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 Motivation 1 P. Comdühr (CAU Kiel) Escaping set of exponential maps 24 November 2015 2 / 16

Outline Motivation 1 Connectivity for real parameters 2 P. Comdühr (CAU Kiel) Escaping set of exponential maps 24 November 2015 2 / 16

Outline Motivation 1 Connectivity for real parameters 2 The general case 3 P. Comdühr (CAU Kiel) Escaping set of exponential maps 24 November 2015 2 / 16

Motivation Consider the family f a : C → C , f a ( z ) = e z + a , a ∈ C . P. Comdühr (CAU Kiel) Escaping set of exponential maps 24 November 2015 3 / 16

Motivation Consider the family f a : C → C , f a ( z ) = e z + a , a ∈ C . Question: What can we say about the connectivity of its escaping sets I ( f a ) := { 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 f a : C → C , f a ( z ) = e z + a , a ∈ C . Question: What can we say about the connectivity of its escaping sets I ( f a ) := { 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 f a : C → C , f a ( z ) = e z + a , a ∈ C . Question: What can we say about the connectivity of its escaping sets I ( f a ) := { z ∈ C : f n a ( z ) → ∞ as n → ∞} ? 1.1 The case a = − 1 1.2 The case a = − 0 . 99 + 0 . 0001 i 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 ( f a ) . P. Comdühr (CAU Kiel) Escaping set of exponential maps 24 November 2015 4 / 16

Motivation Remark. For a ∈ ( − 1 , ∞ ) we have R ⊂ I ( f a ) . (Because f a ( 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 ( f a ) . (Because f a ( x ) ≥ x + ( 1 + a ) for all x ∈ R ) (Devaney, Krych 1984) J ( f a ) = C . P. Comdühr (CAU Kiel) Escaping set of exponential maps 24 November 2015 4 / 16

Motivation Remark. For a ∈ ( − 1 , ∞ ) we have R ⊂ I ( f a ) . (Because f a ( x ) ≥ x + ( 1 + a ) for all x ∈ R ) (Devaney, Krych 1984) J ( f a ) = C . Let us look at the Eremenko-Lyubich class B := { f : C → C entire and transcendental : sing ( f − 1 ) is bounded } . a P. Comdühr (CAU Kiel) Escaping set of exponential maps 24 November 2015 4 / 16

Motivation Remark. For a ∈ ( − 1 , ∞ ) we have R ⊂ I ( f a ) . (Because f a ( x ) ≥ x + ( 1 + a ) for all x ∈ R ) (Devaney, Krych 1984) J ( f a ) = C . Let us look at the Eremenko-Lyubich class B := { f : C → C entire and transcendental : sing ( f − 1 ) is bounded } . a For all a ∈ C we have f a ∈ B . P. Comdühr (CAU Kiel) Escaping set of exponential maps 24 November 2015 4 / 16

Motivation Remark. For a ∈ ( − 1 , ∞ ) we have R ⊂ I ( f a ) . (Because f a ( x ) ≥ x + ( 1 + a ) for all x ∈ R ) (Devaney, Krych 1984) J ( f a ) = C . Let us look at the Eremenko-Lyubich class B := { f : C → C entire and transcendental : sing ( f − 1 ) is bounded } . a For all a ∈ C we have f a ∈ 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 ( f a ) . (Because f a ( x ) ≥ x + ( 1 + a ) for all x ∈ R ) (Devaney, Krych 1984) J ( f a ) = C . Let us look at the Eremenko-Lyubich class B := { f : C → C entire and transcendental : sing ( f − 1 ) is bounded } . a For all a ∈ C we have f a ∈ 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 ( f a ) . (Because f a ( x ) ≥ x + ( 1 + a ) for all x ∈ R ) (Devaney, Krych 1984) J ( f a ) = C . Let us look at the Eremenko-Lyubich class B := { f : C → C entire and transcendental : sing ( f − 1 ) is bounded } . a For all a ∈ C we have f a ∈ 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 ( f a ) , then J ( f a ) and hence I ( f a ) 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 ( f a ) . (Because f a ( x ) ≥ x + ( 1 + a ) for all x ∈ R ) (Devaney, Krych 1984) J ( f a ) = C . Let us look at the Eremenko-Lyubich class B := { f : C → C entire and transcendental : sing ( f − 1 ) is bounded } . a For all a ∈ C we have f a ∈ 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 ( f a ) , then J ( f a ) and hence I ( f a ) is disconnected. In which way does the connectivity of I ( f a ) 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 ( f a ) . (Because f a ( x ) ≥ x + ( 1 + a ) for all x ∈ R ) (Devaney, Krych 1984) J ( f a ) = C . Let us look at the Eremenko-Lyubich class B := { f : C → C entire and transcendental : sing ( f − 1 ) is bounded } . a For all a ∈ C we have f a ∈ 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 ( f a ) , then J ( f a ) and hence I ( f a ) is disconnected. In which way does the connectivity of I ( f a ) 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) 1 Robert 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 ( f a ) is connected. 1 Robert 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 ( f a ) is connected. To prove the theorem, we construct a connected and dense subset of I ( f a ) . 1 Robert 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 ( f a ) is connected. To prove the theorem, we construct a connected and dense subset of I ( f a ) . The idea of this continuum is due to Devaney 1 : 1 Robert 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 ( f a ) is connected. To prove the theorem, we construct a connected and dense subset of I ( f a ) . The idea of this continuum is due to Devaney 1 : Denote S + := { z ∈ C : 0 < Im ( z ) < π } S − := { z ∈ C : − π < Im ( z ) < 0 } 1 Robert 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 ( f a ) is connected. To prove the theorem, we construct a connected and dense subset of I ( f a ) . The idea of this continuum is due to Devaney 1 : 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 } 1 Robert 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