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

1

Definitions and preliminary results

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 2 / 14

Outline

1

Definitions and preliminary results

2

Quasiconformal folding

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 2 / 14

Outline

1

Definitions and preliminary results

2

Quasiconformal folding

3

Functions in class S with only one tract

• 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 fz := 1 2 (fx − ify)

• 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 fz := 1 2 (fx − ify) f¯

z := 1

2 (fx + ify) .

• 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 fz := 1 2 (fx − ify) f¯

z := 1

2 (fx + ify) .

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 fz := 1 2 (fx − ify) f¯

z := 1

2 (fx + ify) .

Remark

f¯

z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). Then:

fz = 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 fz := 1 2 (fx − ify) f¯

z := 1

2 (fx + ify) .

Remark

f¯

z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). Then:

fz = 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 fz := 1 2 (fx − ify) f¯

z := 1

2 (fx + ify) .

Remark

f¯

z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). Then:

fz = f ′. chain rule (g ◦ f )z = (gz ◦ f ) fz + (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 fz := 1 2 (fx − ify) f¯

z := 1

2 (fx + ify) .

Remark

f¯

z ≡ 0 iff f is holomorphic (Cauchy-Riemann equations). Then:

fz = f ′. chain rule (g ◦ f )z = (gz ◦ f ) fz + (g¯

z ◦ f ) f¯ z

(g ◦ f )¯

z = (gz ◦ f ) f¯ z + (g¯ z ◦ f ) fz

• 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 µφ(z) = φ¯

z(z)

φz(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 µφ(z) = φ¯

z(z)

φz(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 µφ = 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 µφ = µ? (i.e. φ¯

z = µ · φz, 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 µφ = µ? (i.e. φ¯

z = µ · φz, 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 µφ = µ? (i.e. φ¯

z = µ · φz, 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, µφ = 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 µφ = µ? (i.e. φ¯

z = µ · φz, 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, µφ = 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 µφ = µ? (i.e. φ¯

z = µ · φz, 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, µφ = 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

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) Bishop constructs among other examples

• 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) Bishop constructs among other examples

f ∈ S with arbitrary order of growth (originally due to S. Merenkov)

• 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) Bishop constructs among other examples

f ∈ S with arbitrary order of growth (originally due to S. Merenkov) counterexamples in S for the area conjecture and the strong Eremenko conjecture

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 6 / 14

Quasiconformal folding Idea of quasiconformal folding

The idea behind quasiconformal folding is quite simple.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 7 / 14

Quasiconformal folding Idea of quasiconformal folding

The idea behind quasiconformal folding is quite simple. Let f be a function in class S with no asymptotic value and exactly two critical values (±1).

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 7 / 14

Quasiconformal folding Idea of quasiconformal folding

The idea behind quasiconformal folding is quite simple. Let f be a function in class S with no asymptotic value and exactly two critical values (±1). −1 1

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 7 / 14

Quasiconformal folding Idea of quasiconformal folding

The idea behind quasiconformal folding is quite simple. Let f be a function in class S with no asymptotic value and exactly two critical values (±1). −1 1 f T

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 7 / 14

Quasiconformal folding Idea of quasiconformal folding

The idea behind quasiconformal folding is quite simple. Let f be a function in class S with no asymptotic value and exactly two critical values (±1). −1 1 f T

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 7 / 14

Quasiconformal folding Idea of quasiconformal folding

The idea behind quasiconformal folding is quite simple. Let f be a function in class S with no asymptotic value and exactly two critical values (±1). −1 1 f cosh T

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 7 / 14

Quasiconformal folding Idea of quasiconformal folding

The idea behind quasiconformal folding is quite simple. Let f be a function in class S with no asymptotic value and exactly two critical values (±1). −1 1 f τj cosh T Ωj

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 7 / 14

Quasiconformal folding Idea of quasiconformal folding

The idea behind quasiconformal folding is quite simple. Let f be a function in class S with no asymptotic value and exactly two critical values (±1). −1 1 f τj cosh T Ωj Reverse this procedure!

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 7 / 14

Quasiconformal folding Bounded geometry

Important for Bishop’s construction to work is the bounded geometry condition on T.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 8 / 14

Quasiconformal folding Bounded geometry

Important for Bishop’s construction to work is the bounded geometry condition on T.

Definition

We say that T has bounded geometry if: The edges are C2 with uniform bounds.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 8 / 14

Quasiconformal folding Bounded geometry

Important for Bishop’s construction to work is the bounded geometry condition on T.

Definition

We say that T has bounded geometry if: The edges are C2 with uniform bounds. The angles between adjacent edges are uniformly bounded away from zero.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 8 / 14

Quasiconformal folding Bounded geometry

Important for Bishop’s construction to work is the bounded geometry condition on T.

Definition

We say that T has bounded geometry if: The edges are C2 with uniform bounds. The angles between adjacent edges are uniformly bounded away from zero. Adjacent edges have uniformly comparable lengths.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 8 / 14

Quasiconformal folding Bounded geometry

Important for Bishop’s construction to work is the bounded geometry condition on T.

Definition

We say that T has bounded geometry if: The edges are C2 with uniform bounds. The angles between adjacent edges are uniformly bounded away from zero. Adjacent edges have uniformly comparable lengths. For non-adjacent edges e and f , diam(e)

dist(e,f ) is uniformly bounded.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 8 / 14

Quasiconformal folding Components of C \ T

Let T be an unbounded, locally finite, connected graph.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 9 / 14

Quasiconformal folding Components of C \ T

Let T be an unbounded, locally finite, connected graph. Every component

• f C \ T is one of the following:
• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 9 / 14

Quasiconformal folding Components of C \ T

Let T be an unbounded, locally finite, connected graph. Every component

• f C \ T is one of the following:

R-component: unbounded components, which are mapped onto the right half-plane, σ : Hr → C is essentially cosh

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 9 / 14

Quasiconformal folding Components of C \ T

Let T be an unbounded, locally finite, connected graph. Every component

• f C \ T is one of the following:

R-component: unbounded components, which are mapped onto the right half-plane, σ : Hr → C is essentially cosh L-component: unbounded Jordan domains, which are mapped onto the left half-plane, σ : Hl → C is just exp (these components assign asymptotic values)

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 9 / 14

Quasiconformal folding Components of C \ T

Let T be an unbounded, locally finite, connected graph. Every component

• f C \ T is one of the following:

R-component: unbounded components, which are mapped onto the right half-plane, σ : Hr → C is essentially cosh L-component: unbounded Jordan domains, which are mapped onto the left half-plane, σ : Hl → C is just exp (these components assign asymptotic values) D-component: bounded Jordan domains (they assign other critical values and higher order critical points). We will not use these.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 9 / 14

Quasiconformal folding Bishop’s theorem

Theorem (Bishop, only L- and R-components)

Suppose T is a bounded geometry tree and suppose τ is conformal from each complementary component of T to its standard version (i.e. left/right half-plane).

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 10 / 14

Quasiconformal folding Bishop’s theorem

Theorem (Bishop, only L- and R-components)

Suppose T is a bounded geometry tree and suppose τ is conformal from each complementary component of T to its standard version (i.e. left/right half-plane). Assume that L components only share edges with R components.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 10 / 14

Quasiconformal folding Bishop’s theorem

Theorem (Bishop, only L- and R-components)

Suppose T is a bounded geometry tree and suppose τ is conformal from each complementary component of T to its standard version (i.e. left/right half-plane). Assume that L components only share edges with R components.

• n L components τ maps edges to intervals of length 2π on ∂Hl with

endpoints in 2πiZ,

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 10 / 14

Quasiconformal folding Bishop’s theorem

Theorem (Bishop, only L- and R-components)

Suppose T is a bounded geometry tree and suppose τ is conformal from each complementary component of T to its standard version (i.e. left/right half-plane). Assume that L components only share edges with R components.

• n L components τ maps edges to intervals of length 2π on ∂Hl with

endpoints in 2πiZ,

• n R components the τ-sizes of all edges are ≥ 2π.
• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 10 / 14

Quasiconformal folding Bishop’s theorem

Theorem (Bishop, only L- and R-components)

Suppose T is a bounded geometry tree and suppose τ is conformal from each complementary component of T to its standard version (i.e. left/right half-plane). Assume that L components only share edges with R components.

• n L components τ maps edges to intervals of length 2π on ∂Hl with

endpoints in 2πiZ,

• n R components the τ-sizes of all edges are ≥ 2π.

Then there is an entire function f and a quasiconformal map φ of the plane so that f ◦ φ = σ ◦ τ off T(r0) (a neighbourhood of T).

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 10 / 14

Quasiconformal folding Bishop’s theorem

Theorem (Bishop, only L- and R-components)

Suppose T is a bounded geometry tree and suppose τ is conformal from each complementary component of T to its standard version (i.e. left/right half-plane). Assume that L components only share edges with R components.

• n L components τ maps edges to intervals of length 2π on ∂Hl with

endpoints in 2πiZ,

• n R components the τ-sizes of all edges are ≥ 2π.

Then there is an entire function f and a quasiconformal map φ of the plane so that f ◦ φ = σ ◦ τ off T(r0) (a neighbourhood of T). The only singular values of f are ±1 (critical values coming from the vertices of T) and the singular values assigned by the L components.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 10 / 14

Functions in class S with only one tract Idea

Question

Given a simply connected, unbounded domain G, does there exist f ∈ S such that ”G = {z ∈ C : |f (z)| > R}”?

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 11 / 14

Functions in class S with only one tract Idea

Question

Given a simply connected, unbounded domain G, does there exist f ∈ S such that ”G = {z ∈ C : |f (z)| > R}”? Idea: Use qc-folding

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 11 / 14

Functions in class S with only one tract Idea

Question

Given a simply connected, unbounded domain G, does there exist f ∈ S such that ”G = {z ∈ C : |f (z)| > R}”? Idea: Use qc-folding: G ∧ =R-component, C \ G ∧ =L-component, make ∂G a bounded geometry tree

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 11 / 14

Functions in class S with only one tract Idea

Question

Given a simply connected, unbounded domain G, does there exist f ∈ S such that ”G = {z ∈ C : |f (z)| > R}”? Idea: Use qc-folding: G ∧ =R-component, C \ G ∧ =L-component, make ∂G a bounded geometry tree

Problem

Not all domains G are possible.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 11 / 14

Functions in class S with only one tract Idea

Question

Given a simply connected, unbounded domain G, does there exist f ∈ S such that ”G = {z ∈ C : |f (z)| > R}”? Idea: Use qc-folding: G ∧ =R-component, C \ G ∧ =L-component, make ∂G a bounded geometry tree

Problem

Not all domains G are possible. Assumptions on G:

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 11 / 14

Functions in class S with only one tract Idea

Question

Given a simply connected, unbounded domain G, does there exist f ∈ S such that ”G = {z ∈ C : |f (z)| > R}”? Idea: Use qc-folding: G ∧ =R-component, C \ G ∧ =L-component, make ∂G a bounded geometry tree

Problem

Not all domains G are possible. Assumptions on G: ∂G sufficiently nice (see bounded geometry)

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 11 / 14

Functions in class S with only one tract Idea

Question

Given a simply connected, unbounded domain G, does there exist f ∈ S such that ”G = {z ∈ C : |f (z)| > R}”? Idea: Use qc-folding: G ∧ =R-component, C \ G ∧ =L-component, make ∂G a bounded geometry tree

Problem

Not all domains G are possible. Assumptions on G: ∂G sufficiently nice (see bounded geometry) G symmetric with respect to R (to make the formulation easier)

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 11 / 14

Functions in class S with only one tract Idea

Question

Given a simply connected, unbounded domain G, does there exist f ∈ S such that ”G = {z ∈ C : |f (z)| > R}”? Idea: Use qc-folding: G ∧ =R-component, C \ G ∧ =L-component, make ∂G a bounded geometry tree

Problem

Not all domains G are possible. Assumptions on G: ∂G sufficiently nice (see bounded geometry) G symmetric with respect to R (to make the formulation easier) width of tract ∼ length of edge (see bounded geometry)

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 11 / 14

Functions in class S with only one tract Shape of G

G

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

η G

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

2πin 2πi(n + 1) η G

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

2πin 2πi(n + 1) G η

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

exp

π 2

− π

2

Log 2π 2πin 2πi(n + 1) G η

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

exp

π 2

− π

2

Log 2π 2πin 2πi(n + 1) ∼ log(n) ∼ log(n + 1) G η

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

exp

π 2

− π

2

Log 2π 2πin 2πi(n + 1) ∼ log(n) ∼ log(n + 1) ∼ 2 log(n) ∼ 2 log(n + 1) G η ∼ z → 2z

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

exp

π 2

− π

2

Log ∼ n2 ∼ (n + 1)2 2π 2πin 2πi(n + 1) ∼ log(n) ∼ log(n + 1) ∼ 2 log(n) ∼ 2 log(n + 1) G η ∼ z → 2z

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

exp

π 2

− π

2

Log ∼ n2 ∼ (n + 1)2 2π 2πin 2πi(n + 1) ∼ log(n) ∼ log(n + 1) ∼ 2 log(n) ∼ 2 log(n + 1) G η ∼ z → 2z

⇒ the length of the n-th edge is ∼ n

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Shape of G

exp

π 2

− π

2

Log ∼ n2 ∼ (n + 1)2 2π 2πin 2πi(n + 1) ∼ log(n) ∼ log(n + 1) ∼ 2 log(n) ∼ 2 log(n + 1) G η ∼ z → 2z

⇒ the length of the n-th edge is ∼ n ⇒ width of G at Re = n2 must be n

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 12 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

2.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0
• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise.
• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise. Let k > 0 and

G :=

• reiϕ : r > 1, |ϕ| < k · (log r)γ

rδ

• .
• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise. Let k > 0 and

G :=

• reiϕ : r > 1, |ϕ| < k · (log r)γ

rδ

• .

Then there exist a quasiregular map g and constants k0 > 0, r0 > 1 and R ≥ 1

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise. Let k > 0 and

G :=

• reiϕ : r > 1, |ϕ| < k · (log r)γ

rδ

• .

Then there exist a quasiregular map g and constants k0 > 0, r0 > 1 and R ≥ 1 such that

• reiϕ : r > r0, |ϕ| < k0 · (log r)γ

rδ

• ⊂ {z ∈ C : |g(z)| ≥ R}
• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise. Let k > 0 and

G :=

• reiϕ : r > 1, |ϕ| < k · (log r)γ

rδ

• .

Then there exist a quasiregular map g and constants k0 > 0, r0 > 1 and R ≥ 1 such that

• reiϕ : r > r0, |ϕ| < k0 · (log r)γ

rδ

• ⊂ {z ∈ C : |g(z)| ≥ R} ⊂ G.
• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise. Let k > 0 and

G :=

• reiϕ : r > 1, |ϕ| < k · (log r)γ

rδ

• .

Then there exist a quasiregular map g and constants k0 > 0, r0 > 1 and R ≥ 1 such that

• reiϕ : r > r0, |ϕ| < k0 · (log r)γ

rδ

• ⊂ {z ∈ C : |g(z)| ≥ R} ⊂ G.

Furthermore, there exists a quasiconformal map φ such that f = g ◦ φ−1 is entire

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise. Let k > 0 and

G :=

• reiϕ : r > 1, |ϕ| < k · (log r)γ

rδ

• .

Then there exist a quasiregular map g and constants k0 > 0, r0 > 1 and R ≥ 1 such that

• reiϕ : r > r0, |ϕ| < k0 · (log r)γ

rδ

• ⊂ {z ∈ C : |g(z)| ≥ R} ⊂ G.

Furthermore, there exists a quasiconformal map φ such that f = g ◦ φ−1 is entire, f ∈ S

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise. Let k > 0 and

G :=

• reiϕ : r > 1, |ϕ| < k · (log r)γ

rδ

• .

Then there exist a quasiregular map g and constants k0 > 0, r0 > 1 and R ≥ 1 such that

• reiϕ : r > r0, |ϕ| < k0 · (log r)γ

rδ

• ⊂ {z ∈ C : |g(z)| ≥ R} ⊂ G.

Furthermore, there exists a quasiconformal map φ such that f = g ◦ φ−1 is entire, f ∈ S and φ is asymptotically conformal at infinity.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

Functions in class S with only one tract Theorem

Theorem

Let 0 ≤ δ ≤ 1

• 2. For δ = 0 let γ < 0 , for δ = 1

2 let γ ≥ 1 and let γ ∈ R

• therwise. Let k > 0 and

G :=

• reiϕ : r > 1, |ϕ| < k · (log r)γ

rδ

• .

Then there exist a quasiregular map g and constants k0 > 0, r0 > 1 and R ≥ 1 such that

• reiϕ : r > r0, |ϕ| < k0 · (log r)γ

rδ

• ⊂ {z ∈ C : |g(z)| ≥ R} ⊂ G.

Furthermore, there exists a quasiconformal map φ such that f = g ◦ φ−1 is entire, f ∈ S and φ is asymptotically conformal at infinity.

Remark

φ asymptotically conformal: lim|z|→∞

φ(z) z

= c exists, c = 0, ∞

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 13 / 14

The End

Thank you very much for your attention.

• S. Albrecht (CAU Kiel)

Construction in class S 11 March 2015 14 / 14