Difference between sequences, orders and series

types of sequences

Integer sequences

Fraction sequences

Farey sequence

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.

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.

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 $c=M_{23,2}=-0.743291890852430+0.131240552308798i$

- 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 $\mathbf {MF} _{1/2}=c_{0}=-1.401155189093314712...$
sequence of root points for periods $p=1*3^{n}$ and the limit point of the sequence is the Myrberg-Feigenbaum point $\mathbf {MF} =c_{0}=-1.7864402555636388747...$

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{array}{cccccccc}3&5&7&9&11&\ldots &(2n+1)\cdot 2^{0}&\ldots \\3\cdot 2&5\cdot 2&7\cdot 2&9\cdot 2&11\cdot 2&\ldots &(2n+1)\cdot 2^{1}&\ldots \\3\cdot 2^{2}&5\cdot 2^{2}&7\cdot 2^{2}&9\cdot 2^{2}&11\cdot 2^{2}&\ldots &(2n+1)\cdot 2^{2}&\ldots \\3\cdot 2^{3}&5\cdot 2^{3}&7\cdot 2^{3}&9\cdot 2^{3}&11\cdot 2^{3}&\ldots &(2n+1)\cdot 2^{3}&\ldots \\&\vdots \\\ldots &2^{n}&\ldots &2^{4}&2^{3}&2^{2}&2&1\end{array}}

"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]

(2n+1)\cdot 2^{0}\prec (2n+1)\cdot 2^{1}\prec (2n+1)\cdot 2^{2}\prec \ldots \prec 2^{n}

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

B\prec MF\prec P

B_{0}\prec B_{1}\prec ...\prec MF\prec ...\prec 2^{1}\prec 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\mathop {=} 1}^{n}{\frac {3}{10^{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	$c_{t}={\frac {2e^{2\pi it}-e^{4\pi it}}{4}}$
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

$t=[0;3,10^{n},g]=0+{\cfrac {1}{3+{\cfrac {1}{10^{n}+{\cfrac {1}{g}}}}}}$

n	t	$c_{t}={\frac {2e^{2\pi it}-e^{4\pi it}}{4}}$
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 )

Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers

Golden Mean Quadratic Siegel Disc

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

[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

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