Fractals/Iterations in the complex plane/misiurewicz
Misiurewicz point is the parameter c ( point oc parameter plane) where the critical orbit is preperiodic.
Name
edit postcritically preperiodic parameters
Properities
edit "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]}
multiplier
edit Each Misiurewicz point has a multiplier which is related to the shape of the spiral.
domain
edit Each Misiurewicz point is surrounded by a Misiurewicz domain within which the iterations are influenced by the Misiurewicz point.
 Local Misiurewicz domain coordinates can be calculated, they have magnitude 1 at the boundary of the domain.
 The size of the Misiurewicz domain can be estimated.
external angles
editRational angles measured in turns correspond to external rays landing on (pre)periodic points of the Mandelbrot set. They are conveniently expressed in binary.
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
notation
editMisiurewicz point (parameter, polynomial, map) can be marked by:^{[4]}
 preperiod and period
 the parameter coordinate c ∈ M
 the angle of the external ray
angle
edit"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 of parameter rays:
 Ray for angle lands on the point from the parameter plane. It is tip of the main antenna ( end of 1/2 limb).
 Ray for angle lands on the point from the parameter plane. It is the first tip of wake 1/3.
 Ray for angle lands on the point from the parameter plane. It is the last tip of wake 1/3
 Ray for angle lands on the point from the parameter plane. It is the principle Misiurewicz point ( branch point or hub) of wake 1/4.
 Ray for angle lands on the point from the parameter plane. It is the principle Misiurewicz point ( branch point or hub) of wake 1/3.
 Ray for angle lands on the point from the parameter plane. It is the principle Misiurewicz point ( branch point or hub) of wake 1/7.
On the the dynamic plane all above points c ( parameters of complex quadratic polynomial) gives the dendrite Julia sets.
partition of the dynamic plane by dynamic rays related with the kneading sequence

1/4

1/6

9/56

129/16256
preperiod and period
edit
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
Preperiod:
 preperiod of critical value
 "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^{[6]}
 Wolf Jung uses : "... 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."
 preperiod of critical point
 Pastor uses preperiod of critical point : "all the Misiurewicz points are given with one unit more in their preperiods, therefore this is given as "^{[7]}
 Demidov: libretexts : The Mandelbrot and Julia sets Anatomy (Demidov) or ibiblio
 Claude HeilandAllen
point
edittypes
editperiod
editMisiurewicz points c
 with period 1 are of the type:^{[8]}
 alpha, i.e.
 beta, i.e
 with period > 1
where alfa and beta are fixed points of complex quadratic polynomial
Topological
editVisual types:^{[9]}
 branch tips = terminal points of the branches^{[10]} or tips of the midgets^{[11]}
 centers of spirals ^{[12]}
 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 )^{[13]} = 2 arm spiral = branch point = points where branches meet^{[14]}
spirals
edit
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^{[15]}
 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.^{[16]}
number of external rays
edit 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 period
editIn 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.^{[17]}
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^{[18]}
 not all terminal points have preperiod 1 or period 1
named types
editprincipal
editThe principal Misiurewicz point of the limb :^{[19]}
 hase m external angles, that are preimages (under doubling) of the external angles of
characteristic
editCharacteristic Misiurewicz point of the chaotic band of the Mandelbrot set is :^{[20]}
 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
separators
editprimary separator
edit bandmerging point = point merging 2 chaotic bands and
Examples: ^{[21]}
 = tip of main antenna , external angle = 1/2^{[22]}
 , 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 separator
edit tree separators ( tree is a subset of band )
noncharacteristic
edit have not the same period as the band
Examples
editMisiurewicz Points, part of the Mandelbrot set:
 Centre 0.4244 + 0.200759i; Max. Iterations 100; View radius 0.00479616 ^{[23]}
wakes
editwake 1/2
edit
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
relation with the map
editFor
 complex quadratic function f(z) = z^2 + c, the branching points are Misurewicz points ( simple point not island)
 complex cubic function f(z) = z^3 + c here are islands ( mini mandelbrot set). Example zoom: c = 0.574891209746913 +0.716327145043763 i
Both for M2 and M3, a Misiurewicz point has a finite number of branches and a hyperbolic component has infinitely many antennas.
However, each small Multibrot set in M3 has two long 1/2limbs, and each small Mandelbrot set has only one.
The multiplier map for Multibrot sets: here it is a 2to1 map, for each internal angle there are two boundary points.
In particular, the conformal map from the component onto the disk is something like the squareroot of the multiplier.
videos
edit Kalles Fraktaler  Dive into Misiurewicz
 Embedded Julia set similar to Misiurewicz Julia set by Wolf Jung
demos
edit Mandel demo 6 page 1
Images
editHow 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^{[24]}
How to compute ...?
editconstructing a polynomial whose roots are exactly the Misiurewicz points of multibrots ?
edit fractalforums.org: misiurewiczpointsinthemultibrot
 Misiurewicz Points for Polynomial Maps and Transversality by Benjamin Hutz, Arxiv: Submitted on 16 Sep 2013 (v1), last revised 8 Aug 2014 (this version, v2)] Adam Towsley.
 two errors in the article by marcm200:
 page 2, the definition of phi_f,n(z) misses the subtraction z in every product's term.
 The generalized dynatomic formula is not welldefined for m=0, in that case it should be identical to the classical dynatomic polynomial.)
The maxima script by marcm200 with the principal function misiurewicz_multibrot_dmn(d,m,n) for the multibrot of degree d with preperiod m and period n:
kill(all); numer:false; display2d:false; /* general dynatomic polynomial to arrive at Misiurewicz points for the quadratic or higher degree Mandelbrot set */ /* based on: B Hitz, A Towsley. Misiurewicz points for polynomial maps and transversality, 2014. */ /* nfold composition */ composition_fn(f,n) := ( ret:"Error. ,composition_fn", if n = 0 then ret:z else if n > 0 then ( ret:f, for i from 2 thru n do ( ret:subst(f,z,ret) ) ) else print("Error. composition_fn"), ratsimp(ret) )$ /* the dynatomic polynomial */ dynatomic_fz0n(f,z0,n) := ( erg:"Error. dynatomic_fz0n", if n < 1 then ( erg:"Error. Period must be at least 1.", print("Error. Period must be at least 1.") ) else ( erg:1, for k from 1 thru n do ( if mod(n,k) = 0 then ( co:composition_fn(f,k), co:subst(z0,z,co), erg: erg * ( ( co  z0 ) ^ moebius(n/k) ) ) ) ), ratsimp(erg) )$ /* generalized dynatomic polynomial */ general_dynatomic_fz0mn(f,z0,m,n) := ( ret:"Error. general_dynatomic_fz0mn", if m = 0 and n > 0 then ( ret: dynatomic_fz0n(f, z , n) ) else if m > 0 and n > 0 then ( ret: ratsimp( dynatomic_fz0n(f, composition_fn(f,m) , n) / dynatomic_fz0n(f, composition_fn(f,m1) , n) ), ret:subst(z0,z,ret) ) else print("Error. general_dynatomic_fz0mn"), ratsimp(ret) )$ /* Misiurewicz points for unicritical multibrot z^d+c with exact preperiod m and period n*/ misiurewicz_multibrot_dmn(d,m,n) := ( numer:false, ret: "Error misiurewicz_multibrot_dmn", if m > 0 and n > 0 and d >= 2 then ( ret1: general_dynatomic_fz0mn(z^d+c,z,m,n), ret1: ratsimp(subst(0,z,ret)), if m # 0 and mod(m1,n) = 0 then ( ret2: general_dynatomic_fz0mn(z^d+c,z,0,n), ret2: subst(0,z,ret2) ^ (d1), ret: ret1 / ret2 ) else ret: ret1 ) else print("Error misiurewicz_multibrot_dmn"), ret:ratsimp(ret), print("Misiurewicz points as solution from"), print(ret,"= 0"), sol:solve(ret=0,c), numer:true, for i from 1 thru length(sol) do ( print("solution",realpart(expand(rhs(sol[i])))," + i*", imagpart(expand(rhs(sol[i]))) ) ) )$
Number of Misiurewicz points
edit 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 ^{[25]}
 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 mapping
edit 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^{[26]}
tracing rays
edit "tracing rays of preperiod + period ~= 500 to dwell ~1000, with all 4 methods and varying sharpness" ^{[27]}
 "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..."^{[28]}
Misiurewicz domains
edit misiurewicz_domains by Claude HeilandAllen
 Misiurewicz domain coordinates and size estimates by Claude HeilandAllen
Solving equations
editroots of polynomial
editMisiurewicz points ^{[29]} 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 polynomials
edit" 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 method
edit 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 point
editMethod:
 tracing preperiodic ray (to the limit of double precision )^{[30]}
 compute external angles of principal Misiurewicz point of wake p/q using Devaney's algorithm
 enumerate binary strings
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:^{[31]} 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
enumerate binary strings
editTo find external angle that lands on the Misiurewicz point
enumerate binary strings of length n = preperiod + period, of which there are 2^n, then discard ones which have a different (pre)period when canonicalized.
Example: preperiod 1, period 2
 there are 2^(1 + 2) = 2^3 = 8 candidates
 .000 = 0
 .001
 .010
 .011
 .100
 .101
 .110
 .111
Example: period = 2, preperiod = 5
 there are 2^(2 + 5) = 2^7 = 128 candidates
 half of these can be eliminated immediately because of the 4 length 2 strings for the period ...(00) and ...(11) simplify to ...(0) and ...(1) (i.e. they have period 1 instead of 2). 64 remain
 consider the last preperiodic digit: if it is the same as the last periodic digit, then the true preperiod is less (shift the periodic part to the left). so another half are eliminated, leaving the possibilities for the last 3 digits as ...0(01) and ...1(10). 32 remain
 as the Mandelbrot set is symmetrical, you only need to consider rays in the upper half plane, which start with .0..., which eliminates another half: 16 remain
you want rays beyond the main Feigenbaum point of the period doubling cascade, which has angle .01 10 1001 10010110 ... (nonrepeating, irrational, related to the ThueMorse sequence). so the digits must be strictly between these two:
 .0110100
 .0111111
enumerating them gives
 .01101(10)
 .01110(01)
 .01111(10)
so there are 3 candidates remaining. tracing external rays gives these coordinates:
 1.6975553932375476
 1.8186201342243300
 1.9935450866059059
None of these are in your sequence, and now I remember that the (pre)period of a Misiurewicz point under iteration of z^2+c need not correspond to the (pre)period of the angle under angle doubling modulo 1 turn...
the answer is probably "tuning", i.e. find the angles of the first separation point, then tune them by the external angle pairs of the period doubling cascade
the tip of the antenna has external angles .0(1) = .1(0) : tracing gives 2 the period 2 bulb has external angles .(01) and .(10) replace each 0 with 01 and 1 with 10 gives the external angles of the tip of the period 2 bulb spoke: .01(10) and .10(01) : tracing gives 1.5436890126920764 repeating give the tip of the period 4 bulb spoke .0110(1001) and .1001(1001) : tracing gives 1.4303576324513074 repeating this gives this table of ray end points traced with double precision (imaginary part of all of them is <1e9):
 2
 1.5436890126920764
 1.4303576324513074
 1.4074051181647029
 1.4024921763585667
 1.4014414942535918
 1.4012165015160745
Questions
editReferences
edit ↑ 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
 ↑ Subdivision rule constructions on critically preperiodic quadratic matings by Mary Wilkerson
 ↑ [https://www.hindawi.com/journals/ddns/2007/045920/ 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
 7. Periodic and preperiodic points in the Mandelbrot set by : Professor Douglas C. Ravenel
 M & Jsets similarity for preperiodic points. Lei's theorem by Douglas C. Ravenel
 Misiurewicz points and the Mset selfsimilarity by Douglas C. Ravenel
 Misiurewicz Point by Robert P. Munafo, 2013 May 28
 On Quotients of the Shift associated with Dendrite Julia sets of Quadratic Polynomials by Christopher S. Penrose