          The problem is that we are exploring environments based upon irrational numbers through computer machinery which works with finite rationals ! ( Alessandro Rosa )

"西格爾圓盤是（最大）域，在其上，全純對映與旋轉相共軛，其角度除以一圈稱為旋轉數。" - A Cheritat^[1]

對於西格爾圓盤^[2]^[3] 引數 c

應該位於曼德爾布羅集的雙曲分量的邊界（內半徑 = 1）
內角應該是無理數，介於 0 和 1 之間。^[4]

因為無理數集是不可數的^[5]，所以具有西格爾圓盤的 Julia 集的數量是無限的。

定義

臨界軌道趨向於弱吸引不動點。當點 c 沿無理內角的內射線從中心移動到邊界時，臨界軌道從一個點穿過螺旋形變為圍繞不動點 alpha 的圓形。
黃金分割二次 Julia 集的臨界軌道、內圓和外圓

中心

西格爾圓盤的中心是無理的無差異週期點。

Mane 定理

"... 除其中心外，西格爾圓盤不能包含任何週期點、臨界點，也不能包含任何臨界點或週期點的迭代前像。另一方面，它可以包含臨界點的迭代像。" ^[6]

Julia 集的中心分量

在 c 位於主心形邊界的情況下，Julia 集的中心分量是一個包含西格爾圓盤（及其中心）的分量。臨界軌道是西格爾圓盤和中心分量的邊界。所有其他分量都是此分量的預像（參見使用逆迭代的動畫影像）。

域

By a classical (nontrivial) result of C.L. Siegel, for certain irrational values of  $\alpha$  the quadratic polynomial  $e^{2\pi \alpha i}z+z^{2}:\mathbb {C} \to \mathbb {C}$ 
is locally linearizable at 0. That is, there exists a local conformal change of coordinate near zero on which this quadratic polynomial becomes the linear map   $z\to e^{2\pi \alpha i}z$ .
The maximal domain on which one has such a linearization is called the Sigel disk of that quadratic polynomial.
There are fascinating, and mysterious, relations between the arithmetic properties of  $\alpha$  and the geometry of the Siegel disks. Davoud Cheraghi^[7]

半徑

共形半徑

共形半徑^[8]^[9]

在數學中，共形半徑是一種衡量從其內部一點 z 觀察到的單連通平面域 D 大小的方式。與使用歐幾里得距離（例如，以 z 為中心的最大內接圓的半徑）的概念相反，這種概念非常適合在複分析中使用，特別是在共形對映和共形幾何中。

定義給定一個單連通域 D ⊂ C，和一個點 z ∈ D，根據黎曼對映定理，存在一個唯一的共形對映 f : D → D 到單位圓盤（通常稱為統一化對映），其中 f(z) = 0 ∈ D 和 f′(z) ∈ R₊。然後將 D 從 z 處的共形半徑定義為

\mathrm {rad} (z,D):={\frac {1}{f'(z)}}\,.

最簡單的例子是半徑為r的圓盤以其中心為視角的共形半徑也是r，由一致化對映x ↦ x/r所示。更多示例見下文。

這個概念之所以有用，其中一個原因是它在共形對映下表現良好：如果φ : D → D′是一個共形雙射，且z 在 D 中，則 $\mathrm {rad} (\varphi (z),D')=|\varphi '(z)|\,\mathrm {rad} (z,D)$ .

共形半徑也可以表示為 $\exp(\xi _{x}(x))$ ，其中 $\xi _{x}(y)$ 是 $\log(|x-y|)$ 從 $\partial D$ 到 $D$ 的調和擴充套件。

內半徑

Siegel 圓盤的內半徑 =

內圓的半徑，其中以不動點為中心的內圓是 Siegel 圓盤內最大的圓。^[10]
Siegel 圓盤中心到臨界軌道的最小距離

計算內半徑的程式碼

Maxima CAS 程式碼

這裡軌道是一個複數點 z 的列表。

f(z,c):=z*z+c $

GiveCriticalOrbit(c,iMax):= 
block(
 ER:2.0, /* Escape Radius */
 z:0+0*%i, /* first point = critical point */
 orbit:[z], 
 if (abs(z)>ER) then return(orbit),
 i:0,
 loop, /*  compute forward orbit */
 z:rectform(f(z,c)),
 orbit:endcons(z,orbit),
 i:i+1,
 if ((abs(z)<ER) and (i<iMax)) then go(loop),
 return(orbit) 
)$

/* find fixed point alfa */
GiveFixed(c):= float(rectform((1-sqrt(1-4*c))/2))$

/* distance between point z and fixed point zf */
GiveDistanceFromCenterTo(z):= abs(z-zf)$

/* inner radius of Siegel Disc ; criticla orbit is a boundary of SD */
GiveInnerRadiusOf(orbit):=lmin(map(GiveDistanceFromCenterTo,orbit))$

/* outer radius of Siegel Disc ; criticla orbit is a boundary of SD */
GiveOuterRadiusOf(orbit):=lmax(map(GiveDistanceFromCenterTo,orbit))$

/*------------ const ---------------------------------*/
c:0.113891513213121  +0.595978335936124*%i; /* fc(z) = z*z + c */
NrPoints:400000;

/* ----------- main ---------------------------------------------------*/
zf:GiveFixed(c); /* fixed point = center of Siegel disc */
orbit:GiveCriticalOrbit(c,NrPoints)$
innerRadius: GiveInnerRadiusOf(orbit) ;
outerRadius: GiveOuterRadiusOf(orbit) ;

C 程式碼

double GiveInternalSiegelDiscRadius(complex double c, complex double a)
{ /* compute critical orbit and finds smallest distance from fixed point */
  int i; /* iteration */
  double complex z =0.0; /* critical point */
  
  /* center of Siegel disc  = a */
  
  double d; // distance
  double dMin = 2.0;

  
  for (i=0;i<=40000 ;i++) /* to small number of iMax gives bad result */
    {
      z = z*z + c; 
      /* */
      
     d = cabs(z - a);
     if (d < dMin) dMin = d; /* smallest distance */
    }
    
  return dMin;
}

外半徑

Siegel 圓盤的外半徑 = 外圓的半徑。

以不動點為中心的，包含 Siegel 圓盤的最小圓是外圓。

C 程式碼

double GiveExternalSiegelDiscRadius(complex double c, complex double a)
{ /* compute critical orbit and finds smallest distance from fixed point */
  int i; /* iteration */
  double complex z =0.0; /* critical point */
  
  /* center of Siegel disc  = a */
  
  double d; // distance
  double dMax = 0.0;

  
  for (i=0;i<=40000 ;i++) /* to small number of iMax gives bad result */
    {
      z = z*z + c; 
      /* */
      
     d = cabs(z - a);
     if (d > dMax) dMax = d; /* biggest distance */
    }
    
  return dMax;
}

摺疊

在 scholaredia 中摺疊^[11]

Siegel 圓盤內爆

" ... 對乘子的微小變化可能會導致 Siegel 圓盤內爆 - 它的內半徑坍縮為零 " ^[12]

例子

從旋轉數為 [0;1,1,1,1,1, ...]= 0.618033988957902 的黃金分割 Siegel 圓盤到旋轉數（內角）為 5/8 = [0;1,1,1,1,1] 的拋物線 Julia 集
Siegel 圓盤在拋物線內爆下的半連續性 : P(n) = [0, 2, 2, n + r]，其中 r = (√5−1)/ 2 ，前三張圖片顯示了 P(n) 的 Siegel 圓盤，其中 n 分別為 10、500、10000，最後一張圖片是 p/q = 2/5 的虛擬 Siegel 圓盤，它們趨向於此。這裡 P(n) > p/q。^[13]

型別

對於 c

在主心形的邊界上是圍繞不動點 alfa 的 Siegel 圓盤
在週期為 n 的分量的邊界上是圍繞 n 週期點的週期性 Siegel 圓盤

這些 Siegel 圓盤是

有界的（因為 Julia 集是有界的）
一般情況下，不光滑（不可微）

圍繞不動點

Julia 集由無數條曲線構成，這些曲線將開區域包圍起來
$f_{c}\,$ 將每個區域對映到“更大的”區域，直到到達包含不動點的區域
在包含 Siegel 圓盤的分量內部， $f_{c}\,$ 將不變迴圈上的點繞不動點旋轉^[14]

換句話說

" $K_{c}$ 有無限多個連通分支。其中一個包含了不動點 α。它被 $f_{c}$ 對映到自身。這就是西格爾圓盤。

"西格爾圓盤是一個連通分支，它有無窮多個原像。如果你在 z = 0 處以較大的 nmax 放大，你會看到有兩個分支在 z=0 處相交。"(沃爾夫·榮格)

視覺化

動力學平面的視覺化

臨界點的正向迭代的臨界軌道
透過繪製臨界軌道來得到朱利亞集
- 中心分支的邊界
- 透過臨界軌道的反向迭代來得到整個朱利亞集
內部
- 由克里斯·金透過平均速度獲得的整個內部
- 西格爾圓盤軌道，透過對西格爾圓盤內部或邊界上的點的正向迭代獲得^[15]

克里斯·金的平均速度

非吸引盆地和花瓣的離散速度：對於不逃逸的點，計算軌道上的平均離散速度^[16]

 $\left\vert z_{n+1}-z_{n}\right\vert$

在 c 中
在 octave 中

最佳化

另請參見通用方法

技巧 1

如果點 (z0 或其正向影像 zn) 在內圓內部，則它就是內部點。

示例

方法

連分數
- t = [3,2,1000,1,.] = 0.28573467254058817026888613062003 。它給出 c = -0.096294753554390530825955047963 + 0.648802699422348309293468536773i^[17]
已知的無理數近似值
- 1/pi
- 尤拉數
- 黃金分割
- 數字的根，比如
  - 平方根：sqrt(2), sqrt(3), sqrt(5), sqrt(7), sqrt(8), sqrt(15), sqrt(21), sqrt(35), sqrt(49), SQRT(65), sqrt(99), SQRT(100), SQRT(101), SQRT(121), sqrt(200), sqrt(278),
  - 立方根，cbrt(3),
- 對數，比如 $log_{2}3$
上述數字的後代
- 比如 1/pi
- 無理數和有理數的和 : 如果 t 是一個無理數，r 是一個有理數，那麼 t+r 是無理數
快速收斂的級數 ^[18]

序列

連分數序列:^[19]

[1, 1, 1, . . .]
[1, 1, 1, 20, 1, . . .]
[1, 1, 1, 20, 1, 1, 1, 30, 1, . . .]

從西格爾圓盤到 Lea-Fatou 花的序列

普通
數字化
虛擬
Leau-Fatou 花

cf(t)	t	c(t)	中心 z	週期	內部 R	外部 R
[0; 1, 1, 1,...]	0.618033988957902	-0.390540870218399-0.586787907346969i	-0.368684439039160-0.337745147130762i	1	0.25	0.4999
[0;3,2,1000,1...]	.2857346725405882	0.113891513213121+0.595978335936124i	-0.111322907374331+0.487449700270424i	1	.1414016645283217	.5285729063154562

值

t = 0.3119660900888915 = [0, 3, 4, 1, 6, 1, 1, 65, 7, 1, 2, 63] ; c = -0.01183223669 + 0.63816572702*%i ^[20]
t = 0.3572354109849235 = [0, 2, 1, 3, 1, 54, 2, 1, 17, 1, 1, 1, 12, 1, 1, 1, 2, 2] ; c = -0.256625459 + 0.6345309*%i
t = 0.3573615246360573 = [0, 2, 1, 3, 1, 22, 1, 1, 4, 1, 5, 1, 4, 3, 2, 21] ; c = -0.257341 + 0.634456*%i

主心形曲線的邊界

傑伊·希爾的示例^[21]

黃金分割的動力學平面，包括臨界軌道和西格爾圓盤內部的一些軌道
黃金分割的二次西格爾圓盤，內部用軌道上的平均離散速度著色
高解析度灰色版本
用 IIM/J 繪製的黃金分割的朱利亞集
動畫版本 - 最高到 100 級
具有旋轉數為 [3,2,1000,1...] 的西格爾圓盤的二次多項式的朱利亞集

引數化

主心形曲線的邊界由 2 個聯立方程描述

描述 f 下的不動點（週期 1）
描述不動點的乘子^[22]（穩定性）（它應該是無差異的）

${\begin{cases}z_{1}^{2}+c=z_{1}\\abs(\lambda (z_{1}))=1\end{cases}}$

其中

$z_{1}=\left\{z:f_{c}(z)=z\right\}\,$

$\lambda (z_{1})=2*z_{1}\,$

第一個方程

$z_{1}^{2}+c=z_{1}$

$c=-z_{1}^{2}+z_{1}$

第二個方程

$\lambda (z_{1})=1\,$

$Abs(2z_{1})=1\,$

$2z_{1}=e^{2*\Pi *t*i}\,$

$z_{1}=e^{\Pi *t*i}=cos(t)+i*sin(t)\,$

因此，我們得到一個函式，描述了引數 c 和內部角度 t 之間的關係

$c=c(t)={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right)={\frac {e^{\Pi *t*i}}{2}}-{\frac {e^{2*\Pi *t*i}}{4}}$

此函式用於計算

主心形的邊界點 c
當 t 為無理數時，填充的 Julia 集具有西格爾圓盤

我們可以計算邊界點 c

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

週期為 1 的雙曲分量（主心形）的邊界點，方法是使用給定 t

 t = atan2(); //   is the argument in radians.
 cx = 0.5*cos(t) - 0.25*cos(2*t); 
 cy = 0.5*sin(t) - 0.25*sin(2*t);

使用任意步長增加 t 後，可以計算新的引數

 t += 0.01; //  It is increased by  0.01  so  1/628 turns.
// From the new  t the new parameter c is computed ..

在劉維爾數附近

t=\sum _{k=1}^{\infty }2^{-k!}=2^{-1}+2^{-2}+2^{-6}+2^{-24}+\ldots ={\frac {12845057}{16777216}}\approx 0.7656250596046448

t = 0.7656250596046448 給出

c = 0.294205040086005 -0.448819580822501*i 在主心形上
c = -0.975495621741693 -0.248796172491005*i 在週期為 2 的分量上
c = -0.219419079596579 +0.741123657495634*i 在沿內部射線 1/3 的週期為 3 的分量上
c = -1.749557112879525 -0.008859942836393*i 在主觸角上的週期為 3 的小心形上

t=\sum _{k=1}^{\infty }4^{-k!}=4^{-1}+4^{-2}+4^{-6}+4^{-24}+\ldots \approx 0.3127441406250035

t=\sum _{k=1}^{\infty }8^{-k!}=8^{-1}+8^{-2}+8^{-6}+8^{-24}+\ldots \approx 0.1406288146972656

t=\sum _{k=1}^{\infty }16^{-k!}=16^{-1}+16^{-2}+16^{-6}+16^{-24}+\ldots \approx 0.06640630960464478

其中 ! 表示階乘。

圖片

DarkHeart by Chris King

Maxima cas 程式碼

f(n) := n^-1 + n^-2 +n^-6 +n^-24;
%i2) f(2);
                                   12845057
(%o2)                              --------
                                   16777216
(%i3) float(f(2));
(%o3)                         0.7656250596046448
(%i4) float(f(3));
(%o4)                          0.445816186560468
(%i5) f(3);
                                 125911658926
(%o5)                            ------------
                                 282429536481

(%i6) f(5);
                               14308929443359376
(%o6)                          -----------------
                               59604644775390625
(%i7) f(6);
                              921453486852538369
(%o7)                         -------------------
                              4738381338321616896
(%i8) f(7);
                             31280196802261814842
(%o8)                        ---------------------
                             191581231380566414401
(%i9) f(8);
                            664100801052053340161
(%o9)                       ----------------------
                            4722366482869645213696
(%i10) 8^24;
(%o10)                      4722366482869645213696

弧度

1/(2*pi) = 0.1591549430918953
2/(2*pi) = 0.3183098861837907
3/(2*pi) = 0.477464829275686
4/(2*pi) = 0.6366197723675814
5/(2*pi) = 0.7957747154594768
6/(2*pi) = 0.954929658551372

圖片

Kerry Mitchell 的示例 Kerry Mitchell 的示例

黃金分割

旋轉數 t 等於黃金分割（恰好是黃金分割的共軛，^[23] 比較黃金分割的 Julia 集，它是斷開的 Julia 集 ^[24]

$t={\frac {{\sqrt {5}}-1}{2}}\approx 0.618033988749895$

連分數展開 :^[25]

$t=[0;1,1,1,\dots ]=[1,1,1,\dots ]={\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}$

在 Maxima CAS 中

(%i2) a:[0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%o2) [0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%i3) t:cfdisrep(a)
(%o3) 1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1))))))))))))))))))))))
(%i4) float(t)
(%o4) 0.618033988957902
(%i5) l:%e^(2*%pi*%i*t)
(%o5) %e^(-(17711*%i*%pi)/23184)
(%i6) c:(l*(1-l/2))/2
(%o6) ((1-%e^(-(17711*%i*%pi)/23184)/2)*%e^(-(17711*%i*%pi)/23184))/2
(%i7) float(rectform(c))
(%o7) -.5867879078859505*%i-.3905408691260131

所以

$c\approx -0.390540870218399-0.586787907346969*i$

與進行比較

透過組合方法描繪的二次 Julia 集^[26]
Curtis T McMullen 的黃金分割 Siegel 圓盤 ^[27]
Jim Muth 的 Siegel 圓盤分形 ^[28]
Davoud Cheraghi 的 Siegel 圓盤 ^[29]
來自 Roger Bagula 的 Mathematica 中更好的 Siegel 圓盤程式 ^[30]
Xander 的影像 ^[31]
Arnaud Chéritat 的圖片 ^[32]
Faber 證明的無理數

落在臨界點的射線

有兩條射線落在臨界點 z=0 上，它們的角為：^[33]

  $\omega /2$ 
  $(\omega +1)/2$

這裡^[34]

   $\omega =[0;2^{0},2^{1},2^{1},2^{2},2^{3},...]=0.709803$

其中 2 的冪形成斐波那契數列 :^[35]

  $0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots \;$

這些射線是落在臨界值的射線 $\omega$ 的原像。

數字化

演算法是這樣的

從有理數 p/q 開始（q 是數字的個數）
使用例如：wolfram alpha 找到 p/q 的連分數
讓我們選擇 2/7 = [0; 3, 2]
對於從 0 到 10 的 n，計算旋轉數 = t(n) = [0;3, 2 , 10^n, golden_ratio]
對於每個 t，計算乘子 m = e^(2*pi*i*t) 和 c = (m*(1-m/2))/2,
對於每個 c，繪製臨界軌道

1/2

c = -0.749998153581339 +0.001569040474910 i 的 Siegel 圓盤
內折 Siegel 圓盤
使用外部射線的動畫。最後一幀顯示拋物線 Julia 集

 [0;2,10^0,  GoldenRatio] = 0.3819660112501051517954131656343618822796908201942371378645513772947395371810975502927927958106088625^[36]
 [0;2,10^1,  GoldenRatio] = 0.4775140100981009996157078147705459192853678713412804123092036673992538239565035505055878663397199776
 [0;2,10^2,  GoldenRatio] = 0.4975276418049443654168822249489777631733396751888937557910272646236455872099717190261209452982670182
 [0;2,10^3,  GoldenRatio] = 0.4997502791963461829064406841975638383643273854865268095992275328965717970298619920531024931799175222
 [0;2,10^4,  GoldenRatio] = 0.4999750027947725068093934556288852812751613102402893275180237493139184231871314468863992103480081398
 [0;2,10^5,  GoldenRatio] = 0.4999975000279505372222411883578948194038778053684924551662056284491619191237767514609446551752982620
 [0;2,10^6,  GoldenRatio] ≈ 0.4999997500002795081846878230972820064874630120375394060599983649731088601118927011908445914889153575
 [0;2,10^7,  GoldenRatio] ≈ 0.4999999750000027950846593747720590697092705648336656666550099352279885262457471130077351190950368647
 [0;2,10^8,  GoldenRatio] ≈ 0.4999999975000000279508494062473746989708466400627903143635595029928608488311534417391176743667414601
 [0;2,10^9,  GoldenRatio] = 0.4999999997500000002795084968749737124005323296851266699307427070578490529472905394986264318888387868
 [0;2,10^10, GoldenRatio] ≈ 0.4999999999750000000027950849715622371205464056480586232583076030201151399460674665945145904063730995
 [0;2,10^11, GoldenRatio] ≈ 0.4999999999975000000000279508497184348712051181647153557573019083692657334910230489541137897282568504
 [0;2,10^12, GoldenRatio] ≈ 0.4999999999997500000000002795084971871612120511470579770310133815923588852631868226288826667637222837
 [0;2,10^13, GoldenRatio] ≈ 0.4999999999999750000000000027950849718744246205114671208526574427310807059077461659968395909016599809
 [0;2,10^14, GoldenRatio] ≈ 0.4999999999999975000000000000279508497187470587051146708626348091578511118332834054112155667123629005
 [0;2,10^15, GoldenRatio] ≈ 0.4999999999999997500000000000002795084971874733995511467085917589150515616367008796489196018543201296
 [0;2,10^16, GoldenRatio] ≈ 0.4999999999999999750000000000000027950849718747368080114670859141302328629213837245072986771954833911
 [0;2,10^17, GoldenRatio] ≈ 0.4999999999999999975000000000000000279508497187473708926146708591409564368639443385654331302200729603
 [0;2,10^18, GoldenRatio] ≈ 0.4999999999999999997500000000000000002795084971874737117386467085914095297794629164357828552071705414
 [0;2,10^19, GoldenRatio] ≈ 0.4999999999999999999750000000000000000027950849718747371201989670859140952943357115116628413693411086
 [0;2,10^20, GoldenRatio] ≈ 0.4999999999999999999975000000000000000000279508497187473712048021708591409529430112233513589149747868
 ...
 [0;2] = 1/2              = 0.5

使用四倍精度

t(0)  = 3.81966011250105151795413165634361882279690820194237137864551377e-01 ; c = -3.90540870218400050669762600713798485817583715938583500790716491e-01 ; 5.86787907346968751196714643055715840096745752123320842032245424e-01
t(1)  = 4.77514010098100999615707814770545919285367871341280412309203667e-01 ; c = -7.35103725789203166246364354183070697092321099215970115885457098e-01 ; 1.40112549815936743590686010452273467218741353649054870130630558e-01
t(2)  = 4.97527641804944365416882224948977763173339675188893755791027265e-01 ; c = -7.49819025417749776930986763368388660297294249313031793877713589e-01 ; 1.55327228158391795314525351457527867568101147692293037994954915e-02
t(3)  = 4.99750279196346182906440684197563838364327385486526809599227533e-01 ; c = -7.49998153581339423261635407668242134664913541791449158951684018e-01 ; 1.56904047490965613194389295941701415441031256575211288282884094e-03
t(4)  = 4.99975002794772506809393455628885281275161310240289327518023749e-01 ; c = -7.49999981498629125645545136832080395069079925634197165921129604e-01 ; 1.57062070991569266952536106210409130728490029437750411952649155e-04
t(5)  = 4.99997500027950537222241188357894819403877805368492455166205628e-01 ; c = -7.49999999814949055379035496840776829777984829652802229165703441e-01 ; 1.57077876479293075511736069683515881947782844720535147967721515e-05
t(6)  = 4.99999750000279508184687823097282006487463012037539406059998365e-01 ; c = -7.49999999998149453312747388186444210086067957518121660969151546e-01 ; 1.57079457059156244743576144811256589754652192800174232944684020e-06
t(7)  = 4.99999975000002795084659374772059069709270564833665666655009935e-01 ; c = -7.49999999999981494495885914313759501577872560098720076587742563e-01 ; 1.57079615117453182906803046090425827728536858759594155863970088e-07
t(8)  = 4.99999997500000027950849406247374698970846640062790314363559503e-01 ; c = -7.49999999999999814944921617531908891370473818724298060108792696e-01 ; 1.57079630923285982648802540286825950414334929521039394902627564e-08
t(9)  = 4.99999999750000000279508496874973712400532329685126669930742707e-01 ; c = -7.49999999999999998149449178933702694145524879491347408142864355e-01 ; 1.57079632503869293681972526129998193269783324874397448954896142e-09
t(10) = 4.99999999975000000002795084971562237120546405648058623258307603e-01 ; c = -7.49999999999999999981494491752095410030080568840077963364830374e-01 ; 1.57079632661927625095878938346134344643704271149725816320948648e-10
t(11) = 4.99999999997500000000027950849718434871205118164715355757301908e-01 ; c = -7.49999999999999999999814944917483712483337770357969922549543000e-01 ; 1.57079632677733458240375473417333652732358282974043728923647563e-11
t(12) = 4.99999999999750000000000279508497187161212051147057977031013382e-01 ; c = -7.49999999999999999999998149449174799883216409502184216102177862e-01 ; 1.57079632679314041554856185862662707966424239622019267641300599e-12
t(13) = 4.99999999999975000000000002795084971874424620511467120852657443e-01 ; c = -7.49999999999999999999999981494491747961590547126303840172625197e-01 ; 1.57079632679472099886304567696577418001577729533565273081497266e-13
t(14) = 4.99999999999997500000000000027950849718747058705114670862634809e-01 ; c = -7.49999999999999999999999999814944917479578663854294268739087328e-01 ; 1.57079632679487905719449408985862706763478042275954257084690447e-14
t(15) = 4.99999999999999750000000000000279508497187473399551146708591759e-01 ; c = -7.49999999999999999999999999998149449174795749396925973912562169e-01 ; 1.57079632679489486302763893145850173816965190682509858763917154e-15
t(16) = 4.99999999999999975000000000000002795084971874736808011467085914e-01 ; c = -7.49999999999999999999999999999981494491747957456727642770350276e-01 ; 1.57079632679489644361095341562159509904086589961983101670245908e-16
t(17) = 4.99999999999999997500000000000000027950849718747370892614670859e-01 ; c = -7.49999999999999999999999999999999814944917479574530034810734727e-01 ; 1.57079632679489660166928486403793549406616456447514036426396814e-17
t(18) = 4.99999999999999999750000000000000000279508497187473711738646709e-01 ; c = -7.49999999999999999999999999999999998149449174795745263106490379e-01 ; 1.57079632679489661747511800887956984415807620355720412842525869e-18
t(19) = 4.99999999999999999975000000000000000002795084971874737120198967e-01 ; c = -7.49999999999999999999999999999999999981494491747957452593823287e-01 ; 1.57079632679489661905570132336373328227316118501557102010628498e-19
t(20) = 4.99999999999999999997500000000000000000027950849718747371204802e-01 ; c = -7.49999999999999999999999999999999999999814944917479574525900991e-01 ; 1.57079632679489661921375965481214962611572862167871366529561451e-20
        5.00000000000000000000000000000000000000000000000000000000000000e-01   c = -7.50000000000000000000000000000000000000000000000000000000000000e-01 ; 0.00000000000000000000000000000000000000000000000000000000000000e+00

1/3

1/3 = [0; 3] = 0.3333333..... = 0.(3)

/* Maxima CAS batch file */ 
kill(all)$
remvalue(all)$

/* a =  [1, 1, ...] =  golden ratio */
g: float((1+sqrt(5))/2)$

GiveT(n):= float(cfdisrep([0,3,10^n,g]))$

GiveC(t):= block(
  [l, c],
  l:%e^(2*%pi*%i*t),
  c : (l*(1-l/2))/2,
  c:float(rectform(c)),
  return(c)
)$

compile(all)$

for n:0 step 1 thru 10 do (
  t:GiveT(n),
  c:GiveC(t),
  print("n=" ,n ," ; t= ", t , "; c = ", c) 
 );

結果

n= 0   t=  0.276393202250021    c = 0.5745454151066983 %i + 0.1538380639536643 
n= 1   t=  0.3231874668087892   c = 0.6469145331346998 %i - 0.07039249652637808 
n= 2   t=  0.3322326933513446   c = 0.649488031636116 %i - 0.1190170769366243 
n= 3   t=  0.3332223278292314   c = 0.6495187369145559 %i - 0.1243960357918423 
n= 4   t=  0.3333222232791965   c = 0.6495190496732967 %i - 0.1249395463818514 
n= 5   t=  0.3333322222327929   c = 0.6495190528066728 %i - 0.1249939540657306 
n= 6   t=  0.3333332222223279   c = 0.6495190528380125 %i - 0.1249993954008478 
n= 7   t=  0.3333333222222233   c = 0.6495190528383257 %i - 0.1249999395400275 
n= 8   t=  0.3333333322222222   c = 0.6495190528383289 %i - 0.1249999939540021 
n= 9   t=  0.3333333332222222   c = 0.649519052838329 %i - 0.1249999993954 
n= 10  t=  0.3333333333222222   c = 0.649519052838329 %i - 0.1249999999395399

(%i1) a:[0,3,a,g];
(%o1)                          [0, 3, a, g]
(%i2) cfdisrep(a);
                                    1
(%o2)                           ---------
                                      1
                                3 + -----
                                        1
                                    a + -
                                        g
(%i3)

四精度 (t 和 c)

// git@gitlab.com:adammajewski/InfoldingSiegelDisk_in_c_1over3_quaddouble.git

t(0)  = 2.76393202250021030359082633126872376455938164038847427572910275e-01 c = 1.53838063953664121728826496636884090757364247198001213740163411e-01  ; 5.74545415106698547533205192239966882171974656527110206599567294e-01
t(1)  = 3.23187466808789167460075188398563024584074865168574811248120429e-01 c = -7.03924965263779817446805013170440359580350303484669206412737432e-02 ; 6.46914533134699876497054948648970463828561836818357035393585013e-01
t(2)  = 3.32232693351344645328281111249254263489588693253128095562021379e-01 c = -1.19017076936624259031145944667596002534849695014545153950004772e-01 ; 6.49488031636116063199566861137419722741268462810784951116992269e-01
t(3)  = 3.33222327829231396180972123516749255293978654448636072108442295e-01 c = -1.24396035791842227723833496314974550377900499387491577355907843e-01 ; 6.49518736914555954950215798854805983039885153673988260646103979e-01
t(4)  = 3.33322223279196467461199806968881974980654735721758806121437081e-01 c = -1.24939546381851514024915251452349793356883503222467666248987991e-01 ; 6.49519049673296680160381595728499835459227600854334414827754792e-01
t(5)  = 3.33332222232792869679683559108320427563171464137271516297451542e-01 c = -1.24993954065730675441065991629838614047426783311966520549106346e-01 ; 6.49519052806672859063863722773813069312955136848755592503084124e-01
t(6)  = 3.33333222222327929601887145303917917465162347855208962292566660e-01 c = -1.24999395400848041352293443820430989442841091304288844173462574e-01 ; 6.49519052838012418008903728036348898528045582108441153259758357e-01
t(7)  = 3.33333322222223279296923970287377310413140932425289152909012301e-01 c = -1.24999939540027553391850682937180513243494903336945859380511500e-01 ; 6.49519052838325819396349608234642964505989043061737603462233217e-01
t(8)  = 3.33333332222222232792970144802560581866962241709924341887457124e-01 c = -1.24999993954002182831269818316294274223873134947476282441764677e-01 ; 6.49519052838328953416022146474675036126754173503570601306155105e-01
t(9)  = 3.33333333222222222327929702353125377874576632652638574699840874e-01 c = -1.24999999395400212558047347868208906283402972461774034404632366e-01 ; 6.49519052838328984756224669944909300404586792219705302525018521e-01
t(10) = 3.33333333322222222223279297024436353559326388564904699626466712e-01 c = -1.24999999939540021198553937965723010802762347867693302140297851e-01 ; 6.49519052838328985069626700977700316581410665652851539203616381e-01
t(11) = 3.33333333332222222222232792970245268635374696979418341612961499e-01 c = -1.24999999993954002119282885827879858604573185189849864410972761e-01 ; 6.49519052838328985072760721293826315500672008735227141791647866e-01
t(12) = 3.33333333333222222222222327929702453591453528488135105883396848e-01 c = -1.24999999999395400211922563503100579972022557444713095237600026e-01 ; 6.49519052838328985072792061496993373578630511176383107487756736e-01
t(13) = 3.33333333333322222222222223279297024536819635066408216696619311e-01 c = -1.24999999999939540021192199099513183456854230374819467960534760e-01 ; 6.49519052838328985072792374899025049957498862929395598811798211e-01
t(14) = 3.33333333333332222222222222232792970245369101450445609885075510e-01 c = -1.24999999999993954002119219337443349599800479106023686175774584e-01 ; 6.49519052838328985072792378033045366727085635213738283716661855e-01
t(15) = 3.33333333333333222222222222222327929702453691919604237626654112e-01 c = -1.24999999999999395400211921928019255272520717007610023985861823e-01 ; 6.49519052838328985072792378064385569894787301025348531521608111e-01
t(16) = 3.33333333333333322222222222222223279297024536920101142157794353e-01 c = -1.24999999999999939540021192192744674730377477910267401190215182e-01 ; 6.49519052838328985072792378064698971926464323481553400821453061e-01
t(17) = 3.33333333333333332222222222222222232792970245369201916521359471e-01 c = -1.24999999999999993954002119219273894965069001852640340429175752e-01 ; 6.49519052838328985072792378064702105946781093711913538281273390e-01
t(18) = 3.33333333333333333222222222222222222327929702453692020070313376e-01 c = -1.24999999999999999395400211921927383771427212725879688582341355e-01 ; 6.49519052838328985072792378064702137286984261414222937744638416e-01
t(19) = 3.33333333333333333322222222222222222223279297024536920201608234e-01 c = -1.24999999999999999939540021192192738319891924397994124922164696e-01 ; 6.49519052838328985072792378064702137600386293091246037537360833e-01
t(20) = 3.33333333333333333332222222222222222222232792970245369202016987e-01 c = -1.24999999999999999993954002119219273831416684471053474052374312e-01 ; 6.49519052838328985072792378064702137603520313408016268541086146e-01
1/3   = 3.33333333333333333333333333333333333333333333333333333333333333e-01 c = -1.25000000000000000000000000000000000000000000000000000000000000e-01 ; 6.49519052838328985072792378064702137603551970178892735520927617e-01

9/13

來自 scholarpedia 的示例^[37]

fpprec:60;
a:[4,4,1,2,4,4,4,4,1,1,1,1,1,1,1,1,1];
b:cfdisrep(a);
c:bfloat(b);
e0:bfloat(cfdisrep([0,1,2,4,4,c]));
e1:bfloat(cfdisrep([0,1,2,4,4,10,c]));
e2:bfloat(cfdisrep([0,1,2,4,4,10^2,c]));
e3:bfloat(cfdisrep([0,1,2,4,4,10^3,c]));
e4:bfloat(cfdisrep([0,1,2,4,4,10^4,c]));
e5:bfloat(cfdisrep([0,1,2,4,4,10^5,c]));
e6:bfloat(cfdisrep([0,1,2,4,4,10^6,c]));
e7:bfloat(cfdisrep([0,1,2,4,4,10^7,c]));
e8:bfloat(cfdisrep([0,1,2,4,4,10^8,c]));
e9:bfloat(cfdisrep([0,1,2,4,4,10^9,c]));
e10:bfloat(cfdisrep([0,1,2,4,4,10^10,c]));
d01:e0-e1;
d12:e1-e2;
d23:e2-e3;
d34:e3-e4;
d45:e4-e5;
d56:e5-e6;
d67:e6-e7;
d78:e7-e8;
d89:e8-e9;
d910:e9-e10;

plot2d( [discrete, [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [ e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10] ]);
plot2d( [discrete, [ 1, 2, 3, 4, 5, 6, 7, 8, 9,10], [ d01, d12, d23, d34, d45, d56, d67, d78, d89, d910] ]);

結果

/* http://maxima-online.org/?inc=r972897020*/
(%i1) fpprec:60;
(%o1)                                 60
(%i2) a:[4,4,1,2,4,4,4,4,1,1,1,1,1,1,1,1,1];
(%o2)         [4, 4, 1, 2, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(%i3) b:cfdisrep(a);
                                         1
(%o3)  4 + -------------------------------------------------------------
                                           1
           4 + ---------------------------------------------------------
                                             1
               1 + -----------------------------------------------------
                                               1
                   2 + -------------------------------------------------
                                                 1
                       4 + ---------------------------------------------
                                                   1
                           4 + -----------------------------------------
                                                     1
                               4 + -------------------------------------
                                                       1
                                   4 + ---------------------------------
                                                         1
                                       1 + -----------------------------
                                                           1
                                           1 + -------------------------
                                                             1
                                               1 + ---------------------
                                                               1
                                                   1 + -----------------
                                                                 1
                                                       1 + -------------
                                                                   1
                                                           1 + ---------
                                                                     1
                                                               1 + -----
                                                                       1
                                                                   1 + -
                                                                       1
(%i4) c:bfloat(b);
(%o4)   4.21317492083944457390374624758406097076199745041327978287391b0
(%i5) e0:bfloat(cfdisrep([0,1,2,4,4,c]));
(%o5)  6.90983385919269333377918690283855111536837789999636063622128b-1
(%i6) e1:bfloat(cfdisrep([0,1,2,4,4,10,c]));
(%o6)  6.90940653590858345205900333693231459583929903277216782489572b-1
(%i7) e2:bfloat(cfdisrep([0,1,2,4,4,10^2,c]));
(%o7)  6.90912381108070743953290526690429251279660195160237673633147b-1
(%i8) e3:bfloat(cfdisrep([0,1,2,4,4,10^3,c]));
(%o8)  6.90909421331077674912831355964859580252020147075327146840293b-1
(%i9) e4:bfloat(cfdisrep([0,1,2,4,4,10^4,c]));
(%o9)  6.90909123965376225147306218871548821073377014383362752163859b-1
(%i10) e5:bfloat(cfdisrep([0,1,2,4,4,10^5,c]));
(%o10) 6.90909094214860373154103745626419162194365585020091374566415b-1
(%i11) e6:bfloat(cfdisrep([0,1,2,4,4,10^6,c]));
(%o11) 6.90909091239669264887898182284536032143262445829248943506691b-1
(%i12) e7:bfloat(cfdisrep([0,1,2,4,4,10^7,c]));
(%o12) 6.90909090942148758764580793509997058314465248663422318011972b-1
(%i13) e8:bfloat(cfdisrep([0,1,2,4,4,10^8,c]));
(%o13) 6.90909090912396694199216055201332263713180254323741379104146b-1
(%i14) e9:bfloat(cfdisrep([0,1,2,4,4,10^9,c]));
(%o14)  6.9090909090942148760314918534473410035349593667510644807155b-1
(%i15) e10:bfloat(cfdisrep([0,1,2,4,4,10^10,c]));
(%o15) 6.90909090909123966942147194332785516326702737819678667743891b-1
(%i16) d01:e0-e1;
(%o16) 4.27323284109881720183565906236519529078867224192811325562048b-5
(%i17) d12:e1-e2;
(%o17) 2.82724827876012526098070028022083042697081169791088564253857b-5
(%i18) d23:e2-e3;
(%o18) 2.95977699306904045917072556967102764004808491052679285309819b-6
(%i19) d34:e3-e4;
(%o19) 2.97365701449765525137093310759178643132691964394676434841921b-7
(%i20) d45:e4-e5;
(%o20) 2.97505158519932024732451296588790114293632713775974431973635b-8
(%i21) d56:e5-e6;
(%o21) 2.97519110826620556334188313005110313919084243105972446795396b-9
(%i22) d67:e6-e7;
(%o22) 2.97520506123317388774538973828797197165826625494718990624414b-10
(%i23) d78:e7-e8;
(%o23) 2.97520645653647383086647946012849943396809389078262402933403b-11
(%i24) d89:e8-e9;
(%o24) 2.97520659606686985659816335968431764863493103259592529156942b-12
(%i25) d910:e9-e10;
(%o25) 2.97520661001991011948584026793198855427780327658967717649496b-13

這些西格爾圓盤有 13 位數字。因此它將接近 t=n/13，可能是

  ${\frac {9}{13}}=0.692307692307692307692307692307692307692307692307692307692...=0.{\overline {692307}}$

這個數字是不可約的，它的十進位制展開週期為 6 ^[38]

數值計算接近

  ${\frac {38}{55}}=0.690909090909090909090909090909090909090909090909090909090...=0.6{\overline {90}}$

它是不可約分數。它的十進位制展開週期為 2 ^[39]

重要數字

 8/13  = 0.6153846153846154 = [0; 1, 1, 1, 1, 2]
 38/55 = 0.690909090909090(90) = [0; 1, 2, 4, 4]
 9/13  = 0.692307692307(692307) = [0; 1, 2, 4]

具有光滑邊界的二次西格爾圓盤

Xavier Buff 和 Arnaud Cheritat 的示例^[40]

α = ( 5 + 1)/2 = [1, 1, 1, 1, . . .],

α(1) = [1, 1, 1, 1, 1, 1, 25, 1, 1, 1, . . .]

α(2) = [1, 1, 1, 1, 1, 1, 25, 1010 , 1, 1, 1, . . .]

2/7

$finite continued fractions aproximation to [0;1,1,1,....$

對 [0;1,1,1,.... 的有限連分數近似
[0;3,2,1000,1...] 朱利亞集的臨界軌道、內圓和外圓
朱利亞集 [0;3,2,1000,1... ]
臨界軌道

旋轉數 t 為 :^[41]

$t=[3,2,1000,1,...]=[0;3,2,1000,1\dots ]={\cfrac {1}{3+{\cfrac {1}{2+{\cfrac {1}{1000+{\cfrac {1}{1+\ddots }}}}}}}}$

在 Maxima CAS 中，可以計算它

(%i2) kill(all)
(%o0) done
(%i1)  a:[0,3,2,1000,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%o1)  [0,3,2,1000,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%i2) t:cfdisrep(a)
(%o2) (1)/(3+(1)/(2+(1)/(1000+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+ (1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
(%i3) float(t)
(%o3) .2857346725405882
(%i4) l:%e^(2*%pi*%i*t)
(%o4) %e^((2*%i*%pi)/(3+(1)/(2+(1)/(1000+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
(%i5) c:(l*(1-l/2))/2
(%o5) ((1-(%e^((2*%i*%pi)/(3+(1)/(2+(1)/(1000+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))/(2))*%e^((2*%i*%pi)/(3+(1)/(2+(1)/(1000+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))/(2)
(%i6) float(rectform(c))
(%o6) .5959783359361234*%i+.1138915132131216

所以

t = .2857346725405882
c = 0.113891513213121  +0.595978335936124 i

如何找到 [3,2,1000,1,...] 的極限 ?

以下是 Bill Wood 的解釋

"我不知道 Maxima 是否對連分數的代數了解很多，但它在破解推導的細節方面可能有所幫助。一個最有用的事實是

   [a1, a2, a3, ...] = a1 + 1/[a2, a3, ...]

前提是連分數收斂。如果我們手動對 [3, 2, 1000, 1, 1, ...] 應用三次，我們會得到

   3 + 1/(2 + 1/(1000 + 1/[1, 1, ...]))

現在眾所周知 [1, 1, ...] 收斂到黃金分割 = (1+sqrt(5))/2。所以現在我們可以使用 Maxima 如下

 (%i20) 3+(1/(2+(1/(1000+(1/((1+sqrt(5))/2))))));
                                    1
 (%o20)                    ---------------------- + 3
                                  1
                          ------------------ + 2
                               2
                          ----------- + 1000
                          sqrt(5) + 1
 (%i21) factor(%o13);
                              7003 sqrt(5) + 7017
 (%o21)                        -------------------
                              2001 sqrt(5) + 2005
 (%i22) %o21,numer;
 (%o22)                         3.499750279196346

您將 a 設定為 [0, 3, 2, 1000, 1, 1, ...]，根據我們的有用事實，它必須是 [3, 2, 1000, 1, 1, ...] 的倒數，實際上 3.499750279196346 的倒數是 0.2857346725405882，這正是您的 float(t) 評估到的結果，因此我們似乎得到了一致的結果。

如果為您提供的連結上顯示的所有旋轉數的連分數最終都重複 1，那麼我上面使用的方法可用於確定它們的極限作為 sqrt(5) 中線性表示式的比率。" Bill Wood

檢查分數：^[42]

 2/7 = [0; 3, 2] = 0.285714285714285714285714285714285714285714285714285714285... = 0.(28571)

/* Maxima CAS batch file */ 
kill(all)$
remvalue(all)$

/* a =  [1, 1, ...] =  golden ratio */
g: float((1+sqrt(5))/2)$

GiveT(n):= float(cfdisrep([0,3,2,10^n,g]))$

GiveC(t):= block(
  [l, c],
  l:%e^(2*%pi*%i*t),
  c : (l*(1-l/2))/2,
  c:float(rectform(c)),
  return(c)

)$

compile(all)$

 for n:0 step 1 thru 10 do (
t:GiveT(n),
c:GiveC(t),
print("n=", n , " ; t= ", t, " ; c = ", c)
);

以及結果

"n="" "0" "" ; t= "" "0.2956859994078892" "" ; c = "" "0.6153124581224951*%i+0.06835556662164869" "
"n="" "1" "" ; t= "" "0.2875617458610296" "" ; c = "" "0.599810068302661*%i+0.1057522049785167" "
"n="" "2" "" ; t= "" "0.285916253540726" "" ; c = "" "0.5963646240901801*%i+0.1130872227062027" "
"n="" "3" "" ; t= "" "0.2857346725405881" "" ; c = "" "0.5959783359361234*%i+0.1138915132131216" "
"n="" "4" "" ; t= "" "0.2857163263170416" "" ; c = "" "0.5959392401496606*%i+0.1139727184036316" "
"n="" "5" "" ; t= "" "0.2857144897937824" "" ; c = "" "0.5959353258460209*%i+0.1139808467588366" "
"n="" "6" "" ; t= "" "0.2857143061224276" "" ; c = "" "0.5959349343683512*%i+0.1139816596727942" "
"n="" "7" "" ; t= "" "0.2857142877551018" "" ; c = "" "0.5959348952201112*%i+0.1139817409649742" "
"n="" "8" "" ; t= "" "0.2857142859183673" "" ; c = "" "0.5959348913052823*%i+0.1139817490942002" "
"n="" "9" "" ; t= "" "0.2857142857346939" "" ; c = "" "0.5959348909137995*%i+0.1139817499071228" "
"n="" "10" "" ; t= "" "0.2857142857163265" "" ; c = "" "0.5959348908746511*%i+0.1139817499884153" "

或使用四倍雙精度（qd 庫）

./a.out
t(0)  = 2.95685999407889202537083517598191479880016272621355940112392033e-01
t(1)  = 2.87561745861029587995763349374215634699576366286473943622953355e-01
t(2)  = 2.85916253540726005296290351777281930430489555597203400050155169e-01
t(3)  = 2.85734672540588170268886130620030074831025799037914834916050614e-01
t(4)  = 2.85716326317041655001121670485871240052920659174284332597727197e-01
t(5)  = 2.85714489793782460278353122604001584582831972498394265344044955e-01
t(6)  = 2.85714306122427620319960416932943500595980573209849938259909447e-01
t(7)  = 2.85714287755101827223406574888790339883583192331314008460721020e-01
t(8)  = 2.85714285918367344802846109454092778340593443209054635310671634e-01
t(9)  = 2.85714285734693877529661113954572642424219379874139361821960174e-01
t(10) = 2.85714285716326530612031305016895554055165716643985018539739541e-01
t(11) = 2.85714285714489795918365211009352427817161964062263710683311655e-01
2/7   = 2.85714285714285714285714285714285714285714285714285714285714286e-01

2/9

(%i7) cf(2/9);
(%o7)                              [0, 4, 2]
(%i8) cf(2/9 + 0.1);
(%o8)                           [0, 3, 9, 1, 2]
(%i9) cf(2/9 + 0.01);
(%o9)                        [0, 4, 3, 3, 1, 3, 4]
(%i10) cf(2/9 + 0.001);
(%o10)                      [0, 4, 2, 11, 1, 9, 8]
(%i11) cf(2/9 + 0.0001);
(%o11)                        [0, 4, 2, 123, 81]
(%i12) cf(2/9 + 0.00001);
(%o12)                      [0, 4, 2, 1234, 8, 10]
(%i13) cf(2/9 + 0.000001);
(%o13)                   [0, 4, 2, 12345, 4, 3, 1, 4]
(%i14) cf(2/9 + 0.0000001);
(%o14)                    [0, 4, 2, 123456, 2, 1, 8]
(%i15) cf(2/9 + 0.00000001);
(%o15)                       [0, 4, 2, 1234567, 2]
(%i16) cf(2/9 + 0.000000001);
(%o16)                        [0, 4, 2, 12345678]
(%i17) cf(2/9 + 0.0000000001);
(%o17)                       [0, 4, 2, 123456806]
(%i18) cf(2/9 + 0.00000000001);
(%o18)                       [0, 4, 2, 1234571860]
(%i19) cf(2/9 + 0.000000000001);
(%o19)                      [0, 4, 2, 12345935203]
(%i20) cf(2/9 + 0.0000000000001);
(%o20)                      [0, 4, 2, 123453937153]
(%i21) cf(2/9 + 0.00000000000001);
(%o21)                     [0, 4, 2, 1234539371537]
(%i22) cf(2/9 + 0.000000000000001);
(%o22)                     [0, 4, 2, 12794317123211]
(%i23)

問題

是否有可能找到將圓對映到西格爾圓盤的函式 ?
虛擬西格爾圓盤和拋物線臨界軌道之間會發生什麼 ?

哪些外部射線（引數射線）落在西格爾點 c 上 ? 無理數角的射線 ?

落在克萊默和西格爾引數點 c 上的外部射線的角度是多少 ?

那些不屬於任何閉合尾跡的射線。它們是無理數，我猜沒有更多已知的資訊。您可以用適合尾跡的角度來近似它們......

這類似於透過連續去除開區間來構建中間 1/3 標準康託集。

從 [0,1] 開始。
去除 (1/3, 2/3)。
去除 (1/7, 2/7) 和 (5/7, 6/7)
...

剩下的角度有一個康託集，它們是落在主心形上的所有射線的角度。有理數角度屬於根，無理數角度屬於西格爾和克萊默引數。此外，每個有理數角度都是經過有限步後去除的區間的邊界點。所以在下面的去除閉區間的構造中，你不會得到一個康託集，只有無理數角度會保留下來

從 [0,1] 開始。
去除 [1/3, 2/3]。
去除 [1/7, 2/7] 和 [5/7, 6/7]

進一步閱讀

參考文獻

某些包含具有無理旋轉圓形的Julia集的區域性連通性。/ Petersen, Carsten Lunde. 發表在：Acta Mathematica，第177卷，第2期，1996年，第 163-224頁。
二次Julia集的構建塊。文章發表在：Transactions of the American Mathematical Society 351(3):1171-1201 · 1999年1月 DOI: 10.1090/S0002-9947-99-02346-6

[1] 關於二次有理對映的西格爾圓盤的張預模型。作者：Arnaud Cheritat CNRS，圖盧茲大學 2011 年 2 月

[2] 維基百科 : 西格爾圓盤

[3] 數學百科全書 : 西格爾圓盤（參見離散情況）

[4] 關於無理數內角的更多樂趣，作者：Faber McMullen

[5] 維基百科 : 無理數

[6] 西格爾圓盤，作者：Xavier Buff 和 Arnaud Ch ́ritat e 圖盧茲大學羅馬，2009 年 4 月

[7] Davoud Cheraghi 主頁

[8] wikipedia : 共形半徑

[9] scholarpedia : 二次Siegel圓盤

[10] scholarpedia : 二次Siegel圓盤

[11] Scholarpedia上的圖片InfoldingSiegelDisk.gif

[12] Julia集的可計算性，作者：Mark Braverman，Michael Yampolsky

[13] 拋物線內爆下Siegel圓盤的半連續性，作者：Arnaud Chéritat

[14] 分形幾何學：數學基礎與應用，第二版，作者：Kenneth Falconer

[15] 二次Julia集中的Siegel圓盤相關的正向軌道的視覺化教程，作者：G.Todd Miller

[16] 繪製Julia集的組合方法，作者：Chris King

[17] 無理內角的更多樂趣，作者：Faber

[18] 快速收斂的有理數級數的無理性，作者：Daniel Duverney

[19] 不可計算的Julia集，作者：M. BRAVERMAN，M. YAMPOLSKY

[20] Mandelbrot序列和軌道，作者：Stefan Forcey

[21] Jay Hill : JuliaSetsOrbitsSiegelDisks.txt

[22] wikipedia上的倍乘器

[23] wikipedia上的黃金分割

[24] 黃金分割Julia集影片，作者：fractal

[25] wikipedia : 連分數

[26] 繪製Julia集的組合方法，作者：Chris King

[27] 黃金分割Siegel圓盤，作者：Curtis T McMullen

[28] Siegel圓盤，作者：Jim Muth

[29] Siegel圓盤，作者：Davoud Cheraghi

[30] Mathematica中的更好的Siegel圓盤程式，作者：Roger Bagula

[31] Xander的部落格

[32] Galerie II : 全純動力學與複分析，作者：Arnaud Chéritat

[33] 二次Siegel圓盤的新舊知識，作者：SAEED ZAKERI

[34] 旋轉集與多項式動力學，作者：Zakeri

[35] 整數序列ID=A000045

[36] wolframalpha : 連分數[0;2,1,GoldenRatio]

[37] 內折Siegel圓盤，作者：Arnaud Chéritat

[38] wolfram alpha : 9/13

[39] wolfram alpha : 38/55

[40] 具有光滑邊界的Siegel圓盤，作者：ARTUR AVILA，XAVIER BUFF，ARNAUD CHERITAT

[41] 二次多項式Siegel圓盤的一些例子，作者：Davoud Cheraghi

[42] wolfram alpha : 2/7

[43] 小除數簡介，作者：S. Marmi

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