介紹

曼德勃羅集的灌木模型
引數道，1/3 覺醒，射線落在主米修列維奇點上

關鍵詞定義

引數平面的部分

灌木
覺醒
肢體
米修列維奇點
- 尖端（點）= 終點= 分支尖端= 輻條尖端= 分支的終點^[1]= 小矮人尖端^[2] “曼德勃羅集中的一個點，位於細絲的末端（與分支點相對）；一個從該點到曼德勃羅集中的其他點只有一條路徑的點。”
  - 第一個尖端= ftip= 第一個也是最長的分支的尖端（從左到右計數時第一個）
  - 最後一個尖端= ltip= （從左到右計數時最後一個）
數字
- 二進位制
- 十進位制

符號

前週期

任務

找到米修列維奇點
- 前週期和週期
- c 值
找到落在它上面的外部射線的角度

演算法

帕斯托

 "the external argument can be calculated as the limit of the arguments of the structural components of the branches 1, 11, 111,..., with periods 4, 5, 6,..., that is, the limit of .(0011), .(00111), .(001111),..., or the limit of .(0100), .(01000), .(010000), .... Hence, ftip(1/3) = .00(1) = .01(0), that are two equal values. " ^[3]

克勞德

由克勞德提供的方法

演算法步驟

找到覺醒的角度
找到主米修列維奇點 M 的角度
使用以下方法找到輻條尖端角度：“每個輻條的尖端是相鄰角度的最長匹配字首，後面追加 1”

1/3

落在 M 上的 3 個角度

  0.001(010) 
  0.001(100)
  0.010(001)

每個輻條的尖端是相鄰角度的最長匹配字首，後面追加 1

  0.001(010) // 9/56 = 0.160(714285)
  0.0011    // ltip = 3/16 = 0.1875
  0.001(100) // 11/56 = 0.196(428571)
  0.01  // ftip = 1/4 = 0.25
  0.010(001) // 15/56 = 0.267(857142)

用 Mandel 程式進行檢查

The angle  3/16  or  0011 has  preperiod = 4  and  period = 1. Entropy: e^h = 2^B = λ = 1.59898328
The corresponding parameter ray lands at a Misiurewicz point of preperiod 4 and period dividing 1. 
Do you want to draw the ray and to shift c to the landing point?

 c = -0.017187977338350  +1.037652343793215 i    period = 0

The angle  1/4  or  01 has  preperiod = 2  and  period = 1.
Entropy: e^h = 2^B = λ = 1.69562077
The corresponding parameter ray lands at a Misiurewicz point of preperiod 2 and period dividing 1.
Do you want to draw the ray and to shift c to the landing point?

 M_{2,1) = c = -0.228155493653962  +1.115142508039937 i

The angle  1/6  or  0p01 has  preperiod = 1  and  period = 2.
The corresponding parameter ray lands at a Misiurewicz point of preperiod 1 and period dividing 2.
Do you want to draw the ray and to shift c to the landing point?

 c = -0.000000000000000  +1.000000000000000 i    period = 10000

例子

1/2

主天線尖端 (1/2 覺醒) $c=M_{1,1}$

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

1/3

 ${\begin{cases}0.(001)_{2}={\frac {1}{7}}=0.(142857)_{10}=wake\\0.001(010)_{2}={\frac {9}{56}}=0.160(714285)=principleM\\0.001(100)_{2}={\frac {11}{56}}=0.196(428571)=principleM\\0.01_{2}={\frac {1}{4}}=0.25=ftip\\0.010(001)_{2}={\frac {15}{56}}=0.267(857142)=principleM\\0.(010)_{2}={\frac {2}{7}}=0.(285714)=wake\\\end{cases}}$

尖端

ftip = M_{2,1} = 0.01(0) = 1/4 = c = -0.228155493653962 +1.115142508039937 i
ltip ??? (待辦事項)
- $0.0(01)={\frac {1}{6}}=0.1(6)_{10}=M_{2,1}=c=i$
- $0.0011(0)_{2}={\frac {3}{16}}=0.1875=M_{4,1}$

The angle  1/4  or  01 has  preperiod = 2  and  period = 1.
Entropy: e^h = 2^B = λ = 1.69562077
The corresponding parameter ray lands at a Misiurewicz point of preperiod 2 and period dividing 1.
Do you want to draw the ray and to shift c to the landing point?
c = -0.228155493653962  +1.115142508039937 i  

The angle  1/4  or  01 has  preperiod = 2  and  period = 1.
The corresponding dynamic ray lands at a preperiodic point of preperiod 2 and period dividing 1.
Do you want to draw the ray and to shift z to the landing point?
z = -0.228155493653962  +1.115142508039937 i

The angle  4/7  or  p100 has  preperiod = 0  and  period = 3. 
The dynamic ray lands at a repelling or parabolic point of period dividing 3.
Do you want to draw the ray and to shift z to the landing point?
z = -0.419643377607081  +0.606290729207199 i

The angle  1/8  or  001 has  preperiod = 3  and  period = 1.
The corresponding dynamic ray lands at a preperiodic point of preperiod 3 and period dividing 1.
Do you want to draw the ray and to shift z to the landing point?
z = 0.000000000159395  +0.000000000076028 i

The angle  3/16  or  0011 has  preperiod = 4  and  period = 1.
Entropy: e^h = 2^B = λ = 1.59898328 
The corresponding parameter ray lands at a Misiurewicz point of preperiod 4 and period dividing 1.
Do you want to draw the ray and to shift c to the landing point?
c = -0.017187977338350  +1.037652343793215 i    period = 0

The angle  1/6  or  0p01 has  preperiod = 1  and  period = 2.
The corresponding parameter ray lands at a Misiurewicz point of preperiod 1 and period dividing 2.
Do you want to draw the ray and to shift c to the landing point?
c = -0.000000000000000  +1.000000000000000 i    period = 10000

1/5

c = 0.444556879255044 +0.409933108300984 i 週期= 0 // 1/16 的著陸

2/5

c = -0.636754346582390 +0.685031297083677 i 週期= 0 // 5/16 的著陸

12/25

12/25 覺醒的主心形由具有以下角度的引數射線包圍
- 11184809/33554431 或 p0101010101010101010101001 = 0.(0101010101010101010101001)
- 11184810/33554431 或 p0101010101010101010101010

m-exray-out 100 -0.7432918908524301 0.1312405523087976  8 1000 24 4

結果

.010101010101010101010100(1010)

即

.01010101010101010101010(01)

參見

落在尖端上的射線的外部角度的週期/前週期與尖端（著陸點）的週期/前週期之間有什麼關係？

所有落在同一個週期點上的射線具有相同的週期：射線的公共週期是其著陸點週期的（可能為真）倍數；因此，可以區分：射線週期與軌道週期。^[4]

參考文獻

↑ 終點作者：羅伯特·P·穆納福，2008 年 3 月 9 日。
↑ mathoverflow 問題：除了簡單地四處探查，還有其他方法可以找到曼德勃羅集中的深度區域嗎？
↑ G. Pastor，M. Romera，G. Alvarez，J. Nunez，D. Arroyo，F. Montoya，“操作 Douady 和 Hubbard 的外部引數”，自然和社會中的離散動力學，第 2007 卷，文章 ID 045920，17 頁，2007 年。https://doi.org/10.1155/2007/45920
↑ H. Bruin 和 D. Schleicher，二次多項式的符號動力學，米塔格-萊夫勒研究所，瑞典皇家科學院，7。

[1] 終點作者：羅伯特·P·穆納福，2008 年 3 月 9 日。

[2] thoverflow 問題：除了簡單地四處探查，還有其他方法可以找到曼德勃羅集中的深度區域嗎？

[3] G. Pastor，M. Romera，G. Alvarez，J. Nunez，D. Arroyo，F. Montoya，“操作 Douady 和 Hubbard 的外部引數”，自然和社會中的離散動力學，第 2007 卷，文章 ID 045920，17 頁，2007 年。https://doi.org/10.1155/2007/45920

[4] H. Bruin 和 D. Schleicher，二次多項式的符號動力學，米塔格-萊夫勒研究所，瑞典皇家科學院，7。

[1]

[2]

[3]

[4]