Fractals/Mathematics/group
Here one can ﬁnd examples of relation between fractalsand groups.
[1]
“Group theory is very useful in that it ﬁnds commonalities among disparate things through the power ofabstraction.”
[2]
There are important connections between the algebraicstructure of selfsimilar groups and the dynamical properties of the polynomials .
1 Alphabet
Alphabet
X
it is a ﬁnite set of symbols x :
X
=
{
x
1
,x
2
,..,x
n
}
2 Automaton
Automaton is the basic abstract mathematical model ofsequential machine. Diﬀerent types of automata :
ã
recognition automata,
ã
Turing machines
ã
Moore machines
ã
Mealy machines,
ã
cellular automata
ã
pushdown automataAutomaton has two visual presentations:
ã
a ﬂow table, which describes the transitions to thenext states and the outputs,
ã
a state diagram.
3 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 toa
[
a
]
X
=
{
x
∈
X

x
∼
a
}
.
4 Expansion
padic digit a natural number between 0 and p − 1 (inclusive).A padic integer is a sequence of padic digits :
..a
i
..a
1
a
0
nadic expansion of number
[3]
binary integer or dyadic integer or 2adic integer :
n
∑
i
=0
a
i
2
i
where a is an element of binary alphabet X ={0,1} = {0,.. ,21)
5 Graph
The Schreier graphsMoorediagramoftheautomaton
(orthestatediagramfor a Moore machine) it is a directed labeled graph with :
ã
the vertices ( nodes) identiﬁed with the states of automaton / generators of the fundamental group1
2
11 VIEWER
ã
the edges (lines with arrows) show the state transition,
ã
labels: (input,output)areadirectedpairofelementsin X
6 Group
“A group is a collection of objects that obey a strict set ofrules when acted upon by an operation.”
[4]
A
group
G
is the algebraic structure :
G
=
{
A,
⊥}
, where :
ã
A
is a nonempty set
ã ⊥
denotes a binary operation called the
group operation
:
⊥
:
A
×
A
→
A,
which must obey the following rules (or axioms) :
ã
Closure,
ã
Associativity
ã
Identity
ã
Inverse
ã
Commutativity
[5]
Identity
: extquotedbl the group must have an elementthat serves as the Identity. The characteristic feature ofthe Identity is that when it is combined with any othermemberunderthegroupoperation, itleavesthatmemberunchanged.”
[6]
Inverse
: extquotedbl each member or element of thegroup must have an inverse. When a member is combined with its inverse under the group operation, the result is the Identity extquotedbl
[7]
Closure
: “This meansthat whenever two group members are combined underthe group operation, the result is another member of thegroup”
[8]
Associativity
: “if we take a list of three or more groupmembers and combine them two at a time, it doesn’t matter which end of the list we start with”
[9]
Automaton group
= Group generated by automaton
FR = Functionally Recursive GroupIMG = Iterated Monodromy Group
[10][11]
Selfsimilar
group
7 Machine
Finite State Machines
[12]
ã
Mealy machine
[13]
ã
Moore machine
8 Polynomial
postcritically ﬁnite
polynomial : the orbit of the critical point is ﬁnite. It is when critical point is periodic orpreperiodic.
[14]
9 Relation
Equivalence relation
~ over/on the set X
ã
itisabinaryrelationrelationonXwhichisreﬂexive,symmetric, and transitive
ã
it induces partition P of a set X
[15]
into several disjoint subsets, called equivalence classes
10 Sequence
ks = kneading sequence(s)
11 Viewer
applet viewer
is a standalone, command line programfrom Sun to run Java applets. It should be available in astandard executable path of your computer.
13.1 Draw
3Examples :
ã
ﬁrefox
ã
appletviewer
[16]
12 Word
WordwoveralphabetXisanysequenceofsymbolsfromalphabet X. Word can be :
ã
inﬁnite
ã
ﬁnite
w
=
x
1
x
2
..x
n
ã
empty = the word of length zero :
w
=
∅
=
ϵ
0
w
denotes word beginning with
0
“The iterated monodromy groups of quadratic rationalmaps with size of postcritical set at most 3, arranged in atable.The algebraic structure of most of them is not yet wellunderstood.”
[17]
Where :
ã
σ
is a permutation
13 GAP
GAP
[19]
is a CAS software
[20]
. To run :/usr/share/gap/bin/gap.shIf 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
[21]
LoadPackage(“fr”);Run fr test :Read( Filename( DirectoriesLibrary( “pkg/fr/tst” ),“testall.g” ) );GAPandfrpackagecanuseexternalprogramslikeMandel, ImageMagic, graphviz or rsvgview to draw and display images.
13.1 Draw
Draw
[22]
Draw(NucleusMachine(BasilicaGroup));
ã
creates graph description of the m (Mealy machineor element m ).It is converted to Postscript using theprogram dot from the graphviz package
[23]
ã
displays image in a separate X window using thecommand lin program display ( from ImageMagic)or rsvgview
[24]
. This works on UNIX systems.One can right click on image to see local menu of displayprogram.If a second argument of Draw function ( ﬁlename) ispresent, the graph is saved, in dot format, under that ﬁlename :Draw(NucleusMachine(BasilicaGroup), extquotedbla.dot”);Saves output to a.dot ﬁle. Dot ﬁle is a text ﬁle describinggraph 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= extquotedbl1/1”,color=red]; a > a [label= extquotedbl2/2”,color=blue]; b > a[label= extquotedbl1/1”,color=red]; b > d [label= extquotedbl2/2”,color=blue]; c > a [label= extquotedbl1/1”,color=red]; c > e [label= extquotedbl2/2”,color=blue]; d > a [label= extquotedbl1/2”,color=red]; d > b [label= extquotedbl2/1”,color=blue]; e > c [label= extquotedbl1/2”,color=red]; e > a [label= extquotedbl2/1”,color=blue]; f > a [label= extquotedbl1/2”,color=red]; f > g [label=extquotedbl2/1”,color=blue]; g > f [label= extquotedbl1/2”,color=red]; g > a [label= extquotedbl2/1”,color=blue]; } This a.dot ﬁle can be covertedto other formats using command line program dot. Forexample in ps ﬁle : dot Tps a.dot o a.psor png ﬁle :dot Tpng a.dot o a.pngor svg :dot Tsvg a.dot o a.svg
4
13 GAP
13.2 External angle
Function from FR package :ExternalAngle(machine)Returns: The external angle identifying machine .IncasemachineistheIMGmachineofaunicriticalpolynomial, this function computes the external angle landingat the critical value.gap> z := Indeterminate(COMPLEX_FIELD, extquotedblz”); z gap> r := P1MapRational(z^21); #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 computesthe equivalence class of that external angle underExternalAnglesRelation [ 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 ]
13.3 Machine
13.3.1 PolynomialIMGMachine
PolynomialIMGMachine(d, per[, pre[, formal]])This function creates a IMG machine that describes atopological polynomial. The polynomial is describedsymbolically in the language of external angles.d is the degree of the polynomial.per is the list of anglespre is the list of preangles.angles are rational numbers, considered modulo 1. Eachentry in per or pre is either a rational (interpreted as anangle), or a list of angles [a1 , . . . , ai ] such that da1 =. . . = dai . The angles in per are angles landing at theroot of a Fatou component, and the angles in pre land atthe other points of Julia set.gap> m:=PolynomialIMGMachine(2,[1/3],[ # theBasilica <FR machine with alphabet [ 1, 2 ] andadder 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>,2f3  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^
−
1,2f2*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
13.3.2 PostCriticalMachine
PostCriticalMachine(f)Returns: The Mealy machine of f ’s postcritical orbit.This function constructs a Mealy machine P on the alphabet [1], which describes the postcritical set of f .gap>z:=Indeterminate(Rationals, extquotedblz”);; 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 outdegree 1.Draw(m); gap> m := PostCriticalMachine(z^21);; Display(m); Correspondence(m);  1 ++ a  c,1 b  b,1c  a,1 ++ [
−
1, inﬁnity, 0 ]
13.4 Kneading Sequence
KneadingSequence(angle)“This function converts a rational angle to a kneadingsequence,
[25]
to describe a quadratic polynomial
[26]
.” (from fr doc )KneadingSequence(1/7);gives :[ 1, 1 ]“If angle is in [1/7, 2/7] and the option marked is set, thekneadingsequenceisdecoratedwithmarkingsinA,B,C.”( from fr doc )KneadingSequence(1/5:marked);gives :[ “A1”, “B1”, “B0” ]
13.5 Rays of root points
ExternalAnglesRelation(degree, n)