Fractals/Iterations in the complex plane/misiurewicz

Misiurewicz point is the parameter c ( point oc parameter plane) where the critical orbit is pre-periodic.
 * description at wikipedia

=Name=
 * post-critically pre-periodic parameters

=Properities=
 * "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
 * The Mandelbrot is asymptotically self-similar about pre-periodic Misiurewicz points.
 * the set of all Misiurewicz points is dense on the boundary of the Mandelbrot set

multiplier

 * Each Misiurewicz point has a multiplier which is related to the shape of the spiral.

domain

 * 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
Rational 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: the pre-period and period can differ between the external angle and the landing Misiurewicz point

Examples:
 * the external angle 0.1(0) has pre-period 1 and period 1, and it lands on the point c = −2 = $$M_{2,1}$$ ( pre-period 2 and period 1)
 * external angle .001001(010010100) ray is landing on the point c = 0.026593792304386393+0.8095285579867694i. The external angle have pre-period 6 and period 9, but landing point has pre-period 7 and period 3

=notation= Misiurewicz point (parameter, polynomial, map) can be marked by:
 * preperiod and period
 * the parameter coordinate c ∈ M
 * the angle $$\theta$$ of the external ray

angle
"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

The preperiodic angle $$\theta$$ (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 $$c = \gamma_M(q/p) = M_{t,p}$$ in Mandelbrot set M on the parameter plane

so

$$ z^2 + c = z^2 + \gamma_M(q/p) $$

Examples of parameter rays:
 * Ray for angle $$\frac{1}{2} = 0.0(1)$$ lands on the point $$c = \gamma_M(1/2) = -2 = M_{1,1}$$ from the parameter plane. It is tip of the main antenna ( end of 1/2 limb).
 * Ray for angle $$\frac{1}{4} = 0.00(1)$$ lands on the point $$c = \gamma_M(1/4) = -0.228155493653962+1.115142508039937i = M_{2,1}$$ from the parameter plane. It is the first tip of wake 1/3.
 * Ray for angle $$\frac{1}{6} = 0.0(01)$$ lands on the point $$c = \gamma_M(1/6) = i = M_{1,2}$$ from the parameter plane. It is the last tip of wake 1/3
 * Ray for angle $$\frac{17}{240} = 0001(0010)$$ lands on the point $$c = \gamma_M(17/240) = 0.366362983422764 +0.591533773261445i = M_{4,4}$$ from the parameter plane. It is the principle Misiurewicz point ( branch point or hub) of wake 1/4.
 * Ray for angle $$\frac{9}{56} = 0.001(010)$$ lands on the point $$c = \gamma_M(9/56) = -0.101096363845622  +0.956286510809142i = M_{3,3}$$ from the parameter plane. It is the principle Misiurewicz point ( branch point or hub) of wake 1/3.
 * Ray for angle $$\frac{129}{16256} = 0.0000001(0000010)$$ lands on the point $$c = \gamma_M(129/16256) = 0.397391822296541 +0.133511204871878i = M_{7,7}$$ 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

preperiod and period
$$M_{t,p}$$

where
 * t is preperiod
 * p is period

Preperiod is used in 2 meanings :
 * T =preperiod of critical point $$z_{cr} = 0$$
 * t = preperiod of critical value $$z_{cv} = c$$

Note that :

$$ t = T - 1$$

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
 * 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 $$M_{2,1}$$ is given as $$M_{3,1}$$ "
 * Demidov: libretexts : The Mandelbrot and Julia sets Anatomy (Demidov) or ibiblio
 * Claude Heiland-Allen

point
=types=

period
Misiurewicz points c
 * with period 1 are of the type:
 * alpha, i.e. $$f^k_c(c) = \alpha_{c}$$
 * beta, i.e $$f^k_c(c) = \beta_{c}$$
 * with period > 1

where alfa and beta are fixed points of complex quadratic polynomial

Topological
Visual types:
 * branch tips = terminal points of the branches or tips of the midgets
 * centers of spirals
 * centers of slow spirals with more then 1 arm
 * centers of spirals = fast spiral
 * band-merging points of chaotic bands (the separator of the chaotic bands $$B_{i-1}$$ and $$B_i$$ ) = 2 arm spiral = branch point = points where branches meet

spirals
The Misiurewicz points which are centers of the spirals can be classified according to speed of turning:
 * fast
 * 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.

number of external rays

 * 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
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.

Tips:
 * "It seems the tips of the nth-longest arms of primary spirals have period n, counting from 1, and the tips of some filaments have period 1" Claude Heiland-Allen
 * not all terminal points have preperiod 1 or period 1

principal
The principal Misiurewicz point $$ c= b$$ of the limb $$M_{k/m}$$:
 * $$f^m(b) =\alpha_b$$
 * hase m external angles, that are preimages (under doubling) of the external angles of $$\alpha_b$$

characteristic
Characteristic Misiurewicz point of the chaotic band of the Mandelbrot set is :
 * 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

primary separator

 * band-merging point = point merging 2 chaotic bands $$ B_{i-1}$$ and $$ B_i$$

$$ m_i = M_{2^i, 2^{i-1}}$$ Examples:
 * $$ c = m_0 = M_{ 1, 1} \ \ =    -2$$ = tip of main antenna, external angle = 1/2
 * $$ c = m_1 = M_{ 2, 1} \ \ =    -1.543 689 012 692 076 ...$$, external angles 5 and 7/12
 * $$ c = m_2 = M_{ 4, 2} \ \ =    -1.430 357 632 451 307 ...$$, external angles 33 and 47/80
 * $$ c = m_3 = M_{ 8, 4} \ \ =    -1.407 405 118 164 702 ...$$,  external angles 1795 and 2557/4352
 * $$ c = m_4 = M_{16, 8} \ =      -1.402 492 176 358 564...$$
 * $$ c = m_5 = M_{32, 16} =       -1.401 441 494 253 588...$$
 * $$ c = m_6 = M_{64, 32}  =      -1.401 216 504 309 415..$$
 * $$ c = m_7 = M_{128, 64} =      -1.401 168 320 839 301..$$
 * $$ c = m_8 = M_{256, 128} =     -1.401 158 001 505 211..$$
 * $$ c = m_9 = M_{512, 256} =     -1.401 155 791 424 613  ..$$
 * $$ c = m_{10} = M_{1024, 512} = -1.401 155 318 093 230  ..$$
 * $$ c = m_{11} = M_{2048, 1024} = -1.401 155 216 720 152 ..$$
 * $$ c = m_{\infty} = MF =      -1.401 155 189 0 ... ...$$ = Feigenbaum point = MF = the Myrberg-Feigenbaum
 * $$ c = m_{\infty} = MF =      -1.401 155 189 0 ... ...$$ = Feigenbaum point = MF = the Myrberg-Feigenbaum

In the text form:

secondary separator

 * tree separators ( tree is a subset of band )

$$ m_{i,j} = M_{(j+1)2^i - 1, 2^i}$$

non-characteristic

 * have not the same period as the band

=Examples=

Misiurewicz Points, part of the Mandelbrot set:
 * Centre 0.4244 + 0.200759i; Max. Iterations 100; View radius 0.00479616

wake 1/2


$$\begin{cases} 0.(s_-) = 0.(01) = \frac{1}{3} = \frac{4}{12} = 0.(3) = wake    \\ 0.s_-(s_+) = 0.01(10) = \frac{5}{12} = 0.41(6) = PrincipalMis = M_{2,2}\\ 0.0(1) = \frac{1}{2} = \frac{6}{12} = 0.5 = tip = M_{1,1} = c = -2\\ 0.s_+(s_-) = 0.10(01) = \frac{7}{12} = 0.58(3) = PrincipalMis = M_{2,2}\\ 0.(s_+) = 0.(10) = \frac{2}{3} = \frac{8}{12} = 0.(6)= wake     \\ \end{cases} $$

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 = $$M_{1,1}$$. It is landing point of externa ray for angle $$0.01 = \frac{1}{2} = 0.5 $$
 * c = -1.543689012692076 = Principal Misiurewicz point of wake 1/2 = main node ( branch point) of the wake = $$ M_{2,2}$$ = landing point of external rays 5/12 i 7/12

relation with the map
For
 * 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 1/2-limbs, and each small Mandelbrot set has only one.

The multiplier map for Multibrot sets: here it is a 2-to-1 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

 * Kalles Fraktaler - Dive into Misiurewicz
 * Embedded Julia set similar to Misiurewicz Julia set by Wolf Jung

demos

 * Mandel demo 6 page 1

Images

 * commons : category:Misiurewicz point

How to colour ?
"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 semi-transparent 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 work-in-progress (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 screen-space 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 screen-space distance estimate channels (which zoomasm scales properly when reprojecting the exponential map)." Claude Heiland-Allen

=How to compute ...?=

constructing a polynomial whose roots are exactly the Misiurewicz points of multibrots ?

 * fractalforums.org: misiurewicz-points-in-the-multibrot
 * 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 well-defined 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. */

/* n-fold 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,m-1), 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(m-1,n) = 0 then (			ret2: general_dynatomic_fz0mn(z^d+c,z,0,n),			ret2: subst(0,z,ret2) ^ (d-1),			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

 * math.stackexchange question: counting-misiurewicz-points
 * enumeration of misiurewicz points by 	Claude Heiland-Allen
 * "... 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 non-rigorous method to and Misiurewicz parameter in a reasonable time like Biham-Wenzel's method." HIROYUKI INOU
 * The On-Line Encyclopedia of Integer Sequences (OEIS)
 * $$M_{0,p}$$ is known to be A000740
 * $$M_{2,p}$$ appears to be A038199
 * $$M_{q,1}$$ appears to be A000225
 * $$M_{q,2}$$ appears to be A166920
 * Corollary 3.3. from Misiurewicz Points for Polynomial Maps and Transversality by Benjamin Hutz, Adam Towsley

The number of $$(m,n)$$ Misiurewicz points for $$f_{d,c}$$ is $$M_{m,n}$$:

$$ M_{m,n} = \begin{cases} \sum_{k|n} \mu (\frac{n}{k}) d^{k-1} & m = 0 \\ (d^m - d^{m-1} - d + 1) \sum_{k \mid n} \mu(\frac{n}{k}) d^{k-1} & m \ne 0 \text{ and } n \mid (m - 1) \\ (d^m - d^{m-1})\sum_{k \mid n} \mu(\frac{n}{k}) d^{k-1} & \text{otherwise} \end{cases} $$

Where:
 * m is
 * n is
 * $$f_{d,c}(z) = z^d + c$$
 * d is a degree of $$f$$ function
 * $$\mu(n)$$ is the Moebius function for the natural n
 * $$\sum_{k|n}\;\mu(r)$$ is the sum of $$\mu(r)$$ over all positive integers $$r$$ dividing $$n$$

Implementations
 * Maxima CAS has function moebius (n)

Misiurewicz points of complex quadratic mapping

 * numerical methods
 * solving equations
 * finding landing point of external rays with pre-periodic 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 Heiland-Allen

tracing rays

 * "tracing rays of preperiod + period ~= 500 to dwell ~1000, with all 4 methods and varying sharpness"
 * "I made the database by tracing every pre-periodic ray with pre-period 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 pre-period..."

Misiurewicz domains

 * misiurewicz_domains by Claude Heiland-Allen
 * Misiurewicz domain coordinates and size estimates by Claude Heiland-Allen

roots of polynomial
Misiurewicz points are special boundary points.

Define polynomial in Maxima CAS : P(n):=if n=0 then 0 else P(n-1)^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
" factorizing the polynomials that determine Misiurewicz points. I believe that you should start with

( f^(p+k-1) (c) + f^(k-1) (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

 * Newton's method for Misiurewicz points by Claude Heiland-Allen
 * Preperiodic Mandelbrot set Newton basins by Claude Heiland-Allen

external angles of rays that land on the Misiurewicz point
Method:
 * tracing pre-periodic ray (to the limit of double precision )
 * 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 Heiland-Allen
 * External angles of Misiurewicz points by Claude Heiland-Allen

Relation between preperiod/period of external angles and Misurewicz point: the pre-period 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 pre-period 1 and period 1, and it lands on the point c = −2 = $$M_{2,1}$$ ( pre-period 2 and period 1)
 * external angle .001001(010010100) ray is landing on the point c = 0.026593792304386393+0.8095285579867694i. The external angle have pre-period 6 and period 9, but landing point has pre-period 7 and period 3

enumerate binary strings
To find external angle that lands on the Misiurewicz point $$ M_{p,q}$$

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 ... (non-repeating, irrational, related to the Thue-Morse 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 <1e-9):
 * -2
 * -1.5436890126920764
 * -1.4303576324513074
 * -1.4074051181647029
 * -1.4024921763585667
 * -1.4014414942535918
 * -1.4012165015160745

=Questions=
 * Questions from math.SE related with Misiurewicz point
 * math.stackexchange question: is there a Golden spiral in the Mandelbrot set?
 * Questions from MO related with Misiurewicz point

=References=
 * 7. Periodic and preperiodic points in the Mandelbrot set by : Professor Douglas C. Ravenel
 * M & J-sets similarity for preperiodic points. Lei's theorem by Douglas C. Ravenel
 * Misiurewicz points and the M-set self-similarity 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