分形/数学/群/巴塞利卡群

巴塞利卡朱利亚集和外部射线

巴塞利卡朱利亚集的层化

NucleusMachine(BasilicaGroup)

巴塞利卡群是 :

• 由自动机定义的群
• 多项式 $z^{2}-1$ 的迭代单值群
• 与巴塞利卡朱利亚集相关： "其作用在 T 的 n 级上的 Schreier 图的缩放极限是巴塞利卡"

计算

多项式 $z^{2}-1$ 的临界点是 $\infty$ 和 $0$ 。

后临界集是 $P=\left\{0,-1,\infty \right\}$

FR

由 GAP CAS 的 FR 包预定义。这里 BinaryKneadingGroup("1") 是 BasilicaGroup。

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

或 

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', '*' ]