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

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 ) Finally, Permutable TMFs Ping-pong Orbits Gustavo R. Ferreira School of Mathematics and Statistics The Open University 3 July 2020

The Permutable Outline Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points The Problem of Commuting Functions Permutable TMFs Transcendental Functions and Escaping Points Revisiting A ( f ) Finally, Permutable TMFs Ping-pong Orbits Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs Ping-pong Orbits

The Permutable The Problem of Commuting Functions Mystery Tour The Problem Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs Given two analytic functions f and g , is it true that Ping-pong Orbits f ◦ g = g ◦ f ⇔ J ( f ) = J ( g ) ?

The Permutable The Problem of Commuting Functions Mystery Tour The Answer Gustavo R. Ferreira The Problem of Commuting ◮ Polynomials Functions ⇒ Fatou, 1920; Julia, 1922; Beardon, 1990 Transcendental Functions and Escaping Points ⇐ Beardon, 1990; Schmidt & Steinmetz, 1995 Permutable TMFs * Terms and conditions apply Revisiting A ( f ) Finally, Permutable TMFs Ping-pong Orbits

The Permutable The Problem of Commuting Functions Mystery Tour The Answer Gustavo R. Ferreira The Problem of Commuting ◮ Polynomials Functions ⇒ Fatou, 1920; Julia, 1922; Beardon, 1990 Transcendental Functions and Escaping Points ⇐ Beardon, 1990; Schmidt & Steinmetz, 1995 Permutable TMFs * Terms and conditions apply Revisiting A ( f ) Finally, Permutable TMFs ◮ Rational functions Ping-pong Orbits ⇒ Fatou, 1920; Julia, 1922; F, 2019 ⇐ Levin & Przytycki, 1997; Ye, 2015 * Terms and conditions apply

The Permutable The Problem of Commuting Functions Mystery Tour The Answer Gustavo R. Ferreira The Problem of Commuting ◮ Polynomials Functions ⇒ Fatou, 1920; Julia, 1922; Beardon, 1990 Transcendental Functions and Escaping Points ⇐ Beardon, 1990; Schmidt & Steinmetz, 1995 Permutable TMFs * Terms and conditions apply Revisiting A ( f ) Finally, Permutable TMFs ◮ Rational functions Ping-pong Orbits ⇒ 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

The Permutable The Problem of Commuting Functions Mystery Tour The Answer Gustavo R. Ferreira The Problem of Commuting ◮ Polynomials Functions ⇒ Fatou, 1920; Julia, 1922; Beardon, 1990 Transcendental Functions and Escaping Points ⇐ Beardon, 1990; Schmidt & Steinmetz, 1995 Permutable TMFs * Terms and conditions apply Revisiting A ( f ) Finally, Permutable TMFs ◮ Rational functions Ping-pong Orbits ⇒ 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

The Permutable Outline Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points The Problem of Commuting Functions Permutable TMFs Transcendental Functions and Escaping Points Revisiting A ( f ) Finally, Permutable TMFs Ping-pong Orbits Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs Ping-pong Orbits

The Permutable Notation Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs f , g Transcendental (entire or meromorphic) functions Revisiting A ( f ) Finally, Permutable TMFs Ping-pong Orbits

The Permutable Notation Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs f , g Transcendental (entire or meromorphic) functions Revisiting A ( f ) M ( r , f ) The maximum modulus function Finally, Permutable TMFs Ping-pong Orbits M ( r , f ) := max {| f ( z ) | : | z | = r }

The Permutable Notation Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points Permutable TMFs f , g Transcendental (entire or meromorphic) functions Revisiting A ( f ) M ( r , f ) The maximum modulus function Finally, Permutable TMFs Ping-pong Orbits M ( r , f ) := max {| f ( z ) | : | z | = r } TEF Transcendental entire function TMF Transcendental meromorphic (non-entire) function

The Permutable The problem with escaping points Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points ◮ The escaping set (Eremenko, 1989; Dom´ ınguez, 1998): Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs I ( f ) := { z ∈ C : f n ( z ) → ∞ as n → + ∞} Ping-pong Orbits

The Permutable The problem with escaping points Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points ◮ The escaping set (Eremenko, 1989; Dom´ ınguez, 1998): Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs I ( f ) := { z ∈ C : f n ( z ) → ∞ as n → + ∞} Ping-pong Orbits ◮ f is transcendental ⇒ infinity is an essential singularity

The Permutable The problem with escaping points Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points ◮ The escaping set (Eremenko, 1989; Dom´ ınguez, 1998): Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs I ( f ) := { z ∈ C : f n ( z ) → ∞ as n → + ∞} Ping-pong Orbits ◮ f is transcendental ⇒ infinity is an essential singularity ⇒ points in I ( f ) escape at very different rates

The Permutable The problem with escaping points Mystery Tour Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points ◮ The escaping set (Eremenko, 1989; Dom´ ınguez, 1998): Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs I ( f ) := { z ∈ C : f n ( z ) → ∞ as n → + ∞} Ping-pong Orbits ◮ 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

The Permutable Progress for Transcendental Entire Functions Mystery Tour Baker, 1958 & 1984 Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points 1958: If f ◦ g = g ◦ f , then g ( J ( f )) ⊂ J ( f ) Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs Ping-pong Orbits

The Permutable Progress for Transcendental Entire Functions Mystery Tour Baker, 1958 & 1984 Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points 1958: If f ◦ g = g ◦ f , then g ( J ( f )) ⊂ J ( f ) Permutable TMFs Revisiting A ( f ) ! It suffices to show that f ◦ g = g ◦ f ⇒ g ( F ( f )) ⊂ F ( f ) Finally, Permutable TMFs Ping-pong Orbits

The Permutable Progress for Transcendental Entire Functions Mystery Tour Baker, 1958 & 1984 Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points 1958: If f ◦ g = g ◦ f , then g ( J ( f )) ⊂ J ( f ) Permutable TMFs Revisiting A ( f ) ! It suffices to show that f ◦ g = g ◦ f ⇒ g ( F ( f )) ⊂ F ( f ) Finally, Permutable TMFs Ping-pong Orbits 1984: If f ◦ g = g ◦ f and U is a non-escaping Fatou component of f , then g ( U ) ⊂ F ( f )

The Permutable Progress for Transcendental Entire Functions Mystery Tour Baker, 1958 & 1984 Gustavo R. Ferreira The Problem of Commuting Functions Transcendental Functions and Escaping Points 1958: If f ◦ g = g ◦ f , then g ( J ( f )) ⊂ J ( f ) Permutable TMFs Revisiting A ( f ) ! It suffices to show that f ◦ g = g ◦ f ⇒ g ( F ( f )) ⊂ F ( f ) Finally, Permutable TMFs Ping-pong Orbits 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 ).

The Permutable Progress for Transcendental Entire Functions Mystery Tour Bergweiler & Hinkkanen, 1999 Gustavo R. Ferreira The Problem of Define the fast escaping set as Commuting Functions Transcendental Functions A ( f ) := { z : ∃ ℓ ∈ N s.t. | f n + ℓ ( z ) | ≥ M n ( r , f ) for all n ∈ N } . and Escaping Points Permutable TMFs Revisiting A ( f ) Finally, Permutable TMFs Ping-pong Orbits

The Permutable Progress for Transcendental Entire Functions Mystery Tour Bergweiler & Hinkkanen, 1999 Gustavo R. Ferreira The Problem of Define the fast escaping set as Commuting Functions Transcendental Functions A ( f ) := { z : ∃ ℓ ∈ N s.t. | f n + ℓ ( z ) | ≥ M n ( r , f ) for all n ∈ N } . and Escaping Points Permutable TMFs Revisiting A ( f ) Then, Finally, Permutable TMFs Ping-pong Orbits ◮ A ( f ) ⊂ I ( f )