Fractals/Mathematics/group

Here one can find examples of relation between fractals and groups.

Introduction

"Group theory is very useful in that it finds commonalities among disparate things through the power of abstraction."

There are important connections between the algebraic structure of self-similar groups and the dynamical properties of the polynomials .

Group theory :

• geometric
• combinatorial
• Computational

Definitions

Alphabet

Alphabet $X$ it is a finite set of symbols x :

$X=\left\{x_{1},x_{2},..,x_{n}\right\}$

Automaton

Automaton is the basic abstract mathematical model of sequential machine. Different types of automata :

• recognition automata,
• Turing machines
• Moore machines
• Mealy machines,
• cellular automata
• pushdown automata

Automaton has two visual presentations:

• a flow table, which describes the transitions to the next states and the outputs,
• a state diagram.

Class

equivalence class of an element a in the set X is the subset of all elements x of the set X which are equivalent to a

$[a]_{X}=\{x\in X|x\sim a\}.$

Expansion

p-adic digit a natural number between 0 and p − 1 (inclusive).

A p-adic integer is a sequence of p-adic digits :

$..a_{i}..a_{1}a_{0}$

n-adic expansion of number

binary integer or dyadic integer or 2-adic integer :

$\sum _{i=0}^{n}a_{i}2^{i}$

where a is an element of binary alphabet X ={0,1} = (0, .. ,2-1)

Graph

The Schreier graphs

Moore diagram of the automaton ( or the state diagram for a Moore machine) it is a directed labeled graph with :

• the vertices ( nodes) identified with the states of automaton / generators of the fundamental group
• the edges (lines with arrows) show the state transition,
• labels : (input,output) are a directed pair of elements in X

Group

"A group is a collection of objects that obey a strict set of rules when acted upon by an operation."

A group $G$ is the algebraic structure :

$G=\{A,\perp \}$

, where :

• $A$ is a non-empty set
• $\perp$ denotes a binary operation called the group operation:

$\perp :A\times {}A\rightarrow {}A,$

which must obey the following rules (or axioms) :

• Closure,
• Associativity
• Identity
• Inverse
• Commutativity

Identity : " the group must have an element that serves as the Identity. The characteristic feature of the Identity is that when it is combined with any other member under the group operation, it leaves that member unchanged."

Inverse : " each member or element of the group must have an inverse. When a member is combined with its inverse under the group operation, the result is the Identity " Closure : "This means that whenever two group members are combined under the group operation, the result is another member of the group"

Associativity : "if we take a list of three or more group members and combine them two at a time, it doesn’t matter which end of the list we start with"

Automaton group = Group generated by automaton

FR = Functionally Recursive Group

IMG = Iterated Monodromy Group

IMG

The iterated monodromy group acts by automorphism on the rooted tree of preimages

$T_{f}:=\bigsqcup _{n\geq 0}f^{-n}(t),$

where a vertex $z\in f^{-n}(t)$ is connected by an edge with $f(z)\in f^{-(n-1)}(t)$.

Machine

Finite State Machines

• Mealy machine
• Moore machine

Polynomial

postcritically finite polynomial : the orbit of the critical point is finite. It is when critical point is periodic or preperiodic.

Relation

Equivalence relation ~ over/on the set X

• it is a binary relation relation on X which is reflexive, symmetric, and transitive
• it induces partition P of a set X into several disjoint subsets, called equivalence classes

Sequence

ks = kneading sequence(s)

Word

Word w over alphabet X is any sequence of symbols from alphabet X. Word can be :

• infinite
• finite $w=x_{1}x_{2}..x_{n}$
• empty = the word of length zero : $w=\varnothing =\epsilon$

$0w$ denotes word beginning with $0$

Examples

"The iterated monodromy groups of quadratic rational maps with size of postcritical set at most 3, arranged in a table.

The algebraic structure of most of them is not yet well understood."

f(z)	standard actions	nucleus	machine	group	comment
$z^{2}$	$t=(1,t)\sigma$		The adding machine	Binary adding group
$z^{2}-1$				Basilica group
$z^{2}-2$					conjugate to the Chebyshev polynomial T2
$-{\frac {z^{3}}{2}}+{\frac {3z}{2}}$				the Hanoi Towers group
$z^{2}+-{\frac {16}{27z}}$				the Sierpinski gasket
$z^{2}+i$				Thompson-Like Group	intermediate growth

Where :

• $\sigma$ is a permutation

Software

• CoxeterTilings in JS : Coxeter group kaleidoscope

Group Explorer

Group Explorer is mathematical visualization software for the abstract algebra classroom.

GAP

GAP is a CAS software. To run :

/usr/share/gap/bin/gap.sh

If the system failed to load packages install libraries, packages and compile them ( nq, pargap, fr). Run test :

 Read( Filename( DirectoriesLibrary( "tst" ), "testinstall.g" ) );

Load package fr by Laurent Bartholdi

LoadPackage("fr");

Run fr test :

Read( Filename( DirectoriesLibrary( "pkg/fr/tst" ), "testall.g" ) );

GAP and fr package can use external programs like Mandel, ImageMagic, graphviz or rsvg-view to draw and display images.

Draw

Draw

 Draw(NucleusMachine(BasilicaGroup));

• creates graph description of the m (Mealy machine or element m ). It is converted to Postscript using the program dot from the graphviz package
• displays image in a separate X window using the command lin program display ( from ImageMagic) or rsvg-view. This works on UNIX systems.

One can right click on image to see local menu of display program.

If a second argument of Draw function ( filename) is present, the graph is saved, in dot format, under that filename :

Draw(NucleusMachine(BasilicaGroup),"a.dot");

Saves output to a.dot file. Dot file is a text file describing graph in dot language.

digraph MealyMachine {
a [shape=circle]
b [shape=circle]
c [shape=circle]
d [shape=circle]
e [shape=circle]
f [shape=circle]
g [shape=circle]
 a -> a [label="1/1",color=red];
 a -> a [label="2/2",color=blue];
 b -> a [label="1/1",color=red];
 b -> d [label="2/2",color=blue];
 c -> a [label="1/1",color=red];
 c -> e [label="2/2",color=blue];
 d -> a [label="1/2",color=red];
 d -> b [label="2/1",color=blue];
 e -> c [label="1/2",color=red];
 e -> a [label="2/1",color=blue];
 f -> a [label="1/2",color=red];
 f -> g [label="2/1",color=blue];
 g -> f [label="1/2",color=red];
 g -> a [label="2/1",color=blue];
}

This a.dot file can be converted to other formats using command line program dot. For example in ps file :

dot -Tps a.dot -o a.ps

or png file :

dot -Tpng a.dot -o a.png

or svg :

dot -Tsvg a.dot -o a.svg

External angle

Function from FR package :

ExternalAngle(machine)

Returns: The external angle identifying machine .

In case machine is the IMG machine of a unicritical polynomial, this function computes the external angle landing at the critical value.

gap> z := Indeterminate(COMPLEX_FIELD,"z");
z
gap> r := P1MapRational(z^2-1); #  Basilica Julia set
<P1 mapping of degree 2>
gap> m:=IMGMachine(r);
<FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f2*f1*f3 ]>
gap> ExternalAngle(m); 
{2/3}
Elements(last); # More precisely, it computes the equivalence class of that external angle under ExternalAnglesRelation
[ 1/3, 2/3 ]

Another example :

gap> m:= PolynomialIMGMachine(2,[1/7]); # the machine descibing the rabbit : degree=2, 
gap> ExternalAngle(m); 
{2/7}
gap> Elements(last);
[ 1/7, 2/7 ]

Machine

PolynomialIMGMachine

PolynomialIMGMachine(d, per[, pre[, formal]])

This function creates a IMG machine that describes a topological polynomial. The polynomial is described symbolically in the language of external angles.

d is the degree of the polynomial.

per is the list of angles

pre is the list of preangles.

angles are rational numbers, considered modulo 1. Each entry in per or pre is either a rational (interpreted as an angle), or a list of angles [a1 , . . . , ai ] such that da1 = . . . = dai . The angles in per are angles landing at the root of a Fatou component, and the angles in pre land at the other points of Julia set.

gap> m:=PolynomialIMGMachine(2,[1/3],[]); # the Basilica
<FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f3*f2*f1 ]>
gap> Display(m);
 G  |      1         2   
----+---------+---------+
 f1 | f1^-1,2   f2*f1,1  
 f2 |    f1,1    <id>,2  
 f3 |    f3,2    <id>,1  
----+---------+---------+
Adding element: FRElement(...,f3)
Relator: f3*f2*f1

gap> Display(PolynomialIMGMachine(2,[],[1/6])); # z^2+I
 G  |      1            2
----+---------------+---------+
 f1 | f1^-1*f2^-1,2   f2*f1,1
 f2 |          f1,1   f3,2
 f3 |          f2,1   <id>,2
 f4 |          f4,2   <id>,1
----+---------------+---------+
Adding element: FRElement(...,f4)
Relator: f4*f3*f2*f1

PostCriticalMachine

PostCriticalMachine(f)

Returns: The Mealy machine of f ’s post-critical orbit. This function constructs a Mealy machine P on the alphabet [1], which describes the post-critical set of f .

gap> z := Indeterminate(Rationals,"z");;
gap> m := PostCriticalMachine(z^2);
<Mealy machine on alphabet [ 1 ] with 2 states>
gap> Display(m);
   | 1 
---+-----+
 a | a,1
 b | b,1
---+-----+
gap> Correspondence(m);
[ 0, infinity ]

It is in fact an oriented graph with constant out-degree 1.

Draw(m);

gap> m := PostCriticalMachine(z^2-1);; Display(m); Correspondence(m);
   | 1
---+-----+
 a | c,1
 b | b,1
 c | a,1
---+-----+
[ -1, infinity, 0 ]

Kneading Sequence

KneadingSequence(angle)

"This function converts a rational angle to a kneading sequence, to describe a quadratic polynomial." ( from fr doc )

KneadingSequence(1/7);

gives :

[ 1, 1 ]

"If angle is in [1/7, 2/7] and the option marked is set, the kneading sequence is decorated with markings in A,B,C." ( from fr doc )

KneadingSequence(1/5:marked);

gives :

[ "A1", "B1", "B0" ]

Rays of root points

ExternalAnglesRelation(degree, n)

" This function returns ... all pairs of external angles that land at a common point of period up to n under angle multiplication by by degree ." ( from fr doc) In other words it gives angles in turns of external rays landing on root points of period n hyperbolic components of Mandelbrot set.

For complex quadratic polynomials ( degree = 2) and period 3 :

ExternalAnglesRelation(2,3);
<equivalence relation on Rationals >

It needs one more command :

EquivalenceRelationPartition(last);

and gives :

[ [ 1/7, 2/7 ], [ 1/3, 2/3 ], [ 3/7, 4/7 ], [ 5/7, 6/7 ] ]

This list has 4 elements :

• 3 pairs of period 3 angles i/7
• 1 pair of period 2 angles i/3

Internal Adress

"This function returns internal addresses for all periodic points of period up to n under angle doubling. These internal addresses describe the prominent hyperbolic components along the path from the landing point to the main cardioid in the Mandelbrot set." (from fr doc )

Compare it with angled internall adresses

Note that angle is a fraction with denominator ( odd number ) :

$d=(2^{n}-1)$

where n is period and d denominator

Period 2

Basilica Julia set for $f_{c}(z)=z^{2}-1$ with external rays 1/3 and 2/3 landing on repelling fixed point alpha

AllInternalAddresses(2);
[ [  ], [ [ 1/3, 2/3, 2 ] ] ]

For period 1 list is a empty.

[]

For period 2 it gives :

[ [ 1/3, 2/3, 2 ] ]

which contain one sublist :

[1/3, 2/3, 2]

It describe period 2 hyperbolic component of Mandelbrot set with external rays 1/3 and 2/3 landing on its root point

Period 3

Rabbit Julia set with 3 external rays : ${\mathcal {R}}_{\frac {1}{7}},{\mathcal {R}}_{\frac {2}{7}},{\mathcal {R}}_{\frac {4}{7}}\,$ landing on fixed point $z=\alpha _{c}\,$

lamination of Rabbit Julia set

AllInternalAddresses(3);
[ [  ], [ [ 1/3, 2/3, 2 ] ], [ [ 1/7, 2/7, 3 ], [ 3/7, 4/7, 3, 1/3, 2/3, 2 ], [ 5/7, 6/7, 3 ] ] ]

For period 3 we have previous period 2 adress and new list for period 3 :

[ [ 1/7, 2/7, 3 ], [ 3/7, 4/7, 3, 1/3, 2/3, 2 ], [ 5/7, 6/7, 3 ] ]

It has 3 elements ( sublists).

First element :

[ 1/7, 2/7, 3 ] 

describes period 3 hyberbolic component with rays 1/7 and 2/7 landing on its root c=0.64951905283833*%i-0.125, which lays on the boundary of main cardioid with internal angle = 1/3

Second element :

[ 3/7, 4/7, 3, 1/3, 2/3, 2 ]

describes period 3 component with rays 3/7 and 4/7 landing on its root point. To find it one must go thru period 2 component.

Because for c in wake (3/7, 4/7) dynamic rays 3/7 and 4/7 land together at a repelling periodic point then it also describes the airplane Julia set " with landing angles [1/3, 2/3] and period 2.

Third element :

[ 5/7, 6/7, 3 ]

describe period 3 hyberbolic component with rays 5/7 and 6/7 landing on its root point ( it is c=-0.64951905283833*%i-0.125, which lays on the boundary of main cardioid with internal angle = 2/3 ).

Spider algorithm

The Spider algorithm constructs polynomials with assigned combinatorics.

Papers :

• Spider Algorithm - paper by John H. Hubbard and Dierk Schleicher
• online article by Claude Heiland-Allen
• original paper by Yuval Fisher
• papers by Tore Moller Jonassen
• papers by Gregory A. Kelsey
• THE MEDUSA ALGORITHM FOR POLYNOMIAL MATINGS by SUZANNE HRUSKA BOYD AND CHRISTIAN HENRIKSEN

See also programs :

• program spider by Yuval Fisher :
• Mandel by Wolf Jung

Goole search : "spider algorithm" polynomial

RationalFunction

RationalFunction(m)

It returns a rational function f whose associated machine is m or a record describing the Thurston obstruction to realizability of f.

gap> m := PolynomialIMGMachine(2,[1/3],[]); # the basilica
<FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f3*f2*f1 ]>
gap>  RationalFunction(m);
z^2-1.

gap> m:=PolynomialIMGMachine(2,[1/7]); # the rabbit
<FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f4) on Group( [ f1, f2, f3, f4 ] )/[ f4*f3*f2*f1 ]>
gap> RationalFunction(m);
z^2 + (-0.12256116687667946+I*0.74486176661942738)

Delaunay triangulation

gap> LoadPackage("fr");
gap> z := Indeterminate(COMPLEX_FIELD);
x_1
gap> f := z^2-1;
x_1^2-1.
gap> m := IMGMachine(f);
<FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f3) on Group( [ f1,\ f2, f3 ] )/[ f2*f1*f3 ]>
gap> Spider(m);
<marked sphere on <triangulation with 6 vertices, 24 edges and 8 faces> marked by [ f1, f2, f3 ] -> [ f1^-1*f2^-1, f1, f2 ]>
gap> Draw(last:julia);

or

gap> LoadPackage("fr");
gap>z := Indeterminate(COMPLEX_FIELD,"z");
gap> r := P1MapRational(z^2-1);
<P1 mapping of degree 2>
gap> IMGMachine(r); #I  Post-critical points at [ P1Point("-1+0i"), P1Point("0+0i"), P1Point("P1infinity") ]
<FR machine with alphabet [ 1 .. 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f2*f1*f3 ]>
gap> Spider(last);
<marked sphere on <triangulation with 6 vertices, 24 edges and 8 faces> marked by [ f1, f2, f3 ] -> [ f1^-1*f2^-1, f\  1, f2 ]>
gap> Draw(last:julia);

Draws dynamical plane on the sphere with marked Delaunay triangulation. One can rotate sphere with mouse in real time.

References

Books:

• Adventures in Group Theory Rubik's Cube, Merlin's Machine, and Other Mathematical Toys by David Joyner

See also

• visual group theory by Nathan Carter