynamic plane by dynamic rays related with the kneading sequence

1/4

1/6

9/56

129/16256
preperiod and periodEdit
where
 t is preperiod
 p is period
Preperiod is used in 2 meanings :
 T =preperiod of critical point
 t = preperiod of critical value
Note that :
Period p is the same for critical value and critical point
Wolf Jung uses preperiod of critical value : "... the usual convention is to use the preperiod of the critical value. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane."
Pastor uses preperiod of critical point : "all the Misiurewicz points are given with one unit more in their preperiods, therefore this is given as "^{[6]}
pointEdit
typesEdit
periodEdit
Misiurewicz points c
 with period 1 are of the type:^{[7]}
 alpha, i.e.
 beta, i.e
 with period > 1
where alfa and beta are fixed points of complex quadratic polynomial
TopologicalEdit
Visual types:^{[8]}
 branch tips = terminal points of the branches^{[9]} or tips of the midgets^{[10]}
 centers of spirals ^{[11]}
 centers of slow spirals with more then 1 arm
 centers of spirals = fast spiral
 bandmerging points of chaotic bands (the separator of the chaotic bands and )^{[12]} = 2 arm spiral = branch point = points where branches meet^{[13]}
spiralsEdit

Misiurewicz point is a center of a twoarms spiral

zoom into a logarithmic spiral
The Misiurewicz points which are centers of the spirals can be classified according to speed of turning:
 fast^{[14]}
 slow
 no turn : if the Misiurewicz point is a real number, it does not turn at all
Spirals can also be classified by the number of arms.
Each Misiurewicz point has a multiplier which is related to the shape of the spiral.^{[15]}
number of external raysEdit
 endpoint = tip = 1 angle
 primitive type = 2 angles of primitive cycle
 satellite type = 2 or more angles from satellite cycle
where primitive and satelite are the types of hyperbolic components
preperiod and periodEdit
In general a preperiodic critical value has a preperiod k, a period p, a ray period rp, and v angles. There are three cases,
 tip: r = 1 and v = 1
 primitive: r = 1 and v = 2
 satellite: r > 1 and v = r
The structural Misiurewicz points in an embedded Julia set all have the same period, that of the influencing island.^{[16]}
Tips:
 "It seems the tips of the nthlongest arms of primary spirals have period n, counting from 1, and the tips of some filaments have period 1" Claude HeilandAllen^{[17]}
 not all terminal points have preperiod 1 or period 1
named typesEdit
principalEdit
The principal Misiurewicz point of the limb :^{[18]}
 hase m external angles, that are preimages (under doubling) of the external angles of
characteristicEdit
Characteristic Misiurewicz point of the chaotic band of the Mandelbrot set is :^{[19]}
 the most prominent and visible Misiurewicz point of a chaotic band
 have the same period as the band
 have the same period as the gene of the band
separatorsEdit
primary separatorEdit
 bandmerging point = point merging 2 chaotic bands and
Examples: ^{[20]}
 = tip of main antenna , external angle = 1/2^{[21]}
 , external angles 5 and 7/12
 , external angles 33 and 47/80
 , external angles 1795 and 2557/4352
 ...
 = Feigenbaum point = MF = the MyrbergFeigenbaum
In the text form:
double m[12] = {
2.0,
1.543689012692076,
1.430357632451307,
1.407405118164702,
1.402492176358564,
1.401441494253588,
1.401216504309415,
1.401168320839301,
1.401158001505211,
1.401155791424613,
1.401155318093230,
1.401155216720152
};
const complex double cf = 1.401155189093314712; //the Feigenbaum point 1.401155 = m[infinity]
secondary separatorEdit
 tree separators ( tree is a subset of band )
noncharacteristicEdit
 have not the same period as the band
ExamplesEdit
Misiurewicz Points, part of the Mandelbrot set:
 Centre 0.4244 + 0.200759i; Max. Iterations 100; View radius 0.00479616 ^{[22]}
wakesEdit
wake 1/2Edit
Important points of the wake:
 bond point = root point between period 1 and 2 components = c = 0.75 = 3/4 = birurcation point for internal angle 1/2 = Landing point of 2 external rays 1/3 and 2/3 = start of wake 1/2
 nucleus (center of component ) for period 2 = c = 1
 tip of main antenna c = 2 = . It is landing point of externa ray for angle
 c = 1.543689012692076 = Principal Misiurewicz point of wake 1/2 = main node ( branch point) of the wake = = landing point of external rays 5/12 i 7/12
videosEdit
 Kalles Fraktaler  Dive into Misiurewicz
 Embedded Julia set similar to Misiurewicz Julia set by Wolf Jung
demosEdit
 Mandel demo 6 page 1
ImagesEdit
How to colour ?Edit
"The legendary colour palette technique embeds an image in the iteration bands of an escape time fractal by linearizing it by scanlines and synchronizing the scan rate to the iterations in the fractal spirals so they line up to reconstruct the original image. Historically this has been done by preparing palettes for fractal software using external tools, and mostly only for small images (KF for example has a palette limited to 1024 colour slots). Kalles Fraktaler 2 has an image texture feature, which historically only allowed you to warp a background through the semitransparent fractal. I added the ability to create custom colouring algorithms in OpenGL shader language (GLSL), with which it is possible to repurpose this texture and (for example) use it as a legendary palette. Here I scaled my avatar (originally 256x256) to 128x16 pixels, and fine tuned the iteration count divisor by hand after zooming to a spiral in the Seahorse Valley of the Mandelbrot set. Then the face from the icon is visible in the spirals all the way down to the end of the video. I used a workinprogress (not yet released) build of KF 2.15.3, which has a new setting not to resize the texture to match the frame size: this allows the legendary technique to work much more straightforwardly. I rendered exponential map EXR frames from KF and assembled into a zoom video with zoomasm. From KF I exported just the RGB channels with the legendary palette colouring, and the distance estimate channels. I did not colour the RGB with the distance estimate in KF, because with the exponential map transformation they would not be screenspace correct (the details would be smaller in the center of the reprojected video than at the edges). I could not do all the colouring in zoomasm either, because it does not support image textures. I added the boundary of the fractal in zoomasm afterwards, by mixing pink with the RGB from KF according to the length of the screenspace distance estimate channels (which zoomasm scales properly when reprojecting the exponential map)." Claude HeilandAllen^{[23]}
How to compute ...?Edit
Number of Misiurewicz pointsEdit
 math.stackexchange question: countingmisiurewiczpoints
 enumeration of misiurewicz points by Claude HeilandAllen
 "... we do not know how to compute (...) Misiurewicz parameters (with high (pre)periods) for the family of quadratic rational maps. One might need to and a nonrigorous method to and Misiurewicz parameter in a reasonable time like BihamWenzel's method." HIROYUKI INOU ^{[24]}
 The OnLine Encyclopedia of Integer Sequences (OEIS)
 is known to be A000740
 appears to be A038199
 appears to be A000225
 appears to be A166920
 Corollary 3.3. from Misiurewicz Points for Polynomial Maps and Transversality by Benjamin Hutz, Adam Towsley
The number of Misiurewicz points for is :
Where:
 m is
 n is
 d is a degree of function
 is the Moebius function for the natural n
 is the sum of over all positive integers dividing
Implementations
Misiurewicz points of complex quadratic mappingEdit
 numerical methods
 solving equations
 finding landing point of external rays with preperiodic angles
 graphical method
 Misiurewicz domains
"the best way of being sure you get to the correct point is to trace an external ray with the correct external angle, until you reach close enough (for example, the ray point has a tiny imaginary part, as these points are all on the real axis). then use Newton's method starting from the ray point." Claude HeilandAllen^{[25]}
tracing raysEdit
 "tracing rays of preperiod + period ~= 500 to dwell ~1000, with all 4 methods and varying sharpness" ^{[26]}
 "I made the database by tracing every preperiodic ray with preperiod and period summing to less than or equal to 16. I traced the rays to the limit of double precision, averaging 400 rays per second on my quad core desktop. Then I grouped together the rays landing at the same point (or nearby, with a small threshold radius). I only grouped together rays having the same period and preperiod..."^{[27]}
Misiurewicz domainsEdit
 misiurewicz_domains by Claude HeilandAllen
 Misiurewicz domain coordinates and size estimates by Claude HeilandAllen
Solving equationsEdit
roots of polynomialEdit
Misiurewicz points ^{[28]} are special boundary points.
Define polynomial in Maxima CAS :
P(n):=if n=0 then 0 else P(n1)^2+c;
Define a Maxima CAS function whose roots are Misiurewicz points, and find them.
M(preperiod,period):=allroots(%i*P(preperiod+period)%i*P(preperiod));
Examples of use :
(%i6) M(2,1); (%o6) [c=2.0,c=0.0] (%i7) M(2,2); (%o7) [c=1.0*%i,c=%i,c=2.0,c=1.0,c=0.0]
factorizing the polynomialsEdit
" factorizing the polynomials that determine Misiurewicz points. I believe that you should start with
( f^(p+k1) (c) + f^(k1) (c) ) / c
This should already have exact preperiod k , but the period is any divisor of p . So it should be factorized further for the periods.
Example: For preperiod k = 1 and period p = 2 we have
c^3 + 2c^2 + c + 2 .
This is factorized as
(c + 2)*(c^2 + 1)
for periods 1 and 2 . I guess that these factors appear exactly once and that there are no other factors, but I do not know."Wolf Jung
Newton methodEdit
 Newton's method for Misiurewicz points by Claude HeilandAllen
 Preperiodic Mandelbrot set Newton basins by Claude HeilandAllen
external angles of rays that land on the Misiurewicz pointEdit
Method:
 tracing preperiodic ray (to the limit of double precision )^{[29]}
 compute external angles of principal Misiurewicz point of wake p/q using Devaney's algorithm
See also
 Navigating by spokes in the Mandelbrot set by Claude HeilandAllen
 External angles of Misiurewicz points by Claude HeilandAllen
Relation between preperiod/period of external angles and Misurewicz point:^{[30]} the preperiod and period can differ between the external angle and the landing Misiurewicz point
The dynamics of the external angle under angle doubling is not the same as the dynamics of the landing point under iteration !!!
Examples:
 the external angle 0.1(0) has preperiod 1 and period 1, and it lands on the point c = −2 = ( preperiod 2 and period 1)
 external angle .001001(010010100) ray is landing on the point c = 0.026593792304386393+0.8095285579867694i. The external angle have preperiod 6 and period 9, but landing point has preperiod 7 and period 3
QuestionsEdit
ReferencesEdit
 ↑ mathoverflow question: istheresomeknownwaytocreatethemandelbrotsettheboundarywithanite
 ↑ Local connectivity of the Mandelbrot set at certain infinitely renormalizable points by Yunping Jiang
 ↑ external angles of misiurewicz points by Claude HeilandAllen
 ↑ Finite Subdivision Rules from Matings of Quadratic Functions: Existence and Constructions by Mary E. Wilkerson
 ↑ Subdivision rule constructions on critically preperiodic quadratic matings by Mary Wilkerson
 ↑ G. Pastor, M. Romera, G. Alvarez, J. Nunez, D. Arroyo, F. Montoya, "Operating with External Arguments of Douady and Hubbard", Discrete Dynamics in Nature and Society, vol. 2007, Article ID 045920, 17 pages, 2007. https://doi.org/10.1155/2007/45920
 ↑ W Jung : Homeomorphisms on Edges of the Mandelbrot Set Ph.D. thesis of 2002
 ↑ Fractal Geometry from Yale University by Michael Frame, Benoit Mandelbrot (19242010), and Nial NegerFebruary 2, 2013
 ↑ Terminal Point by Robert P. Munafo, 2008 Mar 9.
 ↑ mathoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
 ↑ math.stackexchange question : valuetouseascenterofmandelbrotsetzoom
 ↑ Symbolic sequences of onedimensional quadratic map points by G Pastor, Miguel Romera, Fausto Montoya Vitini
 ↑ Branch Point by Robert P. Munafo, 1997 Nov 19.
 ↑ Book : Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set, page 461, by HeinzOtto Peitgen, Hartmut Jürgens, Dietmar Saupe
 ↑ mdescribe by Claude HeilandAllen
 ↑ fractalforums.org : findingmisiurewiczpoints
 ↑ misiurewicz domains by Claude HeilandAllen
 ↑ Families of Homeomorphic Subsets of the Mandelbrot Set by Wolf Jung page 7
 ↑ G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya, "On periodic and chaotic regions in the Mandelbrot set", Chaos, Solitons & Fractals, 32 (2007) 1525
 ↑ A scaling constant equal to unity in 1D quadratic maps by M. ROMERA, G. PASTOR and F. MONTOYA
 ↑ G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 5271
 ↑ example
 ↑ Legendary Colour Palette by mathr
 ↑ VISUALIZATION OF THE BIFURCATION LOCUS OF CUBICPOLYNOMIAL FAMILY by HIROYUKI INOU
 ↑ fractalforums.org : primaryseparatorsmisiurewiczpointsofthemandelbrotsetsrealslice
 ↑ external ray tracing by Claude HeilandAllen
 ↑ external angles of misiurewicz points by Claude HeilandAllen
 ↑ MIsiurewicz point in wikipedia
 ↑ external angles of misiurewicz points by Claude HeilandAllen
 ↑ external angles of misiurewicz points by Claude HeilandAllen