分形/數學/群/巴塞利卡群

巴塞利卡朱利亞集和外部射線

巴塞利卡朱利亞集的層化

NucleusMachine(BasilicaGroup)

巴塞利卡群是 :

• 由自動機定義的群
• 多項式 $z^{2}-1$ 的迭代單值群
• 與巴塞利卡朱利亞集相關： "其作用在 T 的 n 級上的 Schreier 圖的縮放極限是巴塞利卡"

計算

多項式 $z^{2}-1$ 的臨界點是 $\infty$ 和 $0$ 。

後臨界集是 $P=\left\{0,-1,\infty \right\}$

FR

由 GAP CAS 的 FR 包預定義。這裡 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', '*' ]