介绍

曼德勃罗集的灌木模型
参数道，1/3 觉醒，射线落在主米修列维奇点上

关键词定义

参数平面的部分

灌木
觉醒
肢体
米修列维奇点
- 尖端（点）= 终点= 分支尖端= 辐条尖端= 分支的终点^[1]= 小矮人尖端^[2] “曼德勃罗集中的一个点，位于细丝的末端（与分支点相对）；一个从该点到曼德勃罗集中的其他点只有一条路径的点。”
  - 第一个尖端= ftip= 第一个也是最长的分支的尖端（从左到右计数时第一个）
  - 最后一个尖端= ltip= （从左到右计数时最后一个）
数字
- 二进制
- 十进制

符号

前周期

任务

找到米修列维奇点
- 前周期和周期
- c 值
找到落在它上面的外部射线的角度

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]

克劳德

由克劳德提供的方法

算法步骤

找到觉醒的角度
找到主米修列维奇点 M 的角度
使用以下方法找到辐条尖端角度：“每个辐条的尖端是相邻角度的最长匹配前缀，后面追加 1”

1/3

落在 M 上的 3 个角度

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

每个辐条的尖端是相邻角度的最长匹配前缀，后面追加 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 程序进行检查

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

主天线尖端 (1/2 觉醒) $c=M_{1,1}$

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

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 ??? (待办事项)
- $0.0(01)={\frac {1}{6}}=0.1(6)_{10}=M_{2,1}=c=i$
- $0.0011(0)_{2}={\frac {3}{16}}=0.1875=M_{4,1}$

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 周期= 0 // 1/16 的着陆

2/5

c = -0.636754346582390 +0.685031297083677 i 周期= 0 // 5/16 的着陆

12/25

12/25 觉醒的主心形由具有以下角度的参数射线包围
- 11184809/33554431 或 p0101010101010101010101001 = 0.(0101010101010101010101001)
- 11184810/33554431 或 p0101010101010101010101010

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

结果

.010101010101010101010100(1010)

即

.01010101010101010101010(01)

参见

落在尖端上的射线的外部角度的周期/前周期与尖端（着陆点）的周期/前周期之间有什么关系？

所有落在同一个周期点上的射线具有相同的周期：射线的公共周期是其着陆点周期的（可能为真）倍数；因此，可以区分：射线周期与轨道周期。^[4]

参考文献

↑ 终点作者：罗伯特·P·穆纳福，2008 年 3 月 9 日。
↑ mathoverflow 问题：除了简单地四处探查，还有其他方法可以找到曼德勃罗集中的深度区域吗？
↑ G. Pastor，M. Romera，G. Alvarez，J. Nunez，D. Arroyo，F. Montoya，“操作 Douady 和 Hubbard 的外部参数”，自然和社会中的离散动力学，第 2007 卷，文章 ID 045920，17 页，2007 年。https://doi.org/10.1155/2007/45920
↑ H. Bruin 和 D. Schleicher，二次多项式的符号动力学，米塔格-莱夫勒研究所，瑞典皇家科学院，7。

[1] 终点作者：罗伯特·P·穆纳福，2008 年 3 月 9 日。

[2] thoverflow 问题：除了简单地四处探查，还有其他方法可以找到曼德勃罗集中的深度区域吗？

[3] G. Pastor，M. Romera，G. Alvarez，J. Nunez，D. Arroyo，F. Montoya，“操作 Douady 和 Hubbard 的外部参数”，自然和社会中的离散动力学，第 2007 卷，文章 ID 045920，17 页，2007 年。https://doi.org/10.1155/2007/45920

[4] H. Bruin 和 D. Schleicher，二次多项式的符号动力学，米塔格-莱夫勒研究所，瑞典皇家科学院，7。

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