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1
2
{ }
1 : : < ∈ = Δ z z C
{ }
1 : : ≤ ∈ = Δ z z C
{ }
1 : = ∈ = Δ ∂ z z C ) , ( Hol D Δ
- the set of all holomorphic functions on Δ
which map Δ into a set C ⊂ D ) , ( Hol : ) ( Hol Δ Δ = Δ
- the set of all holomorphic self-
M.Elin (joint work with M.Levenshtein, S.Reich, D.Shoikhet) 1 - - PowerPoint PPT Presentation
M.Elin (joint work with M.Levenshtein, S.Reich, D.Shoikhet) 1 - - PowerPoint PPT Presentation
INDAM Workshop on Holomorphic Iteration, Semigroups, and Loewner Chains Rome, 9-12 September 2008 M.Elin (joint work with M.Levenshtein, S.Reich, D.Shoikhet) 1 Notations { } = < : z C : z 1 { } = : z
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SLIDE 3
3
) ( ) (
1
z F z F =
) (
1 z
F
z z F = ) (
z
1
( )
) ( ) (
1 z
F F z F
n n −
=
) (z Fn
………
) (
2 z
F
( )
) ( ) (
2
z F F z F =
n k k n
F G G F F G G F
- =
⇒ =
If then
) ( , Δ ∈ Hol G F
Iterations
Let be a holomorphic self-mapping of the unit disk
) (Δ ∈Hol F
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4
hyperbolic parabolic automorphic nonautomorphic
) Hol(Δ ∈ F
with an interior fixed point elliptic automorphisms self-mappings which are not automorphisms
Classification
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5
) (
1 z
F
) (
2 z
F
) (
3 z
F
) (
4 z
F 1
z τ
∞ → n
F is an elliptic automorphism of Δ
F isn’t an elliptic automorphism of Δ
) (
1 z
F
) (
2 z
F
) (
3 z
F
) (
4 z
F
z
1
τ
∞ → n
( )
τ =
∞ →
z Fn
n
lim
Δ ∈ = τ τ τ , ) ( F Fixed point:
Self-mappings with an interior fixed point
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6
1 ) ( ≤ ′ < τ F
Hyperbolic type: 1 ) ( < ′ < τ F
) (
1 z
F
) (
2 z
F
) (
3 z
F
z
τ
∞ → n
Parabolic type: 1 ) ( = ′ τ F
1
τ
) (
1 z
F ) (
2 z
F ) (
3 z
F
z z
) (
1 z
F ) (
2 z
F ) (
3 z
F
Self-mappings with no interior fixed point
( )
τ =
∞ →
z Fn
n
lim There is a boundary point such that
Δ ∂ ∈ τ
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7
F G G F
- =
⇒ G F, are of the same type: parabolic, hyperbolic, or with an interior fixed point.
) Hol( , Δ ∈ G F which are different from the identity mapping
If F is of hyperbolic type, but
) Aut(Δ ∉ F
→ G is of parabolic type
?
F is of hyp.type, ⇒ Δ ∉ ) Aut( F ) ( Aut Δ ∉
par
G
Commuting self-mappings
- C. C. Cowen (1984)
If
F, G
are not automorphisms, then they are of the same type
- M. H. Heins (1941)
If , then
( )
Δ ∈
hyp
F Aut
( )
Δ ∈
hyp
G Aut
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8
Continuous Semigroups
( ) ( ) ( ) ( )
[
)
D z and s t all for z F F z F i
s t s t
∈ ∞ ∈ =
+
, ,
( ) ( )
D z all for z z F ii
t t
∈ =
+
→0
lim
A family is called a one a one-
- parameter continuous semigroup
parameter continuous semigroup if defines a holomorphic function on Δ, which is called the (infinitesimal) generator of S. (infinitesimal) generator of S. The local continuity condition (ii) implies the differentiability of S with respect to the parameter t ≥ 0 (Berkson&Porta (1978)). The limit
( ) ( )
Δ ∈ = −
+
→
z z f t z F z
t t
, : lim
{ } ( )
t
Δ ⊂ =
≥
Hol F S
t
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9
The point τ is the Denjoy-Wolff point (attractive fixed point)
- f the semigroup generated by f.
Generators and Semigroups
( ) ( )( ) ( )
1 , , f z z z p z z τ τ = − − ∈ Δ
There is a unique point such that with for all
Δ ∈ τ
Δ ∈ z
≥ ) ( Re z p
t f t
e F
) ( '
) ( '
τ
τ
−
=
( )
⎪ ⎩ ⎪ ⎨ ⎧ = − ≠ − − =
− −
1 ) ( ' , ) ( ' ' ) ( ' , ) ( ' ) ( ' ' ) ( ' '
) ( ' ) ( '
τ τ τ τ τ τ
τ τ
f t f f e e f f F
t f t f t
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10
Commuting Semigroups
Suppose that
1 1 1 1
F G G F
- =
?
, , ≥ ∀ = t s F F F F
t s s t
- We say that two semigroups commute
two semigroups commute if
, , ≥ ∀ = t s F G G F
t s s t
- Problem:
1 1 1 1
F G G F
- =
⇒
, , ≥ ∀ = t s F G G F
t s s t
- Let
2 1
} { , } {
≥ ≥
= =
t t t t
G S F S
be two continuous semigroups
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11
2 1
} { , } {
≥ ≥
= =
t t t t
G S F S
are of the same type
1 1 1 1 1 1
and G F F G G F ,
- =
are different from the identity
⇒
hyperbolic parabolic automorphic nonautomorphic
{ }
) Hol(Δ ∈
≥0 t t
F
with an interior fixed point elliptic automorphisms self-mappings which are not automorphisms
SLIDE 12
12
Semigroups with an interior fixed point
2 1
} { , } {
≥ ≥
= =
t t t t
G S F S semigroups of self-mappings which are not automorphisms
1 1 1 1
F G G F
- =
⇒
, , ≥ ∀ = t s F G G F
t s s t
- )
(
1 z
F
) (
2 z
F ) (
3 z
F
) (
4 z
F
z
1 τ
∞ → t
( )
) ( ) ( z m e m z G
t i t
1 1
τ ϕ τ
⋅ =
for some
R ∈ ϕ
2 1
} { , } {
≥ ≥
= =
t t t t
G S F S semigroups of elliptic automorphisms:
1 1 1 1
F G G F
- =
⇒ , , ≥ ∀ = t s F G G F
t s s t
- Corollary 1.
1
} {
≥
=
t t
F S a semigroup of elliptic automorphisms with a common fixed point at Δ ∈ τ
2 1 and ≥
=
t t
G S S } { are commuting iff
) (
1 z
F
) (
2 z
F ) (
3 z
F
) (
4 z
F 1
z τ
∞ → t
( )
. ) ( , C , ) ( ) ( z z z m a z m e m z G
at t
τ τ
τ τ τ
− − = ∈ ⋅ =
−
1
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13
Semigroups with an interior fixed point
2 1
} { , } {
≥ ≥
= =
t t t t
G S F S semigroups of self-mappings which are not automorphisms
1 1 1 1
F G G F
- =
⇒
, , ≥ ∀ = t s F G G F
t s s t
- )
(
1 z
F
) (
2 z
F ) (
3 z
F
) (
4 z
F
z
1 τ
∞ → t 1
} {
≥
=
t t
F S
- a semigroup of elliptic automorphisms
with a common fixed point at Δ ∈ τ
2
} {
≥
=
t t
G S
- a semigroup of self-mappings
) (
1 z
F
) (
2 z
F ) (
3 z
F
) (
4 z
F 1
z τ
∞ → t 1 1 1 1
F G G F
- =
⇒
1 1, s s
F G G F s = ∀ ≥
- 1
1 1 1
F G G F
- =
⇒
, , ≥ ∀ = t s F G G F
t s s t
- Corollary 2.
( )
) ( ) ( z m e m z F
t i t τ ϕ τ
⋅ =
for some
R ∈ ϕ If π
ϕ
is an irrational number, then z e z F
t i t ϕ
= ) (
1 1
≥ ∀ = t F G G F
t t
- , but
2 1,S
S do not commute n π ϕ 2 = , Δ ∈ ∀ ≥ = z z p z zp z g
n
) ( Re ), ( ) (
Example.
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14
Hyperbolic type: 1 ) ( < ′ < τ
t
F
) (
1 z
F
) (
2 z
F
) (
3 z
F
z
τ
∞ → t
1 1 1 1
F G G F
- =
⇒ , , ≥ ∀ = t s F G G F
t s s t
- with an interior fixed point
elliptic automorphisms
hyperbolic
{ }
) Hol( Δ ⊂
≥ t t
F
self-mappings which are not automorphisms
) Aut( ) Aut( Δ ⊄ Δ ⊂
2 1
S S
Semigroups of hyperbolic type
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15
parabolic nonautomorphic automorphic )) ( ), ( ( lim
1
>
+ ∞ →
z F z F
n n n
ρ )) ( ), ( ( lim
1
=
+ ∞ →
z F z F
n n n
ρ Δ ∈ w z, The Poincaré hyperbolic metric: z w z w z m z m z m w z
w w w
− − = − + = 1 ) ( , ) ( 1 ) ( 1 log 2 1 : ) , ( ρ
Semigroups of parabolic type
{ }
)) ( ), ( ( z F z F
n n 1 +
ρ
- a non-increasing sequence
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16
Let at least one of the semigroups
2 1,S
S
is of automorphic type.
) ( , τ
2
C g f ⊂
≠ ′ ′ ≠ ′ ′ ) ( , ) ( τ τ g f If
1 1 1 1
F G G F
- =
⇒
, , ≥ ∀ = t s F G G F
t s s t
- then
Semigroups of parabolic type
Let
2 1
} { , } {
≥ ≥
= =
t t t t
G S F S
be semigroups of parabolic nonautomorphic type. Then
1 1 1 1
F G G F
- =
⇒
, , ≥ ∀ = t s F G G F
t s s t
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17
hyperbolic parabolic automorphic nonautomorphic
{ }
) Hol( Δ ⊂
≥ t t
F
with an interior fixed point
elliptic automorphisms self-mappings which are not automorphisms
) Aut( ) Aut( Δ ⊄ Δ ⊂
2 1
S S
Summary
≠ ′ ′ ≠ ′ ′ ) ( ) ( τ τ g f
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18
Open questions
1 1 1 1
F G G F =
- ?
⇒ , 0,
t s s t
F G G F t s = ∀ ≥ ≥
- ( )
( ) lim 0, lim
z z
f z g z z z
τ τ
τ τ
→ →
≠ ≠ − −
- semigroups of hyperbolic type
( ) ( )
2 2
( ) ( ) lim 0, lim
z z
f z g z z z
τ τ
τ τ
→ →
≠ ≠ − −
- semigroups of parabolic type
( ) ( )
3 3
( ) ( ) lim 0, lim
z z
f z g z z z
τ τ
τ τ
→ →
≠ ≠ − −
- semigroups of parabolic
nonautomorphic type
( ) ( )
[ ]
1 1
( ) ( ) lim 0, lim for some 0,2
z z
f z g z z z
α α τ τ
α τ τ
+ + → →
≠ ≠ ∈ − −
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19
Open questions
( ) ( )
[ ]
1 1
( ) ( ) lim 0, lim for some 0,2
z z
f z g z z z
α α τ τ
α τ τ
+ + → →
≠ ≠ ∈ − −
Conjecture
⇓
1 1 1 1
F G G F =
- ⇒
, 0,
t s s t
F G G F t s = ∀ ≥ ≥
- Question Do there exist two semigroups of parabolic type
such that
1 1 1 1
F G G F =
- , but for some
0, t s ≥ ≥
?
t s s t
F G G F ≠
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20
Dilation case
If , then the semigroups coincide up to rescaling and, consequently, commute
( ) ( )
lim arg
t t t
F z G z τ τ
→∞
⎛ ⎞ − = ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠
Angular Asymptotic Characteristics
Hyperbolic case
if and only if the semigroups commute
( ) ( )
lim arg
t t t
F z G z τ τ
→∞
⎛ ⎞ − = ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠
( )
t
G z
( )
t
F z
τ
F.Jacobzon, S.Reich, D. Shoikhet &M.E.
( ) ( )
lim arg
t t t
F z G z τ τ
→∞
⎛ ⎞ − = ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠
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21
−∞ → t D
η τ
w
Repelling points
) (w Ft ) (w Gt
Suppose that there is a point such that f (η) = 0 and exists.
( )
η
η
− ∠
→ z
z f
z
lim
, , τ η η ≠ Δ ∂ ∈
Then for some point , the orbit can be extended for all real t and .
Δ ∈ w
( )
Δ ∈ w Ft
( ) η
=
−∞ →
w F
t tlim
If there is a nonempty open set , where then the semigroups coincide up to rescaling and, consequently, commute.
Δ ⊂ D
( ) ( )
D w w G w F
t t t
∈ = − −
−∞ →
all for η η arg lim
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22