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PureMathematics
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PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1111204
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CunjiYang
CollegeofMathematicsandCompute,DaliUniversity,DaliYunnan
Received:Oct.6
th
,2021;accepted:Nov.9
th
,2021;published:Nov.16
th
,2021
Abstract
Inthispaper,westudytheconnectivityandtheHausdorffdimensionofJuliasetsin
triangularlatticeIsingmodel.
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DOI:10.12677/pm.2021.1111204
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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/
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<λ<
p
4
p
−
1
10
4
ln
p
,
K
dim
H
J
(
f
p,λ
)
≥
1
−
(
30lnln
p
ln
p
).
Ω
•
ü
ë
Ï
•
,
U
•
∂
Ω
•
.
f
•
½
Â
3
U
þ
X
N
,
¡
Ω
•
RB-
•
,
X
J
÷
v
e
^
‡
µ
1
¤
f
(
∂
Ω) =
∂
Ω,
2
¤
f
(
U
T
Ω)
⊂
Ω,
3
¤
T
∞
k
=0
f
−
k
(
U
T
Ω) =
∂
Ω.
Przytycki,Urbanski
Ú
Zdunik[17,18]
X
e
(
J
µ
Ú
n
2.3
e
Ω
•
RB-
•
,
K
dim
H
(Ω)
>
1.
3.
½
n
1
y
²
d
(1)
f
(
z
) = 2
z
exp(4
z
)
−
(
−
1)
exp(4
z
)
−
(
−
3)
2
ü
>
|
f
(
z
)
|
= 2
|
z
|
|
exp(4
z
)
−
(
−
1)
|
2
|
exp(4
z
)
−
(
−
3)
|
2
(2)
Re
(exp(4
z
))
≤−
2
ž
,
|
exp((4
z
))
−
(
−
1)
|≥|
exp((4
z
))
−
(
−
3)
|
,
d
(2)
|
f
(
z
)
|≥
2
|
z
|
,
Ï
d
|
f
n
(
z
)
|≥
2
n
|
z
|
.
Re
(exp(4
z
))
≥−
2
ž
,
|
exp(4
z
)
−
(
−
1)
|≤|
exp(4
z
)
−
(
−
3)
|
,
d
(2)
|
f
(
z
)
|≤
2
|
z
|
,
Ï
d
|
f
n
(
z
)
|≤
2
n
|
z
|
.
-
z
=
x
+
yi
,
K
Re
(exp(4
z
)) = exp(4
x
)cos(4
y
).
(I)
Re
(exp(4
z
))
≤−
2 ,
=
exp(4
x
)cos(4
y
)
≤−
2
ž
,
cos(4
y
)
≤
−
2
exp(4
x
)
,
Ï
d
,cos(4
y
)
<
0
…
exp(4
x
)
>
2,
=
DOI:10.12677/pm.2021.11112041814
n
Ø
ê
Æ
•
Ä
kπ
2
+
π
8
<y<
kπ
2
+
3
π
8
,k
∈
Z
…
x>
1
4
ln2
.
‡
ƒ
,
kπ
2
+
π
8
<y<
kπ
2
+
3
π
8
,k
∈
Z
,x>
1
4
ln2
…
exp(4
x
)cos(4
y
)
≤−
2
,
ž
Re
(exp(4
z
))
≤−
2
,
|
f
n
(
z
)
|≥
2
n
|
z
|→∞
,n
→∞
.
Ï
d
,
z
∈
F
(
f
).
(II)
Re
(exp(4
z
))
>
−
2,
=
exp(4
x
)cos(4
y
)
>
−
2
ž
,
Œ
2
©
üa
?
Ø
1)
cos(4
y
)
≥
0,
=
kπ
2
−
π
8
≤
y
≤
kπ
2
+
π
8
,k
∈
Z
ž
,
é
?
¿
x
∈
R
Ñ
k
exp(4
x
)cos(4
y
)
>
−
2.
2)
cos(4
y
)
<
0,
=
kπ
2
+
π
8
<y<
kπ
2
+
3
π
8
,k
∈
Z
ž
,exp(4
x
)
<
−
2
cos(4
y
)
.
e
¡
?
˜
Ú
·
‚
é
÷
v
exp(4
x
)cos(4
y
)
>
−
2
E
ê
z
=
x
+
yi
?
1
?
Ø
.
d
(2)
|
f
(
z
)
|
= 2
|
z
|
|
exp(4
z
)+1
|
2
|
exp(4
z
)+3
|
2
=
|
x
+
yi
|
2
|
exp(4
x
)cos(4
y
)+1+
i
exp(4
x
)sin(4
y
)
|
2
|
exp(4
x
)cos(4
y
)+3+
i
exp(4
x
)sin(4
y
)
|
2
.
z
{
|
f
(
z
)
|
=
|
x
+
yi
|
2exp(8
x
)+4exp(4
x
)cos(4
y
)+2
exp(8
x
)+6exp(4
x
)cos(4
y
)+9
e
|
f
(
z
)
|
<
|
z
|
,
K
d
þ
ª
exp(8
x
)
−
2exp(4
x
)cos(4
y
)
−
7
<
0.
‡
ƒ
½
,
.
Ï
d
,
z
=
x
+
yi
÷
v
exp(8
x
)
−
2exp(4
x
)cos(4
y
)
−
7
<
0
exp(4
x
)cos(4
y
)
>
−
2
(3)
ž
|
f
(
z
)
|
<
|
z
|
,
|
f
n
(
z
)
|
<
|
z
|
,
z
∈
F
(
f
).
DOI:10.12677/pm.2021.11112041815
n
Ø
ê
Æ
•
Ä
•
§
|
(3)
Œ
Ó
)
C
/
•
exp(4
x
)cos(4
y
)
>
1
2
exp(8
x
)
−
7
2
exp(4
x
)cos(4
y
)
>
−
2
(4)
e
1
2
exp(8
x
)
−
7
2
>
−
2,
=
x>
1
8
ln3
ž
•
§
|
(4)
Œ
Ó
)
C
/
•
exp(4
x
)cos4
y>
1
2
exp(8
x
)
−
7
2
.
Ï
d
,
z
=
x
+
yi
÷
v
x>
1
8
ln3
…
cos4
y>
1
2
exp(4
x
)
−
7
2
exp(
−
4
x
)
ž
,
z
∈
F
(
f
).
e
1
2
exp(8
x
)
−
7
2
<
−
2,
=
x<
1
8
ln3
ž
,
•
§
|
(4)
Œ
Ó
)
C
/
•
exp(4
x
)cos(4
y
)
>
−
2
.
Ï
d
,
z
=
x
+
yi
÷
v
x<
1
8
ln3
…
cos4
y>
−
2exp(
−
4
x
)
ž
,
z
∈
F
(
f
).
(
Ü
þ
¡
?
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
+
3
π
8
,k
∈
Z
…
cos4
y>
−
2exp(
−
4
x
)
ž
,
z
∈
F
(
f
).
kπ
2
+
π
8
<y<
kπ
2
+
3
π
8
,k
∈
Z
,
x>
1
4
ln2
…
exp(4
x
)cos4
y
≤−
2
ž
,
Re
(exp(4
z
))
≤−
2
,
|
f
n
(
z
)
|≥
2
n
|
z
|→∞
,n
→∞
,
k
= 0.
P
3
x>
1
4
ln2
…
π
8
<y<
3
π
8
S
÷
v
exp(4
x
)cos4
y
≤−
2
:
z
¤
3
«
•
•
D
,
K
d
Ú
n
2.1
•
D
⊆
ϕ
(
f
),
Ï
d
,
d
Ú
n
2.2
dim
H
J
(
f
) =
dim
H
∂ϕ
(
f
)
≥
dim
H
∂D
≥
1
.
e
¡
?
˜
Ú
y
²
dim
H
J
(
f
)
>
1.
Ï
•
f
(
z
)
k
á
5
Ø
Ä:
z
1
= 0,
D
0
•
¹
:
O
ü
ë
Ï
á
5
ØC
•
,
U
•
D
0
S
,
•
.
DOI:10.12677/pm.2021.11112041816
n
Ø
ê
Æ
•
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-
•
,
¤
±
d
Ú
n
2.3
dim
H
(
D
0
)
>
1.
q
Ï
•
∂D
0
⊂
J
(
z
),
¤
±
dim
H
J
(
f
)
≥
dim
H
(
D
0
)
>
1
.
4.
½
n
2
y
²
•
y
²
½
n
2,
·
‚
k
‰
Ñ
¤
I
Ú
n
"
Ú
n
4.1
([19])
æ
X
¼
ê
±
Ï
•
´
ü
ë
Ï
,
ë
Ï
½
Ã
¡
ë
Ï
"
Ú
n
4.2
([13])
f
•
‡
æ
X
¼
ê
,
D
•
f
˜
‡
ØC
•
,
=
F
(
f
)
˜
‡
±
Ï
•
1
±
Ï
©
|
,
K
D
•
ë
Ï
ê
•
0,1,
∞
.
d
Ú
n
4.1
Ú
Ú
n
4.2
Œ
Ú
n
4.3
‡
æ
X
¼
ê
-
½
ØC
•
´
ü
ë
Ï
½
Ã
¡
ë
Ï
"
Ú
n
4.4
([20])
b
C
þ
;
f
8
W
3
b
C
þ
´
ë
Ï
…
=
b
C
\
W
z
‡
©
|
ü
ë
Ï
"
é
c
¡
½
Â
X
e
ü
‡
¼
ê
8
M
=
{
f
:
b
C
→
b
C
|
f
´
‡
æ
X
¼
ê
½
´
g
ê
–
•
2
k
n
¼
ê
}
;
P
=
{
f
|
•
3
˜
‡
Mobius
C
†
M
(
z
)
¦
M
−
1
◦
f
◦
M
(
z
) =
z
k
exp(
S
(
z
)+
T
(
1
z
))
}
,
Ù
¥
k
´
ê
,S(z)
Ú
T(z)
´
¼
ê
"
Baker
X
e
'
u
8
Ü
M
Ú
P
(
Ø
µ
Ú
n
4.5
([21])
f
∈
M
\
P
,
K3
b
C
þ
J
(
f
)
½
ö
´
ü
ë
Ï
½
ö
d
Ø
Œ
ê
©
|
¤
"
½
n
2
¼
ê
f
(
z
)
Julia
8
J
(
f
)
d
Ø
Œ
ê
©
|
¤
"
y
²
k
y
²
f
(
z
)
•
¹
x
¶
z
2
(
z
2
=
1
4
ln
(1+2
√
2))
†
>Ü
©
á
5
ØC
•
D
0
´
Ã
¡
ë
Ï
"
du
D
0
´
f
(
z
)
á
5
ØC
•
,
d
f
(
z
)
9
Ú
n
4.3
•
D
0
´
ü
ë
Ï
½
Ã
¡
ë
Ï
"
D
0
f
•
D
0
0
=
{
z
|
z
∈
D
0
,Rez>
−
δ,
−
i
4
−
δ<Imz<
i
4
+
δ
}
.
d
[6]
¥
f
(
z
)
.
:
˜
?
Ø
9
.
:
l
Ñ
5
•
,
·
ê
δ
Œ
¦
D
0
0
=
¹
z
=
±
πi
4
ü
‡
.
:
"
du
D
0
´
f
(
z
)
á
5
ØC
•
,
•
3
ê
k
≥
1,
¦
f
k
(
D
0
0
)
•
D
0
0
S
ü
ë
Ï
«
•
,
P
•
D
00
0
,
K
f
k
(
D
0
0
) =
D
00
0
,
d
Riemann-Hurwitz
ú
ª
k
χ
(
D
0
0
) =
dχ
(
D
00
0
)
−
Σ
f
,
Ù
¥
d
•
CX
-
ê
,Σ
f
•
.
:
‡
ê
.
du
z
=
±
πi
4
•
f
-
"
:
,
z
= 0
•
f
ü
"
:
,
DOI:10.12677/pm.2021.11112041817
n
Ø
ê
Æ
•
Ä
f
k
CX
-
ê
d
≥
5
k
,
Σ
f
= 2.
d
Riemann-Hurwitz
ú
ª
χ
(
D
0
0
) =
d
−
2
>
1
,
D
0
´
Ã
¡
ë
Ï
"
,
˜
•
¡
,
Œ
y
f
(
z
)
/
∈
P
,
d
Ú
n
4.5,
J
(
f
)
½
ö
´
ü
ë
Ï
½
ö
d
Ø
Œ
ê
©
|
¤
"
b
J
(
f
)
´
ü
ë
Ï
,
K
d
Ú
n
4.4
•
b
C
\
J
(
z
)=
F
(
z
)
z
‡
©
|
ü
ë
Ï
,
ù
†
D
0
Ã
¡
ë
Ï
g
ñ
,
J
(
f
)
d
Ø
Œ
ê
©
|
¤
"
d
ã
1
§
ã
2
§
ã
3
?
˜
Ú
A
y
½
n
1
Ú
2
(
Ø
"
Figure1.
Computesimulationimageoftheverticalrotate
J
(
f
)
ã
1.
Julia
8
^
ž
^
=
90
Ý
O
Ž
Å
[
ã
Figure2.
Computesimulationimageofpart
J
(
f
)
ã
2.
Ü
©
Julia
8
O
Ž
Å
[
ã
DOI:10.12677/pm.2021.11112041818
n
Ø
ê
Æ
•
Ä
Figure3.
Computesimulationimageoflo cal
J
(
f
)
ã
3.
Û
Ü
Julia
8
O
Ž
Å
[
ã
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
(11861005)
"
ë
•
©
z
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