Fractals/Mathematics/group/Basilica group

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Basilica JUlia set and external rays
Lamination of Basilica Julia set
NucleusMachine(BasilicaGroup)

Basilica group is :[1]

  • group defined by automatum
  • the iterated monodromy group of the polynomial z^2-1 [2]
  • related with Basilica Julia set : "the scaling limit of the Schreier graphs of its action on level n of T is the basilica"[3]


Computation[edit]

The critical points of the polynomial z^2-1 are \infty and 0 .

The the postcritical set is P = \left \{ 0, -1, \infty \right \}


FR[edit]

predefined by FR package of GAP CAS. Here BinaryKneadingGroup("1") is BasilicaGroup.

gap> BinaryKneadingGroup(1/3)=BasilicaGroup;
true


or :

gap> B := FRGroup("a=<1,b>(1,2)","b=<1,a>",IsFRMealyElement);
<state-closed group over [ 1, 2 ] with 2 generators>
gap> AssignGeneratorVariables(B);
#I  Assigned the global variables [ "a", "b" ]
gap> B=BasilicaGroup;
#I  \=: converting second argument to FR element
#I  \<: converting second argument to FR element
#I  \<: converting second argument to FR element
#I  \=: converting second argument to FR element
#I  \=: converting second argument to FR element
#I  \<: converting second argument to FR element
#I  \<: converting second argument to FR element
#I  \=: converting second argument to FR element
#I  \=: converting first argument to FR element
#I  \=: converting first argument to FR element
#I  \=: converting first argument to FR element
#I  \=: converting first argument to FR element
#I  \=: converting first argument to FR element
#I  \=: converting first argument to FR element
#I  \=: converting first argument to FR element
#I  \=: converting first argument to FR element
true


gap> Size(BasilicaGroup);
infinity
gap> GeneratorsOfGroup(BasilicaGroup);
[ a, b ]
gap> Alphabet(BasilicaGroup);
[ 1, 2 ]
gap> KnownAttributesOfObject(BasilicaGroup);
[ "Name", "Representative", "OneImmutable", "GeneratorsOfMagma", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement", "UnderlyingFRMachine", "Correspondence", 
"AlphabetOfFRSemigroup", "NucleusOfFRSemigroup", "FRGroupPreImageData", "KneadingSequence", "Alphabet" ]
gap> KnownPropertiesOfObject(BasilicaGroup);
[ "IsDuplicateFree", "IsAssociative", "IsSimpleSemigroup", "IsFinitelyGeneratedGroup", "IsStateClosed", "IsBoundedFRSemigroup", "IsAmenableGroup" ]
gap> KneadingSequence(BasilicaGroup);
[/ '1', '*' ]

References[edit]

  1. A Thompson Group for the Basilica by James Belk, Bradley Forrest
  2. R. I. Grigorchuk and A. Zuk (2002a). On a torsion-free weakly branch group defined by a three state automaton. Internat. J. Algebra Comput., 12(1-2):223–246. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000).
  3. Amenability via random walks Laurent Bartholdi and Balint Virag May 19, 2003