Distortion and Distribution of Sets under Inner Functions Artur - PowerPoint PPT Presentation

Distortion and Distribution of Sets under Inner Functions Artur Nicolau Universitat Aut` onoma de Barcelona

Inner Functions Definition f : D → D analytic is inner if | lim r → 1 f ( r ξ ) | = 1 , a.e. ξ ∈ ∂ D .

Inner Functions Definition f : D → D analytic is inner if | lim r → 1 f ( r ξ ) | = 1 , a.e. ξ ∈ ∂ D . Invariant Subspaces n ≥ 0 a n z n : � | a n | 2 < ∞} . H 2 = { g ( z ) = � S : H 2 → H 2 g ( z ) �→ z g ( z )

Inner Functions Definition f : D → D analytic is inner if | lim r → 1 f ( r ξ ) | = 1 , a.e. ξ ∈ ∂ D . Invariant Subspaces n ≥ 0 a n z n : � | a n | 2 < ∞} . H 2 = { g ( z ) = � S : H 2 → H 2 g ( z ) �→ z g ( z ) Theorem (Beurling, 49) M subspace of H 2 . ⇒ M = f H 2 for some f inner. SM ⊆ M ⇐

Motivation Localization

Motivation Localization g : D → C analytic Ω = connected component of g − 1 ( D ) ϕ : D → Ω conformal f = g ◦ ϕ Crutial Case: f inner

Motivation Localization g : D → C analytic Ω = connected component of g − 1 ( D ) ϕ : D → Ω conformal f = g ◦ ϕ Crutial Case: f inner

Dynamics Ω � C simply connected g : Ω → Ω analytic ϕ : D → Ω conformal Then, f = ϕ − 1 ◦ g ◦ ϕ : D → D Dynamics of g ← → Dynamics of f .

Dynamics Ω � C simply connected g : Ω → Ω analytic ϕ : D → Ω conformal Then, f = ϕ − 1 ◦ g ◦ ϕ : D → D Dynamics of g ← → Dynamics of f . If g : C → C ∞ meromorphic and Ω is an invariant Fatou component, then f is inner. (Baranski, Fagella, Jarque, Karpinska)

Examples f : D → D inner if | lim r → 1 f ( r ξ ) | = 1 a.e. ξ ∈ ∂ D .

Examples f : D → D inner if | lim r → 1 f ( r ξ ) | = 1 a.e. ξ ∈ ∂ D . Finite Blaschke products. Given z 1 , . . . , z N ∈ D N z − z k � f ( z ) = 1 − z k z , z ∈ D . k =1

Examples f : D → D inner if | lim r → 1 f ( r ξ ) | = 1 a.e. ξ ∈ ∂ D . Finite Blaschke products. Given z 1 , . . . , z N ∈ D N z − z k � f ( z ) = 1 − z k z , z ∈ D . k =1 Infinite Blaschke products. Given { z k } ⊂ D , � (1 − | z k | ) < + ∞ , ∞ − z k z − z k � B ( z ) = 1 − z k z , z ∈ D . | z k | k =1

Examples f : D → D inner if | lim r → 1 f ( r ξ ) | = 1 a.e. ξ ∈ ∂ D . Finite Blaschke products. Given z 1 , . . . , z N ∈ D N z − z k � f ( z ) = 1 − z k z , z ∈ D . k =1 Infinite Blaschke products. Given { z k } ⊂ D , � (1 − | z k | ) < + ∞ , ∞ − z k z − z k � B ( z ) = 1 − z k z , z ∈ D . | z k | k =1 Singular Inner Functions. Given a positive singular measure µ on ∂ D , � ξ + z � ˆ S µ ( z ) = exp − ξ − z d µ ( ξ ) , z ∈ D . ∂ D

Examples f : D → D inner if | lim r → 1 f ( r ξ ) | = 1 a.e. ξ ∈ ∂ D . Finite Blaschke products. Given z 1 , . . . , z N ∈ D N z − z k � f ( z ) = 1 − z k z , z ∈ D . k =1 Infinite Blaschke products. Given { z k } ⊂ D , � (1 − | z k | ) < + ∞ , ∞ − z k z − z k � B ( z ) = 1 − z k z , z ∈ D . | z k | k =1 Singular Inner Functions. Given a positive singular measure µ on ∂ D , � ξ + z � ˆ S µ ( z ) = exp − ξ − z d µ ( ξ ) , z ∈ D . ∂ D Theorem f inner. Then, f = BS µ .

Singularities f inner. Consider f : ∂ D → ∂ D defined as f ( ξ ) = lim r → 1 f ( r ξ ) , a.e. ξ ∈ ∂ D .

Singularities f inner. Consider f : ∂ D → ∂ D defined as f ( ξ ) = lim r → 1 f ( r ξ ) , a.e. ξ ∈ ∂ D . Definition Sing ( f ) = { ξ ∈ ∂ D : f does not extend analytically at ξ } = { z n } ′ ∪ spt µ if f = B { z n } S µ

Singularities f inner. Consider f : ∂ D → ∂ D defined as f ( ξ ) = lim r → 1 f ( r ξ ) , a.e. ξ ∈ ∂ D . Definition Sing ( f ) = { ξ ∈ ∂ D : f does not extend analytically at ξ } = { z n } ′ ∪ spt µ if f = B { z n } S µ 0 − 1 Law Let ξ ∈ ∂ D . Either (a) There exists an arc J, ξ ∈ J, such that f extends analytically across J or (b) For every arc J, ξ ∈ J, f ( J \ { ξ } ) = ∂ D .

Distortion Definition For z ∈ D , 1 − | z | 2 w z ( E ) = 1 ˆ | ξ − z | 2 | d ξ | , E ⊂ ∂ D . 2 π E w z = harmonic measure from z w 0 = Lebesgue measure on ∂ D

Distortion Definition For z ∈ D , 1 − | z | 2 w z ( E ) = 1 ˆ | ξ − z | 2 | d ξ | , E ⊂ ∂ D . 2 π E w z = harmonic measure from z w 0 = Lebesgue measure on ∂ D Theorem (Lowner) f inner, z ∈ D . Then, w z ( f − 1 ( E )) = w f ( z ) ( E ) , E ⊂ ∂ D . If z = f ( z ) = 0 , | f − 1 ( E ) | = | E | , E ⊂ ∂ D .

Distortion Definition For 0 < α < 1 and z ∈ D , w z ( J k ) α : E ⊂ ∪ J k } � M α ( w z )( E ) = inf { E ⊂ ∂ D If z = 0, M α ( w 0 ) ≡ Hausdorff content

Distortion Definition For 0 < α < 1 and z ∈ D , w z ( J k ) α : E ⊂ ∪ J k } � M α ( w z )( E ) = inf { E ⊂ ∂ D If z = 0, M α ( w 0 ) ≡ Hausdorff content Theorem (Fernandez, Pestana, 92) f inner, 0 < α < 1 and z ∈ D . Then M α ( w z )( f − 1 ( E )) ≥ C α M α ( w f ( z ) ( E )) , E ⊂ ∂ D , (and, consequently, dim f − 1 ( E ) ≥ dim E , for any E ⊂ ∂ D ) If z = f ( z ) = 0 , M α ( f − 1 ( E )) ≥ C α M α ( E ) ,E ⊂ ∂ D .

Distortion with respect to a boundary point Definition f : D → D analytic and p ∈ ∂ D . We say | f ′ ( p ) | < ∞ if f ( p ) = lim r → 1 f ( rp ) ∈ ∂ D exists (p is a Boundary Fatou point) and f ( z ) − f ( p ) f ′ ( p ) = lim exists. z − p Γ ∋ z → p Otherwise | f ′ ( p ) | = ∞ .

Distortion with respect to a boundary point Definition f : D → D analytic and p ∈ ∂ D . We say | f ′ ( p ) | < ∞ if f ( p ) = lim r → 1 f ( rp ) ∈ ∂ D exists (p is a Boundary Fatou point) and f ( z ) − f ( p ) f ′ ( p ) = lim exists. z − p Γ ∋ z → p Otherwise | f ′ ( p ) | = ∞ . Definition | d ξ | ˆ If p ∈ ∂ D , µ p ( E ) = | ξ − p | 2 , E ⊂ ∂ D . E

Distortion with respect to a boundary point Definition f : D → D analytic and p ∈ ∂ D . We say | f ′ ( p ) | < ∞ if f ( p ) = lim r → 1 f ( rp ) ∈ ∂ D exists (p is a Boundary Fatou point) and f ( z ) − f ( p ) f ′ ( p ) = lim exists. z − p Γ ∋ z → p Otherwise | f ′ ( p ) | = ∞ . Definition | d ξ | ˆ If p ∈ ∂ D , µ p ( E ) = | ξ − p | 2 , E ⊂ ∂ D . E J ⊂ ∂ D arc. µ p ( J ) < ∞ ⇐ ⇒ p / ∈ J . µ p measures the size of E and the distribution of E around p . | J k | E = ∪ J k . Then, µ p ( E ) < ∞ ⇐ ⇒ � dist( p , J k ) 2 < ∞ .

Distortion with respect to a boundary point | d ξ | ˆ µ p ( E ) = | ξ − p | 2 , E ⊂ ∂ D . E

Distortion with respect to a boundary point | d ξ | ˆ µ p ( E ) = | ξ − p | 2 , E ⊂ ∂ D . E Theorem (Levi, N., Soler, 18) f inner, p ∈ ∂ D a BFP. Then, µ p ( f − 1 ( E )) = | f ′ ( p ) | µ f ( p ) ( E ) , E ⊂ ∂ D .

Distortion with respect to a boundary point | d ξ | ˆ µ p ( E ) = | ξ − p | 2 , E ⊂ ∂ D . E Theorem (Levi, N., Soler, 18) f inner, p ∈ ∂ D a BFP. Then, µ p ( f − 1 ( E )) = | f ′ ( p ) | µ f ( p ) ( E ) , E ⊂ ∂ D . Extreme cases.

Distortion with respect to a boundary point | d ξ | ˆ µ p ( E ) = | ξ − p | 2 , E ⊂ ∂ D . E Theorem (Levi, N., Soler, 18) f inner, p ∈ ∂ D a BFP. Then, µ p ( f − 1 ( E )) = | f ′ ( p ) | µ f ( p ) ( E ) , E ⊂ ∂ D . Extreme cases. Definition M α ( µ p )( E ) = inf { � µ p ( I j ) α : E \ { p } ⊂ ∪ I j } 0 < α < 1 and p ∈ ∂ D , Theorem (Levi, N., Soler, 18) f inner, p ∈ ∂ D a BFP, 0 < α < 1 . Then, M α ( µ p )( f − 1 ( E )) ≥ C α | f ′ ( p ) | α M α ( µ f ( p ) )( E ) , E ⊂ ∂ D .

Denjoy-Wolff Theorem Theorem (Denjoy-Wolff) f : D → D analytic, not automorphism. Then, there exists p ∈ D such that f n = f ◦ · · · ◦ f − n →∞ p unif. on compacts of D . − − → Moreover, p ∈ D is the unique fixed point of f with | f ′ ( p ) | ≤ 1 .

Denjoy-Wolff Theorem Theorem (Denjoy-Wolff) f : D → D analytic, not automorphism. Then, there exists p ∈ D such that f n = f ◦ · · · ◦ f − n →∞ p unif. on compacts of D . − − → Moreover, p ∈ D is the unique fixed point of f with | f ′ ( p ) | ≤ 1 . p ≡ DWFP Dynamics of f : ∂ D → ∂ D ?

DWFP in D f inner with DWFP 0, not rotation. Lowner: | f − 1 ( E ) | = | E | for any E ⊂ ∂ D .

DWFP in D f inner with DWFP 0, not rotation. Lowner: | f − 1 ( E ) | = | E | for any E ⊂ ∂ D . Theorem (Poincar´ e Recurrence Theorem) f inner, f (0) = 0 . Then lim inf n →∞ | f n ( ξ ) − ξ | = 0 a.e. ξ ∈ ∂ D . Theorem (Ergodic Theorem) Let f inner with f (0) = 0 . Then ( f , | | ) is ergodic and # { 1 ≤ k ≤ n : f k ( ξ ) ∈ J } lim n →∞ = | J | for any J ⊂ ∂ D . n

Shrinking Targets Notation: J ( ξ 0 , r ) = { ξ ∈ ∂ D : | ξ − ξ 0 | < r } .

Shrinking Targets Notation: J ( ξ 0 , r ) = { ξ ∈ ∂ D : | ξ − ξ 0 | < r } . Theorem (Fern´ andez, Meli´ an, Pestana, 07) f inner, not automorphism, with f (0) = 0 . Fix ξ 0 ∈ ∂ D and r k ≥ 0 decreasing. (a) If � r k = ∞ , then lim n →∞ # { 1 ≤ k ≤ n : f k ( ξ ) ∈ J ( ξ 0 , r k ) } = 1 a.e. ξ ∈ ∂ D . � n k =1 r k (b) If � r k < ∞ , then lim inf n →∞ | f n ( ξ ) − ξ 0 | ≥ 1 a.e. ξ ∈ ∂ D . r n