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PureMathematicsnØêÆ,2021,11(11),1810-1820
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1111204
n/¬‚Ising.Julia8
•••ÄÄÄ
ŒnŒÆêƆOŽÅÆ§HŒn
ÂvFϵ2021c106F¶¹^Fϵ2021c119F¶uÙFϵ2021c1116F
Á‡
©•xn/¬‚þIsing.-zC†¼êJulia8ëÏ5ÚHausdorff‘ê"
'…c
Ising.§Julia8§Hausdorf‘ê§ëÏ5
JuliaSetoftheTriangularLatticeIsing
Model
CunjiYang
CollegeofMathematicsandCompute,DaliUniversity,DaliYunnan
Received:Oct.6
th
,2021;accepted:Nov.9
th
,2021;published:Nov.16
th
,2021
Abstract
Inthispaper,westudytheconnectivityandtheHausdorffdimensionofJuliasetsin
triangularlatticeIsingmodel.
©ÙÚ^:•Ä.n/¬‚Ising.Julia8[J].nØêÆ,2021,11(11):1810-1820.
DOI:10.12677/pm.2021.1111204
•Ä
Keywords
IsingModel,JuliaSet,HausdorffDimension,Connectivity
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó9̇(J
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Ÿ,©¼êÿ2•EC¼ê,l«ƒC:3E²¡þŸA"þ-Vl›c“±5
2É-À-z+•{,•ïăC¯KJÑÏ#å»[3]"Cc5,<‚uy-zC†
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Œ±^5?ØÔn¥•kc^ƒCy–"Ùg§Ising.„Œ±^5?ØÜ7k
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ŒUGê\±í2§Œ±,˜‡-‡ÚOÔn.=Potts.[6]"
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^=£Ý•{)ûù˜¯K"‘§w[8]qO(OŽ‘Ising.gu^zrÝ"
•Ä‘>n/¬‚þIsing.§1973c,NiemeijerÚVanLeeuwen[9]ÒJÑ¢˜m
-z+•{"3‘>n/¬‚þn‡g^|¤˜‡¬N,¦‚Ñ-zC†¼ê
f(z) = 2z
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exp(4z)+3
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ùpz=
ε
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,ε,k•~ê,T •§Ý"'u‘>n/¬‚þIsing.-zC†
DOI:10.12677/pm.2021.11112041811nØêÆ
•Ä
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Ù¥Š‰Œ•:x<0ž,
1
16
(exp(4x)−3exp(−4x)) ≤y≤
1
16
(3exp(−4x)−exp(4x));
x>x
2
ž,
1
16
(3exp(−4x)−exp(4x)) ≤y≤
1
16
(exp(4x)−3exp(−4x));
Ù¥x
2
••§16x+ 4 =exp(4x)+
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1
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128z
n
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4
,
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k
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5ÚHausdor‘ê,·‚Xė(Jµ
½n1¼êf(z)Julia8J(f)Hausdor‘êŒu1,=dim
H
J(f) >1"
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S“P•f
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"f5:8Ü¡•fFatou8, P•F(f)"F(f) {8¡•fJulia 8,
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b
C•*¿E²¡,f•æX¼ê,˜‡:z
0
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C¡•f.:§XJf
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0
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C¡•f
ìCŠ§e•3˜^3E²¡þòá-‚Γ§¦÷XΓz→∞ž§f(z)→a"f(z)
k•.ŠÚìCŠ9Ù§‚4•:Ú¡•f(z)ÛÉŠ"k'E)ÛÄåXÚƒ'Vg
Ú5Ÿë„Beardon[10],CarlesonÚGamelin[11],Milnor[12],?4•[13]Úxïu[14]ƒ'©
DOI:10.12677/pm.2021.11112041812nØêÆ
•Ä
z"
PR= {f:
b
C→
b
C|f´gê–•2kn¼ê};
M= {f:
b
C→
b
C|f´‡æX¼ê½´gê–•2kn¼ê};
ϕ(f) = {z∈C: f
n
(z) →∞,n→∞…z/∈J
∞
(f)}Ù¥J
∞
(f) = O
−
(∞)
S
{∞},
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−1
◦f◦M(z) = z
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1
z
))},
Ù¥k´ê,S(z)ÚT(z)´¼ê"
Ún2.1([14])f∈M\R±9U´‡-½•,eU
T
ϕ(f) 6= Φ,KU⊂ϕ(f).
Ún2.2([14])f∈M\R,KJ(f) = ∂ϕ(f).
‰½Cþ˜‡8ÜE,˜
H
s
δ
(E) = inf{
X
i
(diamU
i
)
s
: {U
i
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s
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s
(E)"½Â8ÜEHausdorff‘êdim
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(E)•
dim
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(E) = inf{s: H
s
(E) = 0}= sup{s: H
s
(E) = ∞}.
'u‡æX¼êJulia8Hausdorff‘ê,Stallard[15,16]y²:f(z)•‡æX¼
ê,K0 <dima
H
J(f) ≤2.é–õ•kk•‡4 :æX¼êJulia8Hausdorff‘ê–•1,
=dima
H
J(f)≥1.ù‡(Øéδ(∞,f) >0‡æX¼ê´Ä¤á„–?˜ÚïÄ"f(z)
•k..‡æX¼ê,äká‡4:{a
n
}"‰½˜r
n
3B
n
= B(a
n
,r
n
)þ
|f(z)|∼
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n
|
|z−a
n
|
m
,|f
0
(z)|∼
m|b
n
|
|z−a
n
|
(m+1)
.
XJé¿©Œn
0
,¦B
k
⊂f(B
n
),k,n∈
Q
,Ù¥
Q
={n∈N:n
0
0
≤n≤n
0
}"K
dim
H
J(f) ≥α,Ù¥αdXeØª(½
Σ
∞
n=n
l
0
(r
n
|a
n
|
−
m+1
m
)
α
≥(K
−1
Lmr
1
|b
l
|
−
1
m
(diamB
0
+1))
α
.
Ù¥l∈
Q
,B
0
= B
n
0
,K,L´~ê"

b
B= {f∈B: P(f)
\
J(f) = Φ},f
p,λ
= λΣ
∞
n=p
2
(
1
n
p
−z
−
1
n
p
+z
),p∈N,λ>0,
Stallard[16]y²:
0 <λ<
p
4p−1
10
4
lnp
,p≥6 ž,f
p,λ
∈
b
B;
DOI:10.12677/pm.2021.11112041813nØêÆ
•Ä
XJp≥6,0 <λ<
1
6
p
,Kdim
H
J(f
p,λ
) ≤
1
p
;
XJp≥6,
p
4p−1
10
5
lnp
<λ<
p
4p−1
10
4
lnp
,Kdim
H
J(f
p,λ
) ≥1−(
30lnlnp
lnp
).
Ω•üëÏ•,U•∂Ω•.f•½Â3UþXN,¡Ω•RB-•,XJ÷v
e^‡µ
1¤f(∂Ω) = ∂Ω,
2¤f(U
T
Ω) ⊂Ω,
3¤
T
∞
k=0
f
−k
(U
T
Ω) = ∂Ω.
Przytycki,UrbanskiÚZdunik[17,18]Xe(Jµ
Ún2.3eΩ•RB-•,Kdim
H
(Ω) >1.
3.½n1y²
d(1)
f(z) = 2z

exp(4z)−(−1)
exp(4z)−(−3)

2
ü>
|f(z)|= 2|z|
|exp(4z)−(−1)|
2
|exp(4z)−(−3)|
2
(2)
Re(exp(4z)) ≤−2 ž,
|exp((4z))−(−1)|≥|exp((4z))−(−3)|,
d(2)|f(z)|≥2|z|,Ïd|f
n
(z)|≥2
n
|z|.
Re(exp(4z)) ≥−2 ž,
|exp(4z)−(−1)|≤|exp(4z)−(−3)|,
d(2)|f(z)|≤2|z|,Ïd|f
n
(z)|≤2
n
|z|.
-z= x+yi,KRe(exp(4z)) = exp(4x)cos(4y).
(I)Re(exp(4z)) ≤−2 ,=exp(4x)cos(4y) ≤−2ž,
cos(4y) ≤
−2
exp(4x)
,
Ïd,cos(4y) <0…exp(4x) >2,=
DOI:10.12677/pm.2021.11112041814nØêÆ
•Ä
kπ
2
+
π
8
<y<
kπ
2
+
3π
8
,k∈Z…x>
1
4
ln2.
‡ƒ,
kπ
2
+
π
8
<y<
kπ
2
+
3π
8
,k∈Z,x>
1
4
ln2…exp(4x)cos(4y) ≤−2,
ž
Re(exp(4z)) ≤−2 ,|f
n
(z)|≥2
n
|z|→∞,n→∞.
Ïd,z∈F(f).
(II)Re(exp(4z)) >−2,=exp(4x)cos(4y) >−2ž,Œ2©üa?Ø
1)cos(4y) ≥0,=
kπ
2
−
π
8
≤y≤
kπ
2
+
π
8
,k∈Z
ž,é?¿x∈RÑkexp(4x)cos(4y) >−2.
2)cos(4y) <0,=
kπ
2
+
π
8
<y<
kπ
2
+
3π
8
,k∈Z
ž,exp(4x) <
−2
cos(4y)
.
e¡?˜Ú·‚é÷vexp(4x)cos(4y) >−2Eêz= x+yi?1?Ø.
d(2)
|f(z)|= 2|z|
|exp(4z)+1|
2
|exp(4z)+3|
2
= |x+yi|
2|exp(4x)cos(4y)+1+iexp(4x)sin(4y)|
2
|exp(4x)cos(4y)+3+iexp(4x)sin(4y)|
2
.
z{
|f(z)|= |x+yi|
2exp(8x)+4exp(4x)cos(4y)+2
exp(8x)+6exp(4x)cos(4y)+9
e|f(z)|<|z|,Kdþªexp(8x)−2exp(4x)cos(4y)−7 <0.‡ƒ½,.Ïd,z= x+yi
÷v







exp(8x)−2exp(4x)cos(4y)−7 <0
exp(4x)cos(4y) >−2
(3)
ž|f(z)|<|z|,|f
n
(z)|<|z|,z∈F(f).
DOI:10.12677/pm.2021.11112041815nØêÆ
•Ä
•§|(3)ŒÓ)C/•





exp(4x)cos(4y) >
1
2
exp(8x)−
7
2
exp(4x)cos(4y) >−2
(4)
e
1
2
exp(8x)−
7
2
>−2,=x>
1
8
ln3ž•§|(4)ŒÓ)C/•
exp(4x)cos4y>
1
2
exp(8x)−
7
2
.
Ïd,z= x+yi÷vx>
1
8
ln3…
cos4y>
1
2
exp(4x)−
7
2
exp(−4x)
ž,z∈F(f).
e
1
2
exp(8x)−
7
2
<−2,=x<
1
8
ln3ž,•§|(4) ŒÓ)C/•exp(4x)cos(4y) >−2.Ïd,
z= x+yi÷vx<
1
8
ln3…cos4y>−2exp(−4x)ž,z∈F(f).
(Üþ¡?Ø,x<
1
8
ln3,
kπ
2
−
π
8
≤y≤
kπ
2
+
π
8
,k∈Z
ž,z∈F(f).

kπ
2
+
π
8
<y<
kπ
2
+
3π
8
,k∈Z
…cos4y>−2exp(−4x)ž,z∈F(f).

kπ
2
+
π
8
<y<
kπ
2
+
3π
8
,k∈Z,
x>
1
4
ln2…exp(4x)cos4y≤−2ž,
Re(exp(4z)) ≤−2 ,|f
n
(z)|≥2
n
|z|→∞,n→∞,
k= 0.P3x>
1
4
ln2…
π
8
<y<
3π
8
S÷vexp(4x)cos4y≤−2:z¤3«••D,Kd
Ún2.1•D⊆ϕ(f),Ïd,dÚn2.2
dim
H
J(f) = dim
H
∂ϕ(f) ≥dim
H
∂D≥1.
e¡?˜Úy²dim
H
J(f) >1.
Ï•f(z)ká5ØÄ:z
1
= 0,D
0
•¹:OüëÏá5ØC•,U•D
0
S,•.
DOI:10.12677/pm.2021.11112041816nØêÆ
•Ä
K
1¤f(∂D
0
) = ∂D
0
,
2¤f(U
T
D
0
) ⊂D
0
,
3¤
T
∞
k=0
f
−k
(U
T
D
0
) = ∂D
0
.
D
0
•RB-•,¤±dÚn2.3dim
H
(D
0
)>1.q쥶D
0
⊂J(z),¤±dim
H
J(f)≥
dim
H
(D
0
) >1.
4.½n2y²
•y²½n2,·‚k‰Ñ¤IÚn"
Ún4.1([19])æX¼ê±Ï•´üëÏ,ëϽáëÏ"
Ún4.2([13])f•‡æX¼ê,D•f˜‡ØC•,=F(f)˜‡±Ï•1±Ï
©|,KD•ëÏê•0,1,∞.
dÚn4.1ÚÚn4.2Œ
Ún4.3‡æX¼ê-½ØC•´üëϽáëÏ"
Ún4.4([20])
b
Cþ;f8W3
b
Cþ´ëÏ…=
b
C\Wz‡©|üëÏ"
éc¡½ÂXeü‡¼ê8
M= {f:
b
C→
b
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BakerXe'u8ÜMÚP(ص
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,
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DOI:10.12677/pm.2021.11112041817nØêÆ
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k
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f
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0
0
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0
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Figure1.ComputesimulationimageoftheverticalrotateJ(f)
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Figure2.ComputesimulationimageofpartJ(f)
ã2.Ü©Julia8OŽÅ[ã
DOI:10.12677/pm.2021.11112041818nØêÆ
•Ä
Figure3.Computesimulationimageoflo calJ(f)
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[1]Yang,C.N.andLee,T.D.(1952)StatisticalTheoryofEquationsofStateandPhaseTransi-
tions.I.TheoryofCondensation.PhysicalReview,87,404-409.
https://doi.org/10.1103/PhysRev.87.404
[2]Yang,C.N.andLee,T.D.(1952)StatisticalTheoryofEquationsofStateandPhaseTransi-
tions.II.LatticeGasandIsingModel.PhysicalReview,87,410-419.
https://doi.org/10.1103/PhysRev.87.410
[3]McMullen,C.T.(1994)ComplexDynamicsandRenormalization.PrincetonUniversityPress,
Princeton,NJ.
[4]Qiao,J.Y.(2005)JuliaSetsandComplexSingularitiesinDiamond-LikeHierarchicalPotts
Models.ScienceinChinaSeriesA:Mathematics,48,388-412.
[5]Qiao,J.Y.andLi,Y.H.(2001)OnConnectivityofJuliaSetsofYang-LeeZeros. Communica-
tionsinMathematicalPhysics,222,319-326.https://doi.org/10.1007/s002200100507
[6]Yang,C.J.andWang,S.M.(2020)TheSingularitiesinRenormalizationofTriangularNet
IsingModel.JournalofDaliUniversity,5,1-8.
DOI:10.12677/pm.2021.11112041819nØêÆ
•Ä
[7]Onsager,L.(1944)CrystalStatistics.I.ATwo-DimensionalModelwithanOrder-Disorder
Transition.PhysicalReview,65,117-149.https://doi.org/10.1103/PhysRev.65.117
[8]Yang, C.N.(1952)TheSpontaneousMagnetizationofaTwo-DimensionalIsingModel.Physical
Review,85,808-816.https://doi.org/10.1103/PhysRev.85.808
[9]Niemeijer,Th.and van Leeuwen,J.M.J. (1973) WilsonTheory for SpinSystems ona Triangu-
larLattice.PhysicalReviewLetters,31,1411.https://doi.org/10.1103/PhysRevLett.31.1411
[10]Beardon,A.F.(1991)IterationofRationalFunctions.Springer-Verlag,NewYork,Berlin.
[11]Carleson, L.andGamelin, T.W.(1993)ComplexDynamics.Springer-Verlag, NewYork, Berlin.
https://doi.org/10.1007/978-1-4612-4364-9
[12]Milnor, J. (2006) Dynamicsin OneComplexVariable. 3rdEdition, PrincetonUniversity Press,
NewJersey.
[13]Ren, F.Y. (1996)ComplexAnalysisDynamics. Fudan UniversityPress, Shanghai.(InChinese)
[14]Zheng,J.H.(2006)DynamicsofMeromorphicFunctions.TsinghuaUniversityPress,Beijing.
[15]Stallard,G.M. (1994) The Hausdorff Dimension of Julia Sets of Meromorphic Functions. Jour-
naloftheLondonMathematicalSociety,49,218-295.https://doi.org/10.1112/jlms/49.2.281
[16]Stallard,G.M.(1999)TheHausdorffDimensionofJuliaSetsofMeromorphicFunctionsII.
JournaloftheLondonMathematicalSociety,60,874-859.
https://doi.org/10.1112/S0024610799008029
[17]Prztycki,F.,Urbanski,M.andZdunik,A.(1990)Harmonic,GibbsandHausdorffMeasures
onRepellersforHolomorphicMaps,II.StudiaMathematica,97,189-225.
https://doi.org/10.4064/sm-97-3-189-225
[18]Zdunik,A. (1991) Harmonic Measure versus Hausdorff Measures on Repellers for Holomorphic
Maps.TransactionsoftheAmericanMathematicalSociety,326,633-652.
https://doi.org/10.1090/S0002-9947-1991-1031980-0
[19]Bolsch, A.(1999)Periodic FatouComponents ofMeromorphicFunctions.BulletinoftheLon-
donMathematicalSociety,31,543-555.https://doi.org/10.1112/S0024609399005950
[20]Noshiro,K.(1960)ClusterSets.Spring-Verlag,Berlin,144.
https://doi.org/10.1007/978-3-642-85928-1
[21]Baker,I.N.(1987)WanderingDomainsforMapsofthePuncturedPlane.AnnalesAcademiae
ScientiarumFennicae:SeriesA.I.Mathematica,12,191-198.
https://doi.org/10.5186/aasfm.1987.1204
DOI:10.12677/pm.2021.11112041820nØêÆ

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