Difference between sequences, orders and series

types of sequences

Integer sequences

Fraction sequences

Farey sequence

The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 (= 0 / 1 ) and 1 (= 1 / 1 ), by taking successive mediants.

The Farey sequence of order n is the sequence of completely reduced vulgar fractions between 0 and 1 which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.

Each Farey sequence starts with the value 0, denoted by the fraction ⁰⁄₁, and ends with the value 1, denoted by the fraction ¹⁄₁ (although some authors omit these terms).

Farey Addition = the mediant of two fractions :

  $\frac{a}{c} \oplus \frac{b}{d} = \frac{a + b}{c + d}$

Terms

next term = child
Previous terms = parents^[1]

Farey tree = Farey sequence as a tree

Sorted

 F1 = {0/1,                                                                                                          1/1}
 F2 = {0/1,                                                   1/2,                                                   1/1}
 F3 = {0/1,                               1/3,                1/2,                2/3,                               1/1}
 F4 = {0/1,                     1/4,      1/3,                1/2,                2/3,      3/4,                     1/1}
 F5 = {0/1,                1/5, 1/4,      1/3,      2/5,      1/2,      3/5,      2/3,      3/4, 4/5,                1/1}
 F6 = {0/1,           1/6, 1/5, 1/4,      1/3,      2/5,      1/2,      3/5,      2/3,      3/4, 4/5, 5/6,           1/1}
 F7 = {0/1,      1/7, 1/6, 1/5, 1/4, 2/7, 1/3,      2/5, 3/7, 1/2, 4/7, 3/5,      2/3, 5/7, 3/4, 4/5, 5/6, 6/7,      1/1}
 F8 = {0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1}

Sequences and orders on the parameter plane

Sequences of Misiurewicz points

external ray for angle 1/(4*2^n) land on the tip of the first branch: 1/4, 1/8, 1/16, 1/32, 1/64, ...
1/(6*2^n) - land on the second branch
principal Misiurewicz point of wake p/q
- How to compute external angles of principal Misiurewicz point of wake p/q using Devaney's algorithm ?
- program Mandel by Wolf Jung
primary_separators

degree

cubic

take the Misiurewicz point for $z^{n} + c$ and increase n ( proposed by Owen Maresh)

The constant (parameter c) for the quadratic (n=2) , cubic ( n=3), and quartic (n=4) polynomials are:

(-0.7432918908524301520519705530861564778806 ,0.1312405523087976002753516038253522297699);
-0.0649150006787816892861875745218343125883 , 0.76821968591243610206311097043854440463 );
(-0.593611822136354943067129147813253628530 ,0.5405019391915187246754930586066158919613 );

Point c is a Misiurewicz point $c = M_{23, 2} = - 0.743291890852430 + 0.131240552308798 i$

- tip of the longest branch ( ftip )
- The angle 8388607/25165824 or 01010101010101010101010p01 has preperiod = 23 and period = 2
- from wake 12/25 , with
  - center c = -0.739829393511579 +0.125072144080321 i and period = 25
  - root point c = -0.738203140939397 +0.124839088573366 i

m-describe 53 30 500 -0.7432918908524301 0.1312405523087976 4
the input point was -7.4329189085243008e-01 + 1.3124055230879761e-01 i
nearby hyperbolic components to the input point:

- a period 1 cardioid
  with nucleus at 0.00000e+00 + 0.00000e+00 i
  the component has size 1.00000e+00 and is pointing west
  the atom domain has size 0.00000e+00
  the atom domain coordinates of the input point are -nan + -nan i
  the atom domain coordinates in polar form are -nan to the east
  the atom coordinates of the input point are -0.74329 + 0.13124 i
  the atom coordinates in polar form are 0.75479 to the west
  the nucleus is 7.54789e-01 to the east of the input point

- a period 2 circle
  with nucleus at -1.00000e+00 + 0.00000e+00 i
  the component has size 5.00000e-01 and is pointing west
  the atom domain has size 1.00000e+00
  the atom domain coordinates of the input point are 0.25671 + 0.13124 i
  the atom domain coordinates in polar form are 0.28831 to the east-north-east
  the atom coordinates of the input point are 0.51342 + 0.26248 i
  the atom coordinates in polar form are 0.57662 to the east-north-east
  the nucleus is 2.88311e-01 to the west-south-west of the input point
  external angles of this component are:
  .(01)
  .(10)
the point escaped with dwell 472.09881

nearby Misiurewicz points to the input point:

- 24p4
  with center at -7.43291890852430202931624325972515e-01 + 1.31240552308797604770845906581477e-01 i
  the Misiurewicz domain has size 1.07586e-03
  the Misiurewicz domain coordinate radius is 1.1395e-13
  the center is 1.21387e-16 to the west of the input point
  the multiplier has radius 1.329970173958942893e+00 and angle 0.150434052944417735 (in turns)

The angle  8388607/25165824  or  01010101010101010101010p01 has  preperiod = 23  and  period = 2.
The corresponding parameter ray lands at a Misiurewicz point of preperiod 23 and period dividing 2.
Do you want to draw the ray and to shift c to the landing point?
c = -0.743291890852430                          +0.131240552308798 i    period = 0

Myrberg-Feigenbaum point

Examples:

sequence of root points for periods $p = 1 * 2^{n}$ ( period doubling cascade) and the limit point of the sequence is the Myrberg-Feigenbaum point ${𝐌 𝐅}_{1 / 2} = c_{0} = - 1.401155189093314712 . . .$
sequence of root points for periods $p = 1 * 3^{n}$ and the limit point of the sequence is the Myrberg-Feigenbaum point $𝐌 𝐅 = c_{0} = - 1.7864402555636388747 . . .$

Sharkovsky ordering

It is the infinite sequence of positive integers ( natural numbers)
It starts from 3 and ends in 1
It contains infinitely many subsequences.^[2]
the number is a period of the miget ( main pseudocardioid of the midget) that appear the first time in that order

\begin{matrix} 3 & 5 & 7 & 9 & 11 & \dots & (2 n + 1) \cdot 2^{0} & \dots \\ 3 \cdot 2 & 5 \cdot 2 & 7 \cdot 2 & 9 \cdot 2 & 11 \cdot 2 & \dots & (2 n + 1) \cdot 2^{1} & \dots \\ 3 \cdot 2^{2} & 5 \cdot 2^{2} & 7 \cdot 2^{2} & 9 \cdot 2^{2} & 11 \cdot 2^{2} & \dots & (2 n + 1) \cdot 2^{2} & \dots \\ 3 \cdot 2^{3} & 5 \cdot 2^{3} & 7 \cdot 2^{3} & 9 \cdot 2^{3} & 11 \cdot 2^{3} & \dots & (2 n + 1) \cdot 2^{3} & \dots \\ ⋮ \\ \dots & 2^{n} & \dots & 2^{4} & 2^{3} & 2^{2} & 2 & 1 \end{matrix}

"The Sharkovski ordering :

begins with the odd numbers >= 3 in increasing order ( n is increasing from left to right ),
then twice these numbers,
then 4 times them,
then 8 times them,
etc.,
ending with the powers of 2 in decreasing order, ending with 2^0 = 1."^[3]

(2 n + 1) \cdot 2^{0} ≺ (2 n + 1) \cdot 2^{1} ≺ (2 n + 1) \cdot 2^{2} ≺ \dots ≺ 2^{n}

chaotic region, which consist of chaotic bands Bm=(2n+1)⋅2m:
- $B_{0} = (2 n + 1) \cdot 2^{0}$
- $B_{1} = (2 n + 1) \cdot 2^{1}$
- $B_{2} = (2 n + 1) \cdot 2^{3}$
- $\dots$
- $B_{\infty}$
MF = Myrberg-Feigenbaum point
periodic region P with period doubling cascade = 2^n

B ≺ M F ≺ P

B_{0} ≺ B_{1} ≺ . . . ≺ M F ≺ . . . ≺ 2^{1} ≺ 2^{0}

Period doubling scenario

1/2 family

Pariod doubling cascade in the Mandelbrot set ( 1/2 family) showed by the exponential mapping
escape route 1/2
period doubling

sequence of fraction in the elephant valley

In the elephant valley^[4]^[5] ( from parameter plane ) there is a sequence of componts with period p : from 1/2 to 1/p

Note that :

internal ray 0/1 = 1/1
internal angle 1/p means that ray goes from period 1 component ( parent period = 1) to period p component ( child period = p)
as child period grows computations are harder
exponential growth^[6] of $2^{p}$ . One can easly create a numeric value that is too large to be represented within the available storage space ( integer overflow^[7] ). For example $2^{34}$ is to big for short ( 16 bit ) and long ( 32 bit) integer.

The upper principal sequence of rotational number around the main cardioid of Mandelbrot set^[8]

n	rotation number = 1/n	parameter c
2	1/2	-0.75
3	1/3	0.64951905283833*i-0.125
4	1/4	0.5*i+0.25
5	1/5	0.32858194507446*i+0.35676274578121
6	1/6	0.21650635094611*i+0.375
7	1/7	0.14718376318856*i+0.36737513441845
8	1/8	0.10355339059327*i+0.35355339059327
9	1/9	0.075191866590218*i+0.33961017714276
10	1/10	0.056128497072448*i+0.32725424859374

See :

Slide show of rescaled limbs converging to the Lavaurs elephant - video by Wolf Jung made with Mandel

sequence of parabolic points on the boundary of main cardioid

t = \sum_{k = 1}^{n} \frac{3}{1 0^{k}}

Here:

t = internal angle ( or rotation number) of main cardioid
q = number of the critical orbit (star) arms. It means that one have to do q iterations around fixed point to move one point toward fixed point along arm.
c is a root point between hyperbolic components of period 1 ( = main cardioid) and period q. This point is at the end ( radius = 1) of internal ray for angle t

k = log10(q)	$t = \frac{p}{q}$	(double) t	$c_{t} = \frac{2 e^{2 π i t} - e^{4 π i t}}{4}$
1	3/10	0.3	+0.047745751406263+0.622474571220695 i
2	33/100	0.33	-0.106920138306109 +0.649235321397436 i
3	333/1000	0.333	-0.123186752260805 +0.649516204880454 i
4	3333/10000	0.3333	-0.124818625550005 +0.649519024348384 i
5	33333/100000	0.33333	-0.124981862061192 +0.649519052553419 i
6	333333/1000000	0.333333	-0.124998186201184 +0.649519052835480 i
7	3333333/10000000	0.3333333	-0.124999818620069 +0.649519052838300 i
8	33333333/100000000	0.33333333	-0.124999981862006 +0.649519052838329 i
9	333333333/1000000000	0.333333333	-0.124999998186201 +0.649519052838329 i
10	3333333333/10000000000	0.3333333333	-0.124999999818620 +0.649519052838329 i

sequence from Siegel disk to Leau-Fatou flower

plain Siegel disk
digitated Siegel disk^[9]
virtual Siegel disk
? Leau-Fatou flower ?

1 over 2

Infolding Siegel Disk for c near internal angle t=1/2 on the boundary of main cardioid of Mandelbrot set

1 over 3

$t = [0; 3, 1 0^{n}, g] = 0 + \frac{1}{3 + \frac{1}{1 0^{n} + \frac{1}{g}}}$

n	t	$c_{t} = \frac{2 e^{2 π i t} - e^{4 π i t}}{4}$
0	0.2763932022500210	+0.1538380639536641 + 0.5745454151066985 i
1	0.3231874668087892	-0.0703924965263780 + 0.6469145331346999 i
2	0.3322326933513446	-0.1190170769366243 + 0.6494880316361160 i
3	0.3332223278292314	-0.1243960357918422 + 0.6495187369145560 i
4	0.3333222232791965	-0.1249395463818515 + 0.6495190496732967 i
5	0.3333322222327929	-0.1249939540657307 + 0.6495190528066729 i
6	0.3333332222223279	-0.1249993954008480 + 0.6495190528380124 i
7	0.3333333222222233	-0.1249999395400276 + 0.6495190528383258 i
8	0.3333333322222222	-0.1249999939540022 + 0.6495190528383290 i
9	0.3333333332222223	-0.1249999993954002 + 0.6495190528383290 i
10	0.3333333333222222	-0.1249999999395400 + 0.6495190528383290 i
11	0.3333333333322222	-0.1249999999939540 + 0.6495190528383290 i

sequence of fractions tending to the golden mean ( Golden Ratio Conjugate )

n =   1 ;  p_n/q_n =  1.0000000000000000000 =                     1 /                    1 
n =   2 ;  p_n/q_n =  0.5000000000000000000 =                     1 /                    2 
n =   3 ;  p_n/q_n =  0.6666666666666666667 =                     2 /                    3 
n =   4 ;  p_n/q_n =  0.6000000000000000000 =                     3 /                    5 
n =   5 ;  p_n/q_n =  0.6250000000000000000 =                     5 /                    8 
n =   6 ;  p_n/q_n =  0.6153846153846153846 =                     8 /                   13 
n =   7 ;  p_n/q_n =  0.6190476190476190476 =                    13 /                   21 
n =   8 ;  p_n/q_n =  0.6176470588235294118 =                    21 /                   34 
n =   9 ;  p_n/q_n =  0.6181818181818181818 =                    34 /                   55 
n =  10 ;  p_n/q_n =  0.6179775280898876404 =                    55 /                   89 
n =  11 ;  p_n/q_n =  0.6180555555555555556 =                    89 /                  144 
n =  12 ;  p_n/q_n =  0.6180257510729613734 =                   144 /                  233 
n =  13 ;  p_n/q_n =  0.6180371352785145888 =                   233 /                  377 
n =  14 ;  p_n/q_n =  0.6180327868852459016 =                   377 /                  610 
n =  15 ;  p_n/q_n =  0.6180344478216818642 =                   610 /                  987 
n =  16 ;  p_n/q_n =  0.6180338134001252348 =                   987 /                 1597 
n =  17 ;  p_n/q_n =  0.6180340557275541796 =                  1597 /                 2584 
n =  18 ;  p_n/q_n =  0.6180339631667065295 =                  2584 /                 4181 
n =  19 ;  p_n/q_n =  0.6180339985218033999 =                  4181 /                 6765 
n =  20 ;  p_n/q_n =  0.6180339850173579390 =                  6765 /                10946 
n =  21 ;  p_n/q_n =  0.6180339901755970865 =                 10946 /                17711 
n =  22 ;  p_n/q_n =  0.6180339882053250515 =                 17711 /                28657 
n =  23 ;  p_n/q_n =  0.6180339889579020014 =                 28657 /                46368 
n =  24 ;  p_n/q_n =  0.6180339886704431856 =                 46368 /                75025 
n =  25 ;  p_n/q_n =  0.6180339887802426829 =                 75025 /               121393 
n =  26 ;  p_n/q_n =  0.6180339887383030068 =                121393 /               196418 
n =  27 ;  p_n/q_n =  0.6180339887543225376 =                196418 /               317811 
n =  28 ;  p_n/q_n =  0.6180339887482036214 =                317811 /               514229 
n =  29 ;  p_n/q_n =  0.6180339887505408394 =                514229 /               832040 
n =  30 ;  p_n/q_n =  0.6180339887496481015 =                832040 /              1346269 
n =  31 ;  p_n/q_n =  0.6180339887499890970 =               1346269 /              2178309 
n =  32 ;  p_n/q_n =  0.6180339887498588484 =               2178309 /              3524578 
n =  33 ;  p_n/q_n =  0.6180339887499085989 =               3524578 /              5702887 
n =  34 ;  p_n/q_n =  0.6180339887498895959 =               5702887 /              9227465 
n =  35 ;  p_n/q_n =  0.6180339887498968544 =               9227465 /             14930352 
n =  36 ;  p_n/q_n =  0.6180339887498940819 =              14930352 /             24157817 
n =  37 ;  p_n/q_n =  0.6180339887498951409 =              24157817 /             39088169 
n =  38 ;  p_n/q_n =  0.6180339887498947364 =              39088169 /             63245986 
n =  39 ;  p_n/q_n =  0.6180339887498948909 =              63245986 /            102334155 
n =  40 ;  p_n/q_n =  0.6180339887498948319 =             102334155 /            165580141 
n =  41 ;  p_n/q_n =  0.6180339887498948544 =             165580141 /            267914296 
n =  42 ;  p_n/q_n =  0.6180339887498948458 =             267914296 /            433494437 
n =  43 ;  p_n/q_n =  0.6180339887498948491 =             433494437 /            701408733 
n =  44 ;  p_n/q_n =  0.6180339887498948479 =             701408733 /           1134903170 
n =  45 ;  p_n/q_n =  0.6180339887498948483 =            1134903170 /           1836311903

This is a sequence of rational numbers ( Julia sets are parabolic). It's limit is an irrational number ( Julia set has a Siegel disk).

Sequence on the dynamic plane

More

orbit
- period of the orbit

References

Template:BookCat

[1] Finding parents in the Farey tree by Claude Heiland-Allen

[2] Sharkovskii's theorem in wikipedia

[3] The On-Line Encyclopedia of Integer Sequences : A005408 = The odd numbers: a(n) = 2n+1

[4] uency : elephant valley

[5] Visual Guide To Patterns In The Mandelbrot Set by Miqel

[6] teger number in wikipedia

[7] Integer overflow in wikipedia

[8] Mandel Set Combinatorics : Principal Series

[9] scholarpedia : Siegel_disks , Quadratic_Siegel_disks, Digitation

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

Fractals/Mathematics/sequences

Contents

Difference between sequences, orders and series

types of sequences

Integer sequences

Fraction sequences

Farey sequence

Sequences and orders on the parameter plane

Sequences of Misiurewicz points

degree

Myrberg-Feigenbaum point

Sharkovsky ordering

Period doubling scenario

sequence of fraction in the elephant valley

sequence of parabolic points on the boundary of main cardioid

sequence from Siegel disk to Leau-Fatou flower

1 over 2

1 over 3

sequence of fractions tending to the golden mean ( Golden Ratio Conjugate )

Sequence on the dynamic plane

More

References

Navigation menu

Fractals/Mathematics/sequences

Difference between sequences, orders and series

types of sequences

Integer sequences

Fraction sequences

Farey sequence

Sequences and orders on the parameter plane

Sequences of Misiurewicz points

degree

Myrberg-Feigenbaum point

Sharkovsky ordering

Period doubling scenario

sequence of fraction in the elephant valley

sequence of parabolic points on the boundary of main cardioid

sequence from Siegel disk to Leau-Fatou flower

1 over 2

1 over 3

sequence of fractions tending to the golden mean ( Golden Ratio Conjugate )

Sequence on the dynamic plane

More

References

Navigation menu

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