Fractals/Mathematics/group/Basilica group

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]


ComputationEdit

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

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


FREdit

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', '*' ]

ReferencesEdit

  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
Last modified on 19 January 2013, at 07:54