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Group_theory_for_Fractals.pdf

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Fractals/Mathematics/group Here one can find examples of relation between fractals and groups. [1] “Group theory is very useful in that it finds common- alities among disparate things through the power of abstraction.” [2] There are important connections between the algebraic structure of self-similar groups and the dynamical prop- erties 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 abstrac
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  Fractals/Mathematics/group Here one can find examples of relation between fractalsand groups. [1] “Group theory is very useful in that it finds common-alities among disparate things through the power ofabstraction.” [2] There are important connections between the algebraicstructure of self-similar groups and the dynamical prop-erties 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 sub-set of all elements x of the set X which are equivalent toa [ a ] X  =  { x  ∈  X  | x  ∼  a } . 4 Expansion p-adic digit a natural number between 0 and p − 1 (inclu-sive).A p-adic integer is a sequence of p-adic digits : ..a i ..a 1 a 0 n-adic expansion of number  [3] binary integer or dyadic integer or 2-adic integer : n ∑ i =0 a i 2 i where a is an element of binary alphabet X ={0,1} = {0,.. ,2-1) 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 au-tomaton / generators of the fundamental group1  2  11 VIEWER ã  the edges (lines with arrows) show the state transi-tion, 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 non-empty set ã ⊥ denotes a binary operation called the  group oper-ation : ⊥ :  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 com-bined with its inverse under the group operation, the re-sult 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 mat-ter which end of the list we start with” [9] Automaton group  = Group generated by automaton FR = Functionally Recursive GroupIMG = Iterated Monodromy Group  [10][11] Self-similar  group 7 Machine Finite State Machines [12] ã  Mealy machine [13] ã  Moore machine 8 Polynomial postcritically finite  polynomial : the orbit of the criti-cal 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 dis-joint 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” ), “testin-stall.g” ) );Load package fr by Laurent Bartholdi  [21] LoadPackage(“fr”);Run fr test :Read( Filename( DirectoriesLibrary( “pkg/fr/tst” ),“testall.g” ) );GAPandfrpackagecanuseexternalprogramslikeMan-del, ImageMagic, graphviz or rsvg-view to draw and dis-play 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 rsvg-view [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 file-name :Draw(NucleusMachine(BasilicaGroup), extquoted-bla.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 [la-bel= extquotedbl2/2”,color=blue]; c -> a [la-bel= extquotedbl1/1”,color=red]; c -> e [la-bel= extquotedbl2/2”,color=blue]; d -> a [la-bel= extquotedbl1/2”,color=red]; d -> b [la-bel= extquotedbl2/1”,color=blue]; e -> c [la-bel= extquotedbl1/2”,color=red]; e -> a [la-bel= extquotedbl2/1”,color=blue]; f -> a [la-bel= extquotedbl1/2”,color=red]; f -> g [label=extquotedbl2/1”,color=blue]; g -> f [label= ex-tquotedbl1/2”,color=red]; g -> a [label= extquot-edbl2/1”,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 .IncasemachineistheIMGmachineofaunicriticalpoly-nomial, this function computes the external angle landingat the critical value.gap> z := Indeterminate(COMPLEX_FIELD, ex-tquotedblz”); 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 computesthe equivalence class of that external angle underExternalAnglesRelation [ 1/3, 2/3 ]Another example :gap> m:= PolynomialIMGMachine(2,[1/7 # the ma-chine descibing the rabbit : degree=2, gap> ExternalAn-gle(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 ele-ment: FRElement(...,f3) Relator: f3*f2*f1 gap> Dis-play(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: FREle-ment(...,f4) Relator: f4*f3*f2*f1 13.3.2 PostCriticalMachine PostCriticalMachine(f)Returns: The Mealy machine of f ’s post-critical orbit.This function constructs a Mealy machine P on the al-phabet [1], which describes the post-critical set of f .gap>z:=Indeterminate(Rationals, extquotedblz”);; gap>m := PostCriticalMachine(z^2); <Mealy machine on al-phabet [ 1 ] with 2 states> gap> Display(m); | 1 ---+-----+a | a,1 b | b,1 ---+-----+ gap> Correspondence(m); [ 0, in-finity ]It is in fact an oriented graph with constant out-degree 1.Draw(m); gap> m := PostCriticalMachine(z^2-1);; Dis-play(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)
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