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 { } Δ = ∈ < : z C : z 1 { } Δ = ∈ ≤ : z C : z 1 { } ∂ Δ = ∈ = z C : z 1 Δ - the set of all holomorphic functions on Δ Hol ( , D ) which map Δ into a set ⊂ D C Δ = Δ Δ Hol ( ) : Hol ( , ) - the set of all holomorphic self- mappings of Δ 2

Iterations ∈ Hol ( Δ F ) Let be a holomorphic self-mapping of the unit disk = F ( z ) z 0 z F 1 z ( ) = F ( z ) F ( z ) 1 F 2 z ( ) ( ) = F ( z ) F F ( z ) 2 1 ……… ( ) F n ( z ) = F ( z ) F F 1 z ( ) − n n ∈ Hol Δ F , G ( ) If then = ⇒ = F � G G � F F � G G � F n k k n 3

Classification ∈ Hol( Δ F ) with an interior hyperbolic parabolic fixed point self-mappings elliptic automorphic nonautomorphic which are not automorphisms automorphisms 4

Self-mappings with an interior fixed point τ = τ τ ∈ Δ Fixed point: F ( ) , F isn’t an elliptic F is an elliptic automorphism of Δ automorphism of Δ → ∞ → ∞ n n F 1 z ( ) z z F 1 z ( ) τ F 2 z ( ) τ F 2 z ( ) 1 1 F 3 z ( ) F 4 z ( ) F 3 z ( ) F 4 z ( ) ( ) = τ lim F n z → ∞ n 5

Self-mappings with no interior fixed point ( ) = τ τ ∈ ∂ Δ lim F n z There is a boundary point such that → ∞ n ′ < τ ≤ 0 F ( ) 1 → ∞ n Hyperbolic type: τ ′ < τ < 0 F ( ) 1 z F 3 z ( ) F 2 z ( ) F 1 z ( ) z Parabolic type: z ′ τ = F ( ) 1 0 1 F 1 z ( ) F 1 z ( ) τ F 3 z ( ) F 2 z ( ) 6 F 3 z ( ) F 2 z ( )

Commuting self-mappings ∈ Δ F , G Hol( ) which are different from the identity mapping = ⇒ F � G G � F F , are of the same type: G parabolic, hyperbolic, or with an interior fixed point. M. H. Heins (1941) ( ) ( ) ∈ Δ ∈ Δ F Aut G Aut If , then hyp hyp ? ∉ Aut( Δ If F is of hyperbolic type, but F ) → G is of parabolic type C. C. Cowen (1984) F, G If are not automorphisms, then they are of the same type ∉ Δ ⇒ ∉ Δ F Aut( ) G Aut ( ) F is of hyp.type, par 7

Continuous Semigroups { } ( ) = ⊂ Δ S F Hol A family is called ≥ t t 0 parameter continuous semigroup if a one- -parameter continuous semigroup a one [ ( ) ( ) ( ( ) ) ) = ∈ ∞ ∈ i F z F F z for all t , s 0 , and z D + t s t s ( ) ( ) = ∈ ii lim F z z for all z D t + → 0 t The local continuity condition (ii) implies the differentiability of S with respect to the parameter t ≥ 0 (Berkson&Porta (1978)). The limit ( ) − z F z ( ) = ∈ Δ lim t : f z , z t + → t 0 defines a holomorphic function on Δ , which is called the (infinitesimal) generator of S. (infinitesimal) generator of S. 8

Generators and Semigroups τ ∈ Δ There is a unique point such that ( ) ( )( ) ( ) = − τ − τ ∈ Δ f z z 1 z p z , z , ∈ Δ ≥ z Re p ( z ) 0 with for all The point τ is the Denjoy-Wolff point (attractive fixed point) of the semigroup generated by f. − τ τ = f ' ( ) t F ' ( ) e t τ ( ) ⎧ f ' ' ( ) − τ − τ − − τ ≠ f ' ( ) t f ' ( ) t e 1 e , f ' ( ) 0 ⎪ τ = τ F ' ' ( ) f ' ( ) ⎨ t ⎪ − τ τ = f ' ' ( ) t , f ' ( ) 0 ⎩ 9

Commuting Semigroups = ∀ ≥ F � F F � F , s , t 0 t s s t = = S { F } , S { G } Let be two continuous semigroups ≥ ≥ 1 t t 0 2 t t 0 We say that two semigroups commute two semigroups commute if = ∀ ≥ F � G G � F , s , t 0 t s s t = F � G G � F Suppose that 1 1 1 1 Problem: ? ⇒ = ∀ ≥ = F � G G � F F � G G � F , s , t 0 t s s t 1 1 1 1 10

= = = F � G G � F and F , G are S { F } , S { G } ⇒ ≥ ≥ 1 1 1 1 1 1 1 t t 0 2 t t 0 different from the identity are of the same type { } ∈ Hol( Δ F ) t ≥ 0 t with an interior hyperbolic parabolic fixed point self-mappings elliptic automorphic nonautomorphic which are not automorphisms 11 automorphisms

Semigroups with an interior fixed point → ∞ t = = S { F } , S { G } semigroups of self-mappings ≥ ≥ 1 t t 0 2 t t 0 z F 1 z ( ) which are not automorphisms τ F 2 z ( ) 1 = = ∀ ≥ ⇒ F 4 z ( ) F � G G � F F � G G � F , s , t 0 F 3 z ( ) 1 1 1 1 t s s t = S { F } a semigroup of elliptic automorphisms → ∞ t ≥ 1 t t 0 F 1 z ( ) τ ∈ Δ z with a common fixed point at τ F 2 z ( ) = S 1 and S { G } are commuting iff 1 ≥ 2 t t 0 F 3 z ( ) ( ) τ − z F 4 z ( ) − = ⋅ ∈ = at G ( z ) m e m ( z ) , a C , m ( z ) . τ τ τ t − τ 1 z ( ) = ϕ ⋅ i t ϕ ∈ Corollary 1. G ( z ) m e m ( z ) R for some τ τ t 1 1 = = S { F } , S { G } semigroups of elliptic automorphisms: ≥ ≥ 1 t t 0 2 t t 0 = ⇒ = ∀ ≥ F � G G � F F � G G � F , s , t 0 1 1 1 1 t s s t 12

Semigroups with an interior fixed point → ∞ t = = S { F } , S { G } semigroups of self-mappings ≥ ≥ 1 t t 0 2 t t 0 z F 1 z ( ) which are not automorphisms τ F 2 z ( ) 1 = = ∀ ≥ ⇒ F 4 z ( ) F � G G � F F � G G � F , s , t 0 F 3 z ( ) 1 1 1 1 t s s t = S { F } - a semigroup of elliptic automorphisms → ∞ t ≥ 1 t t 0 F 1 z ( ) τ ∈ Δ z with a common fixed point at τ F 2 z ( ) = S { G } - a semigroup of self-mappings 1 ≥ 2 t t 0 F 3 z ( ) F 4 z ( ) = ⇒ = ∀ ≥ � � F G G F F G � G � F 1 , s 0 1 1 1 1 1 s s ( ) ϕ = i t ⋅ ϕ ∈ F ( z ) m e m ( z ) for some R Corollary 2. Example. ϕ = i t F ( z ) e z τ τ t t π 2 ϕ ϕ = = ≥ ∀ ∈ Δ n , g ( z ) zp ( z ), Re p ( z ) 0 z If π is an irrational number, then n = ∀ ≥ = ⇒ = ∀ ≥ F � G G � F t 0 , but S 1 , S do not commute F � G G � F F � G G � F , s , t 0 1 t t 1 2 1 1 1 1 t s s t 13

Semigroups of hyperbolic type → ∞ t Hyperbolic type: ′ < τ < 0 F ( ) 1 τ t z F 3 z ( ) = = ∀ ≥ ⇒ F � G G � F F � G G � F , s , t 0 F 2 z ( ) F 1 z ( ) 1 1 1 1 t s s t { } ⊂ Δ F Hol( ) ≥ t t 0 with an interior fixed point hyperbolic elliptic self-mappings which automorphisms are not automorphisms ⊂ Δ S Aut( ) 1 ⊄ Δ S Aut( ) 2 14

Semigroups of parabolic type parabolic automorphic nonautomorphic ρ > ρ = lim ( F ( z ), F ( z )) 0 lim ( F ( z ), F ( z )) 0 + + n n 1 n n 1 → ∞ → ∞ n n ∈ Δ z , w The Poincaré hyperbolic metric: + − 1 m ( z ) 1 w z ρ = = w ( z , w ) : log , m ( z ) w − − 2 1 m ( z ) 1 w z w { } ρ ( F ( z ), F ( z )) - a non-increasing sequence + n n 1 15

Semigroups of parabolic type = = S { F } , S { G } Let be semigroups of parabolic ≥ ≥ 1 t t 0 2 t t 0 nonautomorphic type. Then = = ∀ ≥ F � G G � F F � G G � F , s , t 0 ⇒ 1 1 1 1 t s s t Let at least one of the semigroups is of automorphic type. S 1 , S 2 ⊂ τ 2 f , g C ( ) If ′ ′ ′ ′ τ ≠ τ ≠ f ( ) 0 , g ( ) 0 = = ∀ ≥ then F � G G � F F � G G � F , s , t 0 ⇒ 1 1 1 1 t s s t 16

Summary { } ⊂ Δ F Hol( ) t ≥ t 0 with an interior fixed point hyperbolic parabolic elliptic self-mappings which automorphic nonautomorphic automorphisms are not automorphisms ′ ′ τ ≠ f ( ) 0 ⊂ Δ S Aut( ) ′ ′ τ ≠ 1 g ( ) 0 ⊄ Δ S Aut( ) 2 17

Open questions ? = = ∀ ≥ ≥ ⇒ F G � G � F F G � G � F , t 0, s 0 1 1 1 1 t s s t f z ( ) g z ( ) ≠ ≠ - semigroups of hyperbolic type lim 0, lim 0 τ − τ − z z → τ → τ z z f z ( ) g z ( ) ≠ ≠ -semigroups of parabolic type lim 0, lim 0 ( ) ( ) 2 2 τ − τ − → τ → τ z z z z f z ( ) g z ( ) ≠ ≠ -semigroups of parabolic lim 0, lim 0 ( ) ( ) 3 3 τ − τ − → τ → τ z z z z nonautomorphic type f z ( ) g z ( ) [ ] ≠ ≠ α ∈ lim 0, lim 0 for some 0,2 ( ) ( ) α + α + 1 1 τ − τ − → τ → τ z z z z 18