The Permutable Mystery Tour Transcendental Functions and Escaping - - PowerPoint PPT Presentation

▶
the permutable mystery tour
SMART_READER_LITE
LIVE PREVIEW

The Permutable Mystery Tour Transcendental Functions and Escaping - - PowerPoint PPT Presentation

Nov 27, 2022 46 likes •710 views

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions The Permutable Mystery Tour Transcendental Functions and Escaping Points For transcendental meromorphic functions Permutable TMFs Revisiting A ( f )

slide-1
SLIDE 1

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Permutable Mystery Tour

For transcendental meromorphic functions Gustavo R. Ferreira

School of Mathematics and Statistics The Open University

3 July 2020

slide-2
SLIDE 2

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outline

The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

slide-3
SLIDE 3

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem of Commuting Functions

The Problem

Given two analytic functions f and g, is it true that f ◦ g = g ◦ f ⇔ J(f ) = J(g) ?

slide-4
SLIDE 4

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem of Commuting Functions

The Answer

◮ Polynomials

⇒ Fatou, 1920; Julia, 1922; Beardon, 1990 ⇐ Beardon, 1990; Schmidt & Steinmetz, 1995

* Terms and conditions apply

slide-5
SLIDE 5

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem of Commuting Functions

The Answer

◮ Polynomials

⇒ Fatou, 1920; Julia, 1922; Beardon, 1990 ⇐ Beardon, 1990; Schmidt & Steinmetz, 1995

* Terms and conditions apply

◮ Rational functions

⇒ Fatou, 1920; Julia, 1922; F, 2019 ⇐ Levin & Przytycki, 1997; Ye, 2015

* Terms and conditions apply

slide-6
SLIDE 6

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem of Commuting Functions

The Answer

◮ Polynomials

⇒ Fatou, 1920; Julia, 1922; Beardon, 1990 ⇐ Beardon, 1990; Schmidt & Steinmetz, 1995

* Terms and conditions apply

◮ Rational functions

⇒ Fatou, 1920; Julia, 1922; F, 2019 ⇐ Levin & Przytycki, 1997; Ye, 2015

* Terms and conditions apply

◮ Transcendental entire functions

⇒ Baker, 1984; Bergweiler & Hinkkanen, 1999; Benini, Rippon & Stallard, 2016

slide-7
SLIDE 7

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem of Commuting Functions

The Answer

◮ Polynomials

⇒ Fatou, 1920; Julia, 1922; Beardon, 1990 ⇐ Beardon, 1990; Schmidt & Steinmetz, 1995

* Terms and conditions apply

◮ Rational functions

⇒ Fatou, 1920; Julia, 1922; F, 2019 ⇐ Levin & Przytycki, 1997; Ye, 2015

* Terms and conditions apply

◮ Transcendental entire functions

⇒ Baker, 1984; Bergweiler & Hinkkanen, 1999; Benini, Rippon & Stallard, 2016

◮ Transcendental meromorphic functions

⇒ Tsantaris, 2019; F, 2020

slide-8
SLIDE 8

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outline

The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

slide-9
SLIDE 9

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Notation

f , g Transcendental (entire or meromorphic) functions

slide-10
SLIDE 10

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Notation

f , g Transcendental (entire or meromorphic) functions M(r, f ) The maximum modulus function M(r, f ) := max{|f (z)| : |z| = r}

slide-11
SLIDE 11

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Notation

f , g Transcendental (entire or meromorphic) functions M(r, f ) The maximum modulus function M(r, f ) := max{|f (z)| : |z| = r} TEF Transcendental entire function TMF Transcendental meromorphic (non-entire) function

slide-12
SLIDE 12

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The problem with escaping points

◮ The escaping set (Eremenko, 1989; Dom´ ınguez, 1998): I(f ) := {z ∈ C : f n(z) → ∞ as n → +∞}

slide-13
SLIDE 13

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The problem with escaping points

◮ The escaping set (Eremenko, 1989; Dom´ ınguez, 1998): I(f ) := {z ∈ C : f n(z) → ∞ as n → +∞} ◮ f is transcendental ⇒ infinity is an essential singularity

slide-14
SLIDE 14

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The problem with escaping points

◮ The escaping set (Eremenko, 1989; Dom´ ınguez, 1998): I(f ) := {z ∈ C : f n(z) → ∞ as n → +∞} ◮ f is transcendental ⇒ infinity is an essential singularity

⇒ points in I(f ) escape at very different rates

slide-15
SLIDE 15

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The problem with escaping points

◮ The escaping set (Eremenko, 1989; Dom´ ınguez, 1998): I(f ) := {z ∈ C : f n(z) → ∞ as n → +∞} ◮ f is transcendental ⇒ infinity is an essential singularity

⇒ points in I(f ) escape at very different rates

◮ I(f ) is never empty, and is dense in the Julia set

slide-16
SLIDE 16

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Baker, 1958 & 1984

1958: If f ◦ g = g ◦ f , then g (J(f )) ⊂ J(f )

slide-17
SLIDE 17

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Baker, 1958 & 1984

1958: If f ◦ g = g ◦ f , then g (J(f )) ⊂ J(f )

! It suffices to show that f ◦ g = g ◦ f ⇒ g (F(f )) ⊂ F(f )

slide-18
SLIDE 18

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Baker, 1958 & 1984

1958: If f ◦ g = g ◦ f , then g (J(f )) ⊂ J(f )

! It suffices to show that f ◦ g = g ◦ f ⇒ g (F(f )) ⊂ F(f )

1984: If f ◦ g = g ◦ f and U is a non-escaping Fatou component of f , then g(U) ⊂ F(f )

slide-19
SLIDE 19

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Baker, 1958 & 1984

1958: If f ◦ g = g ◦ f , then g (J(f )) ⊂ J(f )

! It suffices to show that f ◦ g = g ◦ f ⇒ g (F(f )) ⊂ F(f )

1984: If f ◦ g = g ◦ f and U is a non-escaping Fatou component of f , then g(U) ⊂ F(f )

Baker, 1984

Let f and g be commuting TEFs without escaping Fatou

  • components. Then, J(f ) = J(g).
slide-20
SLIDE 20

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Bergweiler & Hinkkanen, 1999

Define the fast escaping set as A(f ) := {z : ∃ ℓ ∈ N s.t. |f n+ℓ(z)| ≥ Mn(r, f ) for all n ∈ N}.

slide-21
SLIDE 21

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Bergweiler & Hinkkanen, 1999

Define the fast escaping set as A(f ) := {z : ∃ ℓ ∈ N s.t. |f n+ℓ(z)| ≥ Mn(r, f ) for all n ∈ N}. Then, ◮ A(f ) ⊂ I(f )

slide-22
SLIDE 22

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Bergweiler & Hinkkanen, 1999

Define the fast escaping set as A(f ) := {z : ∃ ℓ ∈ N s.t. |f n+ℓ(z)| ≥ Mn(r, f ) for all n ∈ N}. Then, ◮ A(f ) ⊂ I(f ) ◮ A(f ) is dense in J(f )

slide-23
SLIDE 23

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Bergweiler & Hinkkanen, 1999

Define the fast escaping set as A(f ) := {z : ∃ ℓ ∈ N s.t. |f n+ℓ(z)| ≥ Mn(r, f ) for all n ∈ N}. Then, ◮ A(f ) ⊂ I(f ) ◮ A(f ) is dense in J(f ) ◮ Every Fatou component in A(f ) is a wandering domain

slide-24
SLIDE 24

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Bergweiler & Hinkkanen, 1999

Define the fast escaping set as A(f ) := {z : ∃ ℓ ∈ N s.t. |f n+ℓ(z)| ≥ Mn(r, f ) for all n ∈ N}. Then, ◮ A(f ) ⊂ I(f ) ◮ A(f ) is dense in J(f ) ◮ Every Fatou component in A(f ) is a wandering domain ◮ If f ◦ g = g ◦ f , then g−1 (A(f )) ⊂ A(f )

slide-25
SLIDE 25

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Bergweiler & Hinkkanen, 1999

Define the fast escaping set as A(f ) := {z : ∃ ℓ ∈ N s.t. |f n+ℓ(z)| ≥ Mn(r, f ) for all n ∈ N}. Then, ◮ A(f ) ⊂ I(f ) ◮ A(f ) is dense in J(f ) ◮ Every Fatou component in A(f ) is a wandering domain ◮ If f ◦ g = g ◦ f , then g−1 (A(f )) ⊂ A(f )

Bergweiler & Hinkkanen, 1999

Let f and g be commuting TEFs without fast escaping wandering domains. Then, J(f ) = J(g).

slide-26
SLIDE 26

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Benini, Rippon & Stallard, 2016

BRS, 2016

If U is a multiply connected wandering domain of f and f ◦ g = g ◦ f , then g(U) is also a multiply connected wandering domain of f .

slide-27
SLIDE 27

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Progress for Transcendental Entire Functions

Benini, Rippon & Stallard, 2016

BRS, 2016

If U is a multiply connected wandering domain of f and f ◦ g = g ◦ f , then g(U) is also a multiply connected wandering domain of f .

So, where are we now?

Let f and g be commuting transcendental entire functions without simply connected fast escaping wandering domains. Then, J(f ) = J(g).

slide-28
SLIDE 28

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outline

The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

slide-29
SLIDE 29

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outer Sequences

Rippon & Stallard, 2005

Let f be a transcendental meromorphic function with finitely many poles.

slide-30
SLIDE 30

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outer Sequences

Rippon & Stallard, 2005

Let f be a transcendental meromorphic function with finitely many poles. ◮ Let γ be a Jordan curve. Its outer set is defined as its unbounded complementary component

slide-31
SLIDE 31

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outer Sequences

Rippon & Stallard, 2005

Let f be a transcendental meromorphic function with finitely many poles. ◮ Let γ be a Jordan curve. Its outer set is defined as its unbounded complementary component ◮ Outer sequence for f : Jordan curves γn with outer sets En s.t.

slide-32
SLIDE 32

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outer Sequences

Rippon & Stallard, 2005

Let f be a transcendental meromorphic function with finitely many poles. ◮ Let γ be a Jordan curve. Its outer set is defined as its unbounded complementary component ◮ Outer sequence for f : Jordan curves γn with outer sets En s.t.

◮ Every γn surrounds all the poles of f

slide-33
SLIDE 33

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outer Sequences

Rippon & Stallard, 2005

Let f be a transcendental meromorphic function with finitely many poles. ◮ Let γ be a Jordan curve. Its outer set is defined as its unbounded complementary component ◮ Outer sequence for f : Jordan curves γn with outer sets En s.t.

◮ Every γn surrounds all the poles of f ◮ γn → ∞ as n → +∞

slide-34
SLIDE 34

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outer Sequences

Rippon & Stallard, 2005

Let f be a transcendental meromorphic function with finitely many poles. ◮ Let γ be a Jordan curve. Its outer set is defined as its unbounded complementary component ◮ Outer sequence for f : Jordan curves γn with outer sets En s.t.

◮ Every γn surrounds all the poles of f ◮ γn → ∞ as n → +∞ ◮ γn+1 ⊂ f (γn)

slide-35
SLIDE 35

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outer Sequences

Rippon & Stallard, 2005

Let f be a transcendental meromorphic function with finitely many poles. ◮ Let γ be a Jordan curve. Its outer set is defined as its unbounded complementary component ◮ Outer sequence for f : Jordan curves γn with outer sets En s.t.

◮ Every γn surrounds all the poles of f ◮ γn → ∞ as n → +∞ ◮ γn+1 ⊂ f (γn) ◮ Every component of f −1(En+1) either lies in En or is surrounded by γ1

slide-36
SLIDE 36

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

A(f ) Via Outer Sequences

Rippon & Stallard, 2005

Let f be a TMF with finitely many poles, and En an outer sequence for f . Then, A(f ) := {z : ∃ ℓ ∈ N s.t. f n+ℓ(z) ∈ En for all n ∈ N}.

slide-37
SLIDE 37

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

A(f ) Via Outer Sequences

Rippon & Stallard, 2005

Let f be a TMF with finitely many poles, and En an outer sequence for f . Then, A(f ) := {z : ∃ ℓ ∈ N s.t. f n+ℓ(z) ∈ En for all n ∈ N}. ◮ A(f ) is independent of the choice of outer sequence

slide-38
SLIDE 38

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

A(f ) Via Outer Sequences

Rippon & Stallard, 2005

Let f be a TMF with finitely many poles, and En an outer sequence for f . Then, A(f ) := {z : ∃ ℓ ∈ N s.t. f n+ℓ(z) ∈ En for all n ∈ N}. ◮ A(f ) is independent of the choice of outer sequence ◮ If f is entire, this definition agrees with the previous one

slide-39
SLIDE 39

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

A(f ) Via Outer Sequences

Rippon & Stallard, 2005

Let f be a TMF with finitely many poles, and En an outer sequence for f . Then, A(f ) := {z : ∃ ℓ ∈ N s.t. f n+ℓ(z) ∈ En for all n ∈ N}. ◮ A(f ) is independent of the choice of outer sequence ◮ If f is entire, this definition agrees with the previous one ◮ A(f ) is dense in the Julia set

slide-40
SLIDE 40

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outline

The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

slide-41
SLIDE 41

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Hold Our Horses

Osborne & Sixsmith, 2016

◮ TMFs f and g commute iff, for every z ∈ C, either f (g(z)) = g (f (z)) or neither side is defined.

slide-42
SLIDE 42

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Hold Our Horses

Osborne & Sixsmith, 2016

◮ TMFs f and g commute iff, for every z ∈ C, either f (g(z)) = g (f (z)) or neither side is defined.

⇒ If f and g commute, they have the same poles

slide-43
SLIDE 43

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Hold Our Horses

Osborne & Sixsmith, 2016

◮ TMFs f and g commute iff, for every z ∈ C, either f (g(z)) = g (f (z)) or neither side is defined.

⇒ If f and g commute, they have the same poles

◮ A TMF f commutes with at most countably many

  • ther TMFs
slide-44
SLIDE 44

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Hold Our Horses

Osborne & Sixsmith, 2016

◮ TMFs f and g commute iff, for every z ∈ C, either f (g(z)) = g (f (z)) or neither side is defined.

⇒ If f and g commute, they have the same poles

◮ A TMF f commutes with at most countably many

  • ther TMFs

◮ If a TMF commutes with a rational function R, then R is a M¨

  • bius transformation
slide-45
SLIDE 45

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Plan

We need meromorphic versions of the following results:

Baker, 1984

Let f and g be permutable TEFs. If U is a non-escaping Fatou component of f , then g(U) ⊂ F(f ).

slide-46
SLIDE 46

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Plan

We need meromorphic versions of the following results:

Baker, 1984

Let f and g be permutable TEFs. If U is a non-escaping Fatou component of f , then g(U) ⊂ F(f ).

Bergweiler & Hinkkanen, 1999

Let f and g be permutable TEFs. If z ∈ C \ A(f ), then g(z) / ∈ A(f ).

slide-47
SLIDE 47

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Outcome

F, 2020

Let f and g be permutable TMFs. If U is a non-escaping Fatou component of f , then g(U) ⊂ F(f ).

slide-48
SLIDE 48

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Outcome

F, 2020

Let f and g be permutable TMFs. If U is a non-escaping Fatou component of f , then g(U) ⊂ F(f ).

F, 2020

Let f and g be permutable TMFs with finitely many poles. If z ∈ I(f ) \ A(f ), then g(z) / ∈ A(f ).

slide-49
SLIDE 49

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Outcome

F, 2020

Let f and g be permutable TMFs. If U is a non-escaping Fatou component of f , then g(U) ⊂ F(f ).

F, 2020

Let f and g be permutable TMFs with finitely many poles. If z ∈ I(f ) \ A(f ), then g(z) / ∈ A(f ).

F, 2020

Let f and g be TMFs with finitely many poles s.t. A(f ) ⊂ J(f ) and A(g) ⊂ J(g). Then, f ◦ g = g ◦ f ⇒ J(f ) = J(g).

slide-50
SLIDE 50

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

All Together Now

Tsantaris, 2019

Let f and g be permutable TMFs not in class P. Then, J(f ) = J(g).

slide-51
SLIDE 51

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

All Together Now

Tsantaris, 2019

Let f and g be permutable TMFs not in class P. Then, J(f ) = J(g).

  • Class P: f (z) = z0 +

eg(z) (z − z0)m

slide-52
SLIDE 52

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

All Together Now

Tsantaris, 2019

Let f and g be permutable TMFs not in class P. Then, J(f ) = J(g).

  • Class P: f (z) = z0 +

eg(z) (z − z0)m

  • So, where are we now?

Let f and g be permutable TMFs. Then, J(f ) = J(g) except possibly when f and g are in class P and have simply connected fast escaping wandering domains.

slide-53
SLIDE 53

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Outline

The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

slide-54
SLIDE 54

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem With Poles

◮ Let f and g be permutable TEFs

slide-55
SLIDE 55

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem With Poles

◮ Let f and g be permutable TEFs

◮ Then, if z / ∈ I(f ), we have g(z) / ∈ I(f )

slide-56
SLIDE 56

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem With Poles

◮ Let f and g be permutable TEFs

◮ Then, if z / ∈ I(f ), we have g(z) / ∈ I(f )

◮ What about permutable TMFs?

slide-57
SLIDE 57

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem With Poles

◮ Let f and g be permutable TEFs

◮ Then, if z / ∈ I(f ), we have g(z) / ∈ I(f )

◮ What about permutable TMFs?

◮ If z is s.t. f nk(z) → p, then it is possible that g(z) ∈ I(f )

slide-58
SLIDE 58

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

The Problem With Poles

p . . . . . . . ...

slide-59
SLIDE 59

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Ping-pong Orbits

Definition

We say that z ∈ C has a ping-pong orbit if there exist a pole p of f , subsequences (f mk)k≥1 and (f nk)k≥1 and a natural number M ≥ 1 s.t.

slide-60
SLIDE 60

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Ping-pong Orbits

Definition

We say that z ∈ C has a ping-pong orbit if there exist a pole p of f , subsequences (f mk)k≥1 and (f nk)k≥1 and a natural number M ≥ 1 s.t. (i) f mk(z) → p and f nk(z) → ∞;

slide-61
SLIDE 61

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Ping-pong Orbits

Definition

We say that z ∈ C has a ping-pong orbit if there exist a pole p of f , subsequences (f mk)k≥1 and (f nk)k≥1 and a natural number M ≥ 1 s.t. (i) f mk(z) → p and f nk(z) → ∞; (ii) |mk − nk| ≤ M and |nk − mk+1| ≤ M.

slide-62
SLIDE 62

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Ping-pong Orbits

Definition

We say that z ∈ C has a ping-pong orbit if there exist a pole p of f , subsequences (f mk)k≥1 and (f nk)k≥1 and a natural number M ≥ 1 s.t. (i) f mk(z) → p and f nk(z) → ∞; (ii) |mk − nk| ≤ M and |nk − mk+1| ≤ M. The set of all points with a ping-pong orbit is denoted by BUP(f ).

slide-63
SLIDE 63

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Ping-pong Orbits

And where to find them

F, 2020

Let f be a TMF with finitely many poles. Then, BUP(f ) is dense in J(f ). Furthermore, if f / ∈ P, then points in BUP(f ) can “escape” arbitrarily fast.

slide-64
SLIDE 64

The Permutable Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions

Transcendental Functions and Escaping Points

Permutable TMFs

Revisiting A(f ) Finally, Permutable TMFs Ping-pong Orbits

Ping-pong Orbits

And where to find them

F, 2020

Let f be a TMF with finitely many poles. Then, BUP(f ) is dense in J(f ). Furthermore, if f / ∈ P, then points in BUP(f ) can “escape” arbitrarily fast.

F, 2020

There exist TMFs with a single pole, both in class P and

  • utside of it, with ping-pong wandering domains.

* This theorem was brought to you by David Mart´ ı-Pete

slide-65
SLIDE 65

The Permutable Mystery Tour Gustavo R. Ferreira Appendix

References

References I

I.N. Baker Wandering domains in the iteration of entire functions.

  • Proc. London Math. Soc., 49:563–576, 1984.

A.F. Beardon Symmetries of Julia sets.

  • Bull. London Math. Soc., 22:576–582, 1990.
  • W. Bergweiler and A. Hinkkanen

On semiconjugation of entire functions.

  • Math. Proc. Camb. Phil. Soc., 126:565–574, 1999.

A.M. Benini, P.J. Rippon and G.M. Stallard Permutable entire functions and multiply connected wandering domains. Advances in Mathematics, 287:451–462, 2016.

slide-66
SLIDE 66

The Permutable Mystery Tour Gustavo R. Ferreira Appendix

References

References II

G.R. Ferreira Escaping points of commuting meromorphic functions with finitely many poles. Available on arXiv, 2020.