Parabolic implosion - from discontinuity to renormalization

α z + . . .
e 2πiα z + λ . f(z) f. = 0 .
e e q z a 2 z + . . . a
Fatou coordinates ,
i ∈ N a
and Horn
i ≥ N
T (w) T (w) 2πi = qe = w 2πiα λ z = + 1O(z a
w + 1 1 2 ) 2 ≠ 0
α ∈ C {0} small
Φand Φ
| arg α|
map
a i ∈ N attr a attr Φ< i ≥ N rep π Φ rep C/Z C/Z mod CZ C E
= e 2πiα z + O(z 2 4
f0
α z + O(z 2 )
)
e 2πiα 0 (z) z + a 2 z 2 + . . .
F 0 (w) = w + 1 + o(1) w = − c f 0 (z) z + holomorphic f(z) O(z f ′ (0) 2 = ) = αe near 2πiα ∈e 2πiα R0 z , + Q
f (0) = 0 f ′ 0
α. (z) . . small holomorphic 0 (0) = near λ 0 f 0 (0) = 0 f 0 ′(0) = w
near 0 near mod Z E
λ = e 2πi ∞ p T (w) F 0 (w) w = 1w Φ1 + o(1) attr Φ w = rep C/Z − c λ
) iα = , fα 0
e(z) 2πiα small =
,
λz
α+ asmall
2 z 2 + . . . f 0 (z) = λz + a 2 z 2 + . . .
f0
w Cnear 0 near ∞ T
e f
q λ = mod 1 Za 2 ≠E0
0 R 0 f 0 R 0 f
2πiα , αf(z) αsmall
∈= a
C e {0} small and | arg α| < π i 2πiα ∈ Nz +
a i O(z
≥ N 2 )
f0
f f 0 R 0 f 0 R 0 f 0 (z) = z + O(z 2 0 (z) = e 2πi p q z + a 2 z 2 + . . . f
C {0} small and | arg α| < π 0 (z) = e 2πi p q z + a 2 z 2 + . . .
} small and | arg α| < π 4
)
{0} small f αf(z) ′ F 0
(0) ∈and R= e | 2πiα (w) = w + 1 2πiα arg4α| z + < f. 0
. π + o(1) w = − c near 0 near ∞ T (w) = w + 1 Φ attr
Q , α small . 4
z
f R 0 f 0 R 0 f 0 (z) = z + O(z 2 0 (z) = z + a 2 z 2 + mod . . . Z Ef 0 (z) = z + a 2 z 2 f0
4
E f + . . .
)
R Q
f Rf
fRf(z) Rf
= e 2πiβ z
Rf(z)
+ O(z 2 )
λ = e f(z) = e 2πiα z + O(z 2 )
Z
2πi p q
β = − 1 λ = 1 2 ≠ λ = e 2πi p q
Q αa ∈ i ∈C N f {0} a i ≥small N and λ | arg 1 α| a 2 < π 0 R 0 f 0 R 0 f 0 (z) = z + O(z ≠ 2 0)
α
mod Z f Rf Rf(z) 4= e 2πiβ z + O(z 2 ) β = − 1 ≥ N
α mod Z
N f(z) a i = ≥e 2πiα N
f ′ z + . . .
(0) e 2πiα f(z) = e 2πiα z + . . .
f Φ attr (z)
, α small
Φ attr (z)
attr (z)
+ 1 f(z) + o(1) = e
near 0 2πiα z w + = O(z− c Rf Rf(z) = e
a i ≥ N αF ∈ 0 (w) R = Qw + 1 + o(1)
2πiβ z + O(z
w = − c 2 ) β = −
near 0 1
near ∞ 2 )
) = w + 1 + o(1) w T (w) = − c α mod near Z ∞ T (w) = w + 1
near f(z) = 0 e
w 2πiα near z + O(z ∞
+ near 1 0Φ 2 ) T w
mod Z wE (w) = w + 1 Φ attr Φ rep C/Z C
Φ f0 attr (f(z)) attr near
α C {0} small and | arg α| < π Φ∞ rep Φ attr
T (f(z)) C/Z (w) = Cw w
+ attr
1(z) Φ+ attr 1
Φ rep rep (f(
= f 0 w
Z
f ′ (0) + 1
E= + e 2πiα o(1) , α small w = − c = Φ attr (z) + 1 Φ rep (f(z)) = Φ rep (z) + 1 Φ ... (f 0 (z)) = Φ ... (z) + 1
a i ∈ N a i ≥ Nf ′ (0) = near e
(z)) = Φ f0
rep (z) 2πiα , 0α small near ∞ T (w) = w + 1 Φ attr Φ rep C
Φnear rep ∞C/Z T (w) C = w + 1 Φ attr
w Φ rep attr (f(z)) C/Z C= Φ attr (z) 4 + Φ rep (f(z)) = Φ rep (z) + 1
E E f (z) = Φ attr attr Φ(f(z)) = Φ
R −1
f0
f 0 R
rep
att
α ∈ C {0} small and | argα α| ∈< C π
R α ∈ R Q 4
1 0 f 0 R 0 f 0 (z) = z + O(z E 2 {0} small and | arg
f (z) ) = Φ attr ◦ Φ −1
α| < π 0 f 0 (z) = z + E 0 f 0 R
O(z 2 0 f 0 (z) = z + O(z 2 )
)
F 0 (w) = w + 1 + o(1) w = − c f0 (z) = Φ attr ◦ Φ −1
rep χ f χ f (z) = z − 1 Rf = χ f ◦ E f
near40α
near ∞ T (w) = w + 1 Φ
w rep
0f 0 α ∈ R 0 mod
Q
α ∈ R Q
Rf(z) = e f P ◦ ϕ
E −1
1
f = P ◦ ϕ −1
f (z) = Φ attr ◦ Φ
Rf β = − 1 2πiβ f 0 (z) Z= z + O(z 2 )
z +
Rf f = EP O(z 2 )
Rf(z)
β =
=
−
e 12πiβ a
α
mod Z
i ∈ NRf(z) a i ≥ N= e 2πiβ z + O(z 2 ) β = − 1 α
mod Z
α mod z + Z O(z 2 ) β = − 1 f0 ◦ ϕ −1
α mod Z
a
α mod 1
=
f Z
i ∈ N a i ≥ N
f= ΦRf(z) f Q ◦ ϕ −1
attr Φ+ = 1e 2πiβ Φ rep
z + (f(z)) O(z 2 = ) where a i ∈ N
F f Q ◦ ϕ −1 0 (w) w + 1 + o(1) w = F− c Φ rep
β (z) = − 1 0 Rf + α1 mod Φ(f(z)) Z
attr (f(z)) 0 f= Q ◦ 0 Rϕ −1
= Φ 0 f 0 (z) = z + O(z
attr (z) + 1 Φ 2 )
rep 0 (w) = near w + 01 + near o(1) ∞w = T (w) − c = Φ(z) Φ rep + (z) 1 + 1 Φ ... (f 0 (z)) = Φ ... (
0 ∞ (−∞, = w near + 0 1Φnear attr ∞Φ rep T (w) C/Z = = Φ a 1 ±
0 ∞ f (−∞, 1 −1] P ◦ ϕ −1 w + C1 Φ attr
r(f(z)) = rep Φ(z) 1 Φ(f(z)) w −1]
Φ(z) 1
mod Z
attr (z) + 1 Φ rep (f(z)) = Φ rep (z) + w1 Φ(f(z)) = Φ(z) + 1
f0
mod Z E f0
rep(z) (z)) ttr ◦ + Φ= 1
−1
rep Φ attr
fE f0 (z) (z) Rf + = 1 ΦRf(z) attr Φ rep
◦(f(z)) Φ= −1 e 2πiβ rep = z Φ+ rep
O(z (z) 2 + ) 1 β = Φ
a ...
− 1 (f
2 ± α 0 (z)) mod = Z Φ ... (z) + 1
C R Z Q D H R Q C/Z C ∗ D ∗ C C D (−∞, −1]