On the escaping set of exponential maps Patrick Comdhr - - PowerPoint PPT Presentation

On the escaping set of exponential maps

Patrick Comdühr

Christian-Albrechts-Universität zu Kiel

Barcelona, 24 November 2015

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 1 / 16

Outline

1

Motivation

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 2 / 16

Outline

1

Motivation

2

Connectivity for real parameters

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 2 / 16

Outline

1

Motivation

2

Connectivity for real parameters

3

The general case

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 2 / 16

Motivation

Consider the family fa : C → C, fa(z) = ez + a, a ∈ C.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 3 / 16

Motivation

Consider the family fa : C → C, fa(z) = ez + a, a ∈ C. Question: What can we say about the connectivity of its escaping sets I(fa) := {z ∈ C : f n

a (z) → ∞ as n → ∞} ?

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 3 / 16

Motivation

Consider the family fa : C → C, fa(z) = ez + a, a ∈ C. Question: What can we say about the connectivity of its escaping sets I(fa) := {z ∈ C : f n

a (z) → ∞ as n → ∞} ?

1.1 The case a = −1

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 3 / 16

Motivation

Consider the family fa : C → C, fa(z) = ez + a, a ∈ C. Question: What can we say about the connectivity of its escaping sets I(fa) := {z ∈ C : f n

a (z) → ∞ as n → ∞} ?

1.1 The case a = −1 1.2 The case a = −0.99 + 0.0001i

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 3 / 16

Motivation

Remark.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa).

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R)

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R) (Devaney, Krych 1984) J (fa) = C.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R) (Devaney, Krych 1984) J (fa) = C. Let us look at the Eremenko-Lyubich class B := {f : C → C entire and transcendental: sing(f −1

a

) is bounded}.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R) (Devaney, Krych 1984) J (fa) = C. Let us look at the Eremenko-Lyubich class B := {f : C → C entire and transcendental: sing(f −1

a

) is bounded}. For all a ∈ C we have fa ∈ B.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R) (Devaney, Krych 1984) J (fa) = C. Let us look at the Eremenko-Lyubich class B := {f : C → C entire and transcendental: sing(f −1

a

) is bounded}. For all a ∈ C we have fa ∈ B. (Eremenko-Lyubich 1992) If f ∈ B, then I(f ) ⊂ J (f ).

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R) (Devaney, Krych 1984) J (fa) = C. Let us look at the Eremenko-Lyubich class B := {f : C → C entire and transcendental: sing(f −1

a

) is bounded}. For all a ∈ C we have fa ∈ B. (Eremenko-Lyubich 1992) If f ∈ B, then I(f ) ⊂ J (f ). In fact I(f ) = J (f ), because J (f ) = ∂I(f ).

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R) (Devaney, Krych 1984) J (fa) = C. Let us look at the Eremenko-Lyubich class B := {f : C → C entire and transcendental: sing(f −1

a

) is bounded}. For all a ∈ C we have fa ∈ B. (Eremenko-Lyubich 1992) If f ∈ B, then I(f ) ⊂ J (f ). In fact I(f ) = J (f ), because J (f ) = ∂I(f ). If a ∈ C and a / ∈ J (fa), then J (fa) and hence I(fa) is disconnected.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R) (Devaney, Krych 1984) J (fa) = C. Let us look at the Eremenko-Lyubich class B := {f : C → C entire and transcendental: sing(f −1

a

) is bounded}. For all a ∈ C we have fa ∈ B. (Eremenko-Lyubich 1992) If f ∈ B, then I(f ) ⊂ J (f ). In fact I(f ) = J (f ), because J (f ) = ∂I(f ). If a ∈ C and a / ∈ J (fa), then J (fa) and hence I(fa) is disconnected. In which way does the connectivity of I(fa) depend on the parameter a?

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Motivation

• Remark. For a ∈ (−1, ∞) we have

R ⊂ I(fa). (Because fa(x) ≥ x + (1 + a) for all x ∈ R) (Devaney, Krych 1984) J (fa) = C. Let us look at the Eremenko-Lyubich class B := {f : C → C entire and transcendental: sing(f −1

a

) is bounded}. For all a ∈ C we have fa ∈ B. (Eremenko-Lyubich 1992) If f ∈ B, then I(f ) ⊂ J (f ). In fact I(f ) = J (f ), because J (f ) = ∂I(f ). If a ∈ C and a / ∈ J (fa), then J (fa) and hence I(fa) is disconnected. In which way does the connectivity of I(fa) depend on the parameter a? Lasse Rempe-Gillen has given an answer to this question:

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 4 / 16

Connectivity for real parameters

Restriction to real parameters

Theorem (Rempe-Gillen 2008)

1Robert L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory

• Dynam. Systems 13 (1993), no. 4, 627–634.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 5 / 16

Connectivity for real parameters

Restriction to real parameters

Theorem (Rempe-Gillen 2008)

For a ∈ (−1, ∞) the set I(fa) is connected.

1Robert L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory

• Dynam. Systems 13 (1993), no. 4, 627–634.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 5 / 16

Connectivity for real parameters

Restriction to real parameters

Theorem (Rempe-Gillen 2008)

For a ∈ (−1, ∞) the set I(fa) is connected. To prove the theorem, we construct a connected and dense subset of I(fa).

1Robert L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory

• Dynam. Systems 13 (1993), no. 4, 627–634.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 5 / 16

Connectivity for real parameters

Restriction to real parameters

Theorem (Rempe-Gillen 2008)

For a ∈ (−1, ∞) the set I(fa) is connected. To prove the theorem, we construct a connected and dense subset of I(fa). The idea of this continuum is due to Devaney1:

1Robert L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory

• Dynam. Systems 13 (1993), no. 4, 627–634.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 5 / 16

Connectivity for real parameters

Restriction to real parameters

Theorem (Rempe-Gillen 2008)

For a ∈ (−1, ∞) the set I(fa) is connected. To prove the theorem, we construct a connected and dense subset of I(fa). The idea of this continuum is due to Devaney1: Denote S+ := {z ∈ C: 0 < Im(z) < π} S− := {z ∈ C: − π < Im(z) < 0}

1Robert L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory

• Dynam. Systems 13 (1993), no. 4, 627–634.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 5 / 16

Connectivity for real parameters

Restriction to real parameters

Theorem (Rempe-Gillen 2008)

For a ∈ (−1, ∞) the set I(fa) is connected. To prove the theorem, we construct a connected and dense subset of I(fa). The idea of this continuum is due to Devaney1: Denote S+ := {z ∈ C: 0 < Im(z) < π} S− := {z ∈ C: − π < Im(z) < 0} H+ := {z ∈ C: Im(z) > 0} H− := {z ∈ C: Im(z) < 0}

1Robert L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory

• Dynam. Systems 13 (1993), no. 4, 627–634.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 5 / 16

Connectivity for real parameters

Restriction to real parameters

Theorem (Rempe-Gillen 2008)

For a ∈ (−1, ∞) the set I(fa) is connected. To prove the theorem, we construct a connected and dense subset of I(fa). The idea of this continuum is due to Devaney1: Denote S+ := {z ∈ C: 0 < Im(z) < π} S− := {z ∈ C: − π < Im(z) < 0} H+ := {z ∈ C: Im(z) > 0} H− := {z ∈ C: Im(z) < 0} La,σ : Hσ → Sσ the branch of f −1

a

in Sσ, where σ ∈ {+, −}

1Robert L. Devaney, Knaster-like continua and complex dynamics, Ergodic Theory

• Dynam. Systems 13 (1993), no. 4, 627–634.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 5 / 16

Connectivity for real parameters Construction of the continuum

We can extend La,σ to a homeomorphism ˜ La,σ : Hσ \ {a} → Sσ, which we denote again by La,σ for simplicity.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 6 / 16

Connectivity for real parameters Construction of the continuum

We can extend La,σ to a homeomorphism ˜ La,σ : Hσ \ {a} → Sσ, which we denote again by La,σ for simplicity. S+ S− H+ H− La,+ La,− a

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 6 / 16

Connectivity for real parameters Construction of the continuum

Now we are ready to construct our continuum.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 7 / 16

Connectivity for real parameters Construction of the continuum

Now we are ready to construct our continuum. Define γσ

a,0 := (−∞, a) and γσ a,k+1 := La,σ(γσ a,k) for all k ∈ N0

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 7 / 16

Connectivity for real parameters Construction of the continuum

Now we are ready to construct our continuum. Define γσ

a,0 := (−∞, a) and γσ a,k+1 := La,σ(γσ a,k) for all k ∈ N0

Γσ

a :=

• k≥0

γσ

a,k

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 7 / 16

Connectivity for real parameters Construction of the continuum

Now we are ready to construct our continuum. Define γσ

a,0 := (−∞, a) and γσ a,k+1 := La,σ(γσ a,k) for all k ∈ N0

Γσ

a :=

• k≥0

γσ

a,k

For simplicity we just write γσ

k and Γσ.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 7 / 16

Connectivity for real parameters Construction of the continuum

Now we are ready to construct our continuum. Define γσ

a,0 := (−∞, a) and γσ a,k+1 := La,σ(γσ a,k) for all k ∈ N0

Γσ

a :=

• k≥0

γσ

a,k

For simplicity we just write γσ

k and Γσ.

γ+ γ+

1

γ+

2

γ+

3

γ+

4

γ+

5

Construction of Γ+.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 7 / 16

Connectivity for real parameters Construction of the continuum

By continuing this procedure, we obtain:

The set Γ+. (Borrowed from Lasse Rempe-Gillen2)

• 2L. Rempe, The escaping set of the exponential., Ergodic Theory & Dynam. Systems

30 (2010), 595-599

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 8 / 16

Connectivity for real parameters Construction of the continuum

By continuing this procedure, we obtain:

The set Γ+. (Borrowed from Lasse Rempe-Gillen2)

It seems that γσ

k is "close" to Γσ for large k.

• 2L. Rempe, The escaping set of the exponential., Ergodic Theory & Dynam. Systems

30 (2010), 595-599

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 8 / 16

Connectivity for real parameters Construction of the continuum

Definition (Hausdorff distance/limit)

Let (X, d) be a metric space and A, B nonempty subsets of X.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 9 / 16

Connectivity for real parameters Construction of the continuum

Definition (Hausdorff distance/limit)

Let (X, d) be a metric space and A, B nonempty subsets of X. Then we define dH(A, B) := max

• sup

a∈A

inf

b∈B d(a, b), sup b∈B

inf

a∈A d(a, b)

• as the Hausdorff distance between A and B.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 9 / 16

Connectivity for real parameters Construction of the continuum

Definition (Hausdorff distance/limit)

Let (X, d) be a metric space and A, B nonempty subsets of X. Then we define dH(A, B) := max

• sup

a∈A

inf

b∈B d(a, b), sup b∈B

inf

a∈A d(a, b)

• as the Hausdorff distance between A and B.

Moreover, we call a closed set C ⊂ X the Hausdorff limit of the sequence (Cn)n∈N ⊂ X N of closed sets with respect to the Hausdorff distance, if lim

n→∞ dH(Cn, C) = 0.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 9 / 16

Connectivity for real parameters Construction of the continuum

Definition (Hausdorff distance/limit)

Let (X, d) be a metric space and A, B nonempty subsets of X. Then we define dH(A, B) := max

• sup

a∈A

inf

b∈B d(a, b), sup b∈B

inf

a∈A d(a, b)

• as the Hausdorff distance between A and B.

Moreover, we call a closed set C ⊂ X the Hausdorff limit of the sequence (Cn)n∈N ⊂ X N of closed sets with respect to the Hausdorff distance, if lim

n→∞ dH(Cn, C) = 0.

Lemma (Hausdorff limit for γσ

k )

The set Γσ ∪ {∞} is the Hausdorff limit of the sequence

• γσ

k ∪ {∞}

• k∈N0

with respect to the chordal metric χ.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 9 / 16

Connectivity for real parameters Construction of the continuum

Definition (Hausdorff distance/limit)

Let (X, d) be a metric space and A, B nonempty subsets of X. Then we define dH(A, B) := max

• sup

a∈A

inf

b∈B d(a, b), sup b∈B

inf

a∈A d(a, b)

• as the Hausdorff distance between A and B.

Moreover, we call a closed set C ⊂ X the Hausdorff limit of the sequence (Cn)n∈N ⊂ X N of closed sets with respect to the Hausdorff distance, if lim

n→∞ dH(Cn, C) = 0.

Lemma (Hausdorff limit for γσ

k )

The set Γσ ∪ {∞} is the Hausdorff limit of the sequence

• γσ

k ∪ {∞}

• k∈N0

with respect to the chordal metric χ. In particular,

k≥k0 γσ k is a dense

subset of Γσ for all k0 ∈ N0.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 9 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ C.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.

We need to show: Γσ ⊂ U

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.

We need to show: Γσ ⊂ U Take z0 ∈ Γσ ∩ U.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.

We need to show: Γσ ⊂ U Take z0 ∈ Γσ ∩ U. = ⇒ ∃k0 ∈ N0 ∀k ≥ k0 : γσ

k ∩ U = ∅

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.

We need to show: Γσ ⊂ U Take z0 ∈ Γσ ∩ U. = ⇒ ∃k0 ∈ N0 ∀k ≥ k0 : γσ

k ∩ U = ∅

γσ

k connected

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.

We need to show: Γσ ⊂ U Take z0 ∈ Γσ ∩ U. = ⇒ ∃k0 ∈ N0 ∀k ≥ k0 : γσ

k ∩ U = ∅

γσ

k connected =

⇒ γσ

k ⊂ U for k ≥ k0

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.

We need to show: Γσ ⊂ U Take z0 ∈ Γσ ∩ U. = ⇒ ∃k0 ∈ N0 ∀k ≥ k0 : γσ

k ∩ U = ∅

γσ

k connected =

⇒ γσ

k ⊂ U for k ≥ k0

Thus Γσ ⊂ Γσ

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.

We need to show: Γσ ⊂ U Take z0 ∈ Γσ ∩ U. = ⇒ ∃k0 ∈ N0 ∀k ≥ k0 : γσ

k ∩ U = ∅

γσ

k connected =

⇒ γσ

k ⊂ U for k ≥ k0

Thus Γσ ⊂ Γσ =

• k≥k0

γσ

k

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Lemma

Let X ⊂ ˆ

• C. Suppose for all U ⊂ C open we have

(X ∩ U = ∅ ∧ X ∩ ∂U = ∅) = ⇒ X ⊂ U. Then X is connected.

Lemma (Connectivity of Γσ)

The sets Γ+ =

k≥0 γ+ k and Γ− = k≥0 γ− k are connected.

• Proof. Let U ⊂ C be open such that Γσ ∩ U = ∅ and Γσ ∩ ∂U = ∅.

We need to show: Γσ ⊂ U Take z0 ∈ Γσ ∩ U. = ⇒ ∃k0 ∈ N0 ∀k ≥ k0 : γσ

k ∩ U = ∅

γσ

k connected =

⇒ γσ

k ⊂ U for k ≥ k0

Thus Γσ ⊂ Γσ =

• k≥k0

γσ

k ⊂ U.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 10 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−. So how should we proceed?

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−. So how should we proceed? We "glue" our sets together with there 2πi-translates.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−. So how should we proceed? We "glue" our sets together with there 2πi-translates. Therefore define Y :=

• σ∈{+,−}

k∈Z

(Γσ + 2πik).

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−. So how should we proceed? We "glue" our sets together with there 2πi-translates. Therefore define Y :=

• σ∈{+,−}

k∈Z

(Γσ + 2πik). Because Γ+ ∪ Γ− is connected, Y is connected as well.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−. So how should we proceed? We "glue" our sets together with there 2πi-translates. Therefore define Y :=

• σ∈{+,−}

k∈Z

(Γσ + 2πik). Because Γ+ ∪ Γ− is connected, Y is connected as well. Put Y0 := Y and Yj+1 := f −1(Yj) ∪ Yj for all j ∈ N0.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−. So how should we proceed? We "glue" our sets together with there 2πi-translates. Therefore define Y :=

• σ∈{+,−}

k∈Z

(Γσ + 2πik). Because Γ+ ∪ Γ− is connected, Y is connected as well. Put Y0 := Y and Yj+1 := f −1(Yj) ∪ Yj for all j ∈ N0. Then Yj is connected and Z :=

• j≥0

f −j(−1)

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−. So how should we proceed? We "glue" our sets together with there 2πi-translates. Therefore define Y :=

• σ∈{+,−}

k∈Z

(Γσ + 2πik). Because Γ+ ∪ Γ− is connected, Y is connected as well. Put Y0 := Y and Yj+1 := f −1(Yj) ∪ Yj for all j ∈ N0. Then Yj is connected and Z :=

• j≥0

f −j(−1) ⊂

• j≥0

Yj

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Now we got an idea of the contruction in S+ ∪ S−. So how should we proceed? We "glue" our sets together with there 2πi-translates. Therefore define Y :=

• σ∈{+,−}

k∈Z

(Γσ + 2πik). Because Γ+ ∪ Γ− is connected, Y is connected as well. Put Y0 := Y and Yj+1 := f −1(Yj) ∪ Yj for all j ∈ N0. Then Yj is connected and Z :=

• j≥0

f −j(−1) ⊂

• j≥0

Yj ⊂ I(f ).

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 11 / 16

Connectivity for real parameters Construction of the continuum

Because −1 ∈ J (f ) = C

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 12 / 16

Connectivity for real parameters Construction of the continuum

Because −1 ∈ J (f ) = C , we know from complex dynamics that Z is dense in C and in I(f ).

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 12 / 16

Connectivity for real parameters Construction of the continuum

Because −1 ∈ J (f ) = C , we know from complex dynamics that Z is dense in C and in I(f ). So

j≥0 Yj is the dense and connected subset of I(f ) we were looking for.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 12 / 16

Connectivity for real parameters Construction of the continuum

Because −1 ∈ J (f ) = C , we know from complex dynamics that Z is dense in C and in I(f ). So

j≥0 Yj is the dense and connected subset of I(f ) we were looking for.

Thus I(f ) is connected for all a ∈ (−1, ∞), which is our Theorem 1.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 12 / 16

The general case

The general case

To formulate the theorem for the general case, we need some preparation:

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 13 / 16

The general case

The general case

To formulate the theorem for the general case, we need some preparation:

Definition (Accessibility of a)

We say that the asymptotic value a of fa is accessible

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 13 / 16

The general case

The general case

To formulate the theorem for the general case, we need some preparation:

Definition (Accessibility of a)

We say that the asymptotic value a of fa is accessible if a ∈ J (fa)

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 13 / 16

The general case

The general case

To formulate the theorem for the general case, we need some preparation:

Definition (Accessibility of a)

We say that the asymptotic value a of fa is accessible if a ∈ J (fa) and there is an injective curve γ : [0, ∞) → J (fa) such that γ(0) = a

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 13 / 16

The general case

The general case

To formulate the theorem for the general case, we need some preparation:

Definition (Accessibility of a)

We say that the asymptotic value a of fa is accessible if a ∈ J (fa) and there is an injective curve γ : [0, ∞) → J (fa) such that γ(0) = a γ(t) ∈ I(fa) for all t > 0

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 13 / 16

The general case

The general case

To formulate the theorem for the general case, we need some preparation:

Definition (Accessibility of a)

We say that the asymptotic value a of fa is accessible if a ∈ J (fa) and there is an injective curve γ : [0, ∞) → J (fa) such that γ(0) = a γ(t) ∈ I(fa) for all t > 0 Re(γ(t)) → ∞ as t → ∞.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 13 / 16

The general case

The general case

To formulate the theorem for the general case, we need some preparation:

Definition (Accessibility of a)

We say that the asymptotic value a of fa is accessible if a ∈ J (fa) and there is an injective curve γ : [0, ∞) → J (fa) such that γ(0) = a γ(t) ∈ I(fa) for all t > 0 Re(γ(t)) → ∞ as t → ∞. We call such a curve a dynamic ray.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 13 / 16

The general case

The general case

To formulate the theorem for the general case, we need some preparation:

Definition (Accessibility of a)

We say that the asymptotic value a of fa is accessible if a ∈ J (fa) and there is an injective curve γ : [0, ∞) → J (fa) such that γ(0) = a γ(t) ∈ I(fa) for all t > 0 Re(γ(t)) → ∞ as t → ∞. We call such a curve a dynamic ray. Until now it is not known whether every a ∈ J (fa) is accessible or not.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 13 / 16

The general case

Taking preimages of γ under f ,

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 14 / 16

The general case

Taking preimages of γ under f , we get a partition of C \ f −1

a

(γ) into strips Sk,

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 14 / 16

The general case

Taking preimages of γ under f , we get a partition of C \ f −1

a

(γ) into strips Sk, where S0 is the strip containing r + iπ for r large and Sk = S0 + 2πik for all k ∈ Z.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 14 / 16

The general case

Taking preimages of γ under f , we get a partition of C \ f −1

a

(γ) into strips Sk, where S0 is the strip containing r + iπ for r large and Sk = S0 + 2πik for all k ∈ Z. S0 S1 S2 S−1 S−2 a γ f −1

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 14 / 16

The general case

Definition (External address)

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 15 / 16

The general case

Definition (External address)

For γ and Sk as before we call for z ∈ C the sequence u = u0u1u2 · · · ∈ ZN0 such that f j(z) ∈ Suj for all j ∈ N0 the external address of z.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 15 / 16

The general case

Definition (External address)

For γ and Sk as before we call for z ∈ C the sequence u = u0u1u2 · · · ∈ ZN0 such that f j(z) ∈ Suj for all j ∈ N0 the external address of z. The external address of a is called the kneading sequence

• f fa.
• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 15 / 16

The general case

Definition (External address)

For γ and Sk as before we call for z ∈ C the sequence u = u0u1u2 · · · ∈ ZN0 such that f j(z) ∈ Suj for all j ∈ N0 the external address of z. The external address of a is called the kneading sequence

• f fa.

Theorem (Rempe-Gillen 2011)

Let a ∈ C.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 15 / 16

The general case

Definition (External address)

For γ and Sk as before we call for z ∈ C the sequence u = u0u1u2 · · · ∈ ZN0 such that f j(z) ∈ Suj for all j ∈ N0 the external address of z. The external address of a is called the kneading sequence

• f fa.

Theorem (Rempe-Gillen 2011)

Let a ∈ C. If a ∈ I(fa)

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 15 / 16

The general case

Definition (External address)

For γ and Sk as before we call for z ∈ C the sequence u = u0u1u2 · · · ∈ ZN0 such that f j(z) ∈ Suj for all j ∈ N0 the external address of z. The external address of a is called the kneading sequence

• f fa.

Theorem (Rempe-Gillen 2011)

Let a ∈ C. If a ∈ I(fa) or a ∈ J (fa) \ I(fa) and is accessible with non periodic kneading sequence,

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 15 / 16

The general case

Definition (External address)

For γ and Sk as before we call for z ∈ C the sequence u = u0u1u2 · · · ∈ ZN0 such that f j(z) ∈ Suj for all j ∈ N0 the external address of z. The external address of a is called the kneading sequence

• f fa.

Theorem (Rempe-Gillen 2011)

Let a ∈ C. If a ∈ I(fa) or a ∈ J (fa) \ I(fa) and is accessible with non periodic kneading sequence, then I(fa) is connected.

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 15 / 16

The End

Thank you for your attention!

• P. Comdühr (CAU Kiel)

Escaping set of exponential maps 24 November 2015 16 / 16