On the construction of entire functions in the Speiser class Simon - PowerPoint PPT Presentation

On the construction of entire functions in the Speiser class Simon Albrecht Christian-Albrechts-Universit¨ at zu Kiel London, 11 March 2015 S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 1 / 14

Outline Definitions and preliminary results 1 S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 2 / 14

Outline Definitions and preliminary results 1 Quasiconformal folding 2 S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 2 / 14

Outline Definitions and preliminary results 1 Quasiconformal folding 2 Functions in class S with only one tract 3 S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 2 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable. S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable.The Wirtinger derivatives are S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable.The Wirtinger derivatives are f z := 1 2 ( f x − if y ) S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable.The Wirtinger derivatives are f z := 1 2 ( f x − if y ) z := 1 f ¯ 2 ( f x + if y ) . S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable.The Wirtinger derivatives are f z := 1 2 ( f x − if y ) z := 1 f ¯ 2 ( f x + if y ) . Remark f ¯ z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable.The Wirtinger derivatives are f z := 1 2 ( f x − if y ) z := 1 f ¯ 2 ( f x + if y ) . Remark f ¯ z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). Then: f z = f ′ . S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable.The Wirtinger derivatives are f z := 1 2 ( f x − if y ) z := 1 f ¯ 2 ( f x + if y ) . Remark f ¯ z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). Then: f z = f ′ . chain rule S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable.The Wirtinger derivatives are f z := 1 2 ( f x − if y ) z := 1 f ¯ 2 ( f x + if y ) . Remark f ¯ z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). Then: f z = f ′ . chain rule ( g ◦ f ) z = ( g z ◦ f ) f z + ( g ¯ z ◦ f ) f ¯ z S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let G ⊂ C be a domain and f : G → C partially differentiable.The Wirtinger derivatives are f z := 1 2 ( f x − if y ) z := 1 f ¯ 2 ( f x + if y ) . Remark f ¯ z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). Then: f z = f ′ . chain rule ( g ◦ f ) z = ( g z ◦ f ) f z + ( g ¯ z ◦ f ) f ¯ z ( g ◦ f ) ¯ z = ( g z ◦ f ) f ¯ z + ( g ¯ z ◦ f ) f z S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 3 / 14

Definitions and preliminary results Definition Let U ⊂ C be open. A measurable function µ : U → C is called a k-Beltrami coefficient of U if � µ � ∞ = k < 1. S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 4 / 14

Definitions and preliminary results Definition Let U ⊂ C be open. A measurable function µ : U → C is called a k-Beltrami coefficient of U if � µ � ∞ = k < 1. Definition Let U , V be open sets in C . S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 4 / 14

Definitions and preliminary results Definition Let U ⊂ C be open. A measurable function µ : U → C is called a k-Beltrami coefficient of U if � µ � ∞ = k < 1. Definition Let U , V be open sets in C . A map φ : U → V is said to be quasiregular if it has locally square integrable weak derivatives S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 4 / 14

Definitions and preliminary results Definition Let U ⊂ C be open. A measurable function µ : U → C is called a k-Beltrami coefficient of U if � µ � ∞ = k < 1. Definition Let U , V be open sets in C . A map φ : U → V is said to be quasiregular if it has locally square integrable weak derivatives and the function z ( z ) µ φ ( z ) = φ ¯ φ z ( z ) is a k -Beltrami coefficient. S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 4 / 14

Definitions and preliminary results Definition Let U ⊂ C be open. A measurable function µ : U → C is called a k-Beltrami coefficient of U if � µ � ∞ = k < 1. Definition Let U , V be open sets in C . A map φ : U → V is said to be quasiregular if it has locally square integrable weak derivatives and the function z ( z ) µ φ ( z ) = φ ¯ φ z ( z ) is a k -Beltrami coefficient. A quasiregular homeomorphism is called quasiconformal . S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 4 / 14

Definitions and preliminary results Question Given a k -Beltrami coefficient µ S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 5 / 14

Definitions and preliminary results Question Given a k -Beltrami coefficient µ , does there exist a quasiconformal map φ such that µ φ = µ ? S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 5 / 14

Definitions and preliminary results Question Given a k -Beltrami coefficient µ , does there exist a quasiconformal map φ such that µ φ = µ ? (i.e. φ ¯ z = µ · φ z , φ is a solution of the Beltrami equation) S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 5 / 14

Definitions and preliminary results Question Given a k -Beltrami coefficient µ , does there exist a quasiconformal map φ such that µ φ = µ ? (i.e. φ ¯ z = µ · φ z , φ is a solution of the Beltrami equation) Answer: Theorem (Measurable Riemann Mapping Theorem (MRMT)) Let µ : C → C be a k-Beltrami coefficient. S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 5 / 14

Definitions and preliminary results Question Given a k -Beltrami coefficient µ , does there exist a quasiconformal map φ such that µ φ = µ ? (i.e. φ ¯ z = µ · φ z , φ is a solution of the Beltrami equation) Answer: Theorem (Measurable Riemann Mapping Theorem (MRMT)) Let µ : C → C be a k-Beltrami coefficient. Then there exists a unique quasiconformal map φ : C → C such that φ ( 0 ) = 0 , φ ( 1 ) = 1 , µ φ = µ . S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 5 / 14

Definitions and preliminary results Question Given a k -Beltrami coefficient µ , does there exist a quasiconformal map φ such that µ φ = µ ? (i.e. φ ¯ z = µ · φ z , φ is a solution of the Beltrami equation) Answer: Theorem (Measurable Riemann Mapping Theorem (MRMT)) Let µ : C → C be a k-Beltrami coefficient. Then there exists a unique quasiconformal map φ : C → C such that φ ( 0 ) = 0 , φ ( 1 ) = 1 , µ φ = µ . Corollary Let g : C → C be quasiregular. S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 5 / 14

Definitions and preliminary results Question Given a k -Beltrami coefficient µ , does there exist a quasiconformal map φ such that µ φ = µ ? (i.e. φ ¯ z = µ · φ z , φ is a solution of the Beltrami equation) Answer: Theorem (Measurable Riemann Mapping Theorem (MRMT)) Let µ : C → C be a k-Beltrami coefficient. Then there exists a unique quasiconformal map φ : C → C such that φ ( 0 ) = 0 , φ ( 1 ) = 1 , µ φ = µ . Corollary Let g : C → C be quasiregular. Then there exists a quasiconformal map φ such that f := g ◦ φ − 1 is holomorphic. S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 5 / 14

Quasiconformal folding Examples Quasiconformal folding is a technique to construct functions in class S with good control of the singular values. S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 6 / 14

Quasiconformal folding Examples Quasiconformal folding is a technique to construct functions in class S with good control of the singular values. introduced by C. Bishop S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 6 / 14

Quasiconformal folding Examples Quasiconformal folding is a technique to construct functions in class S with good control of the singular values. introduced by C. Bishop first preprint in 2011 S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 6 / 14

Quasiconformal folding Examples Quasiconformal folding is a technique to construct functions in class S with good control of the singular values. introduced by C. Bishop first preprint in 2011 appeared in acta mathematica (214:1(2015) 1-60) S. Albrecht (CAU Kiel) Construction in class S 11 March 2015 6 / 14