Criniferous entire functions Criniferous functions Conjugacies - PowerPoint PPT Presentation

Criniferous entire functions Lasse Rempe Hairs Criniferous entire functions Criniferous functions Conjugacies Lasse Rempe Department of Mathematical Sciences, University of Liverpool On geometric complexity of Julia sets - II, Warsaw, August 2020

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Criniferous Conjugacies Adjective. Having hair; hairy. (From Latin crinis (“hair”) + ferre ("to bear").)

Criniferous entire Crinifer piscator functions Lasse Rempe Western plantain-eater Hairs Criniferous functions Conjugacies ( Crinifer piscator - Lotherton Hall, West Yorkshire by snowmanradio / CC BY-SA 2.0)

Criniferous entire Veturius criniferous functions Lasse Rempe Hairs Criniferous functions Conjugacies (Fonseca and Reyes-Castillo, Zootaxa 789 (2004), 1–26)

Criniferous entire Dynamics of entire functions functions Lasse Rempe Hairs Criniferous functions Conjugacies f : C → C non-constant, non-linear entire function Fatou set F ( f ) – set of normality; Julia set J ( f ) = C \ F ( f ) – set of non-normality; . = { z ∈ C : f n ( z ) → ∞} . Escaping set: I ( f ) .

Criniferous entire Dynamics of entire functions functions Lasse Rempe Hairs Criniferous functions Conjugacies f : C → C non-constant, non-linear entire function Fatou set F ( f ) – set of normality; Julia set J ( f ) = C \ F ( f ) – set of non-normality; . = { z ∈ C : f n ( z ) → ∞} . Escaping set: I ( f ) .

Criniferous entire Dynamics of entire functions functions Lasse Rempe Hairs Criniferous functions Conjugacies f : C → C non-constant, non-linear entire function Fatou set F ( f ) – set of normality; Julia set J ( f ) = C \ F ( f ) – set of non-normality; . = { z ∈ C : f n ( z ) → ∞} . Escaping set: I ( f ) .

Criniferous entire Dynamics of entire functions functions Lasse Rempe Hairs Criniferous functions Conjugacies f : C → C non-constant, non-linear entire function Fatou set F ( f ) – set of normality; Julia set J ( f ) = C \ F ( f ) – set of non-normality; . = { z ∈ C : f n ( z ) → ∞} . Escaping set: I ( f ) .

Criniferous entire Hairs functions Lasse Rempe Hairs Criniferous functions Conjugacies In 1926, Fatou observed that the escaping sets of certain functions contain arcs to infinity . In the 1980s, Devaney (with a number of collaborators) observed that there are many such curves for simple transcendental entire functions. These curves are called Devaney hairs , or just hairs .

Criniferous entire Hairs functions Lasse Rempe Hairs Criniferous functions Conjugacies In 1926, Fatou observed that the escaping sets of certain functions contain arcs to infinity . In the 1980s, Devaney (with a number of collaborators) observed that there are many such curves for simple transcendental entire functions. These curves are called Devaney hairs , or just hairs .

Criniferous entire Hairs functions Lasse Rempe Hairs Criniferous functions Conjugacies In 1926, Fatou observed that the escaping sets of certain functions contain arcs to infinity . In the 1980s, Devaney (with a number of collaborators) observed that there are many such curves for simple transcendental entire functions. These curves are called Devaney hairs , or just hairs .

Criniferous entire f ( z ) = e z − 2 functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire f ( z ) = e z − 2 functions Lasse Rempe Hairs 6 Criniferous functions 5 Conjugacies 4 3 2 1 -12 -12 -11 -11 -10 -10 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 1 2 3 4 5 6 7 8 9 10 10 11 11 12 12 13 13 -1 -1 -2 -2 f -3 -3 -4 -4 g

Criniferous entire f ( z ) = e z − 2 functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire f ( z ) = e z − 2 functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire f ( z ) = e z − 2 functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire Dynamic rays of polynomials functions Lasse Rempe Hairs Devaney–Goldberg–Hubbard (1980s): Think of hairs as analogues (and limits) Criniferous of dynamic rays of polynomials. functions Conjugacies

Criniferous entire Cantor bouquets functions Lasse Rempe f ( z ) = e z − 2 Hairs Criniferous functions Conjugacies

Criniferous entire Cantor bouquets functions Lasse Rempe f ( z ) = e z − 2 Hairs Criniferous J ( f ) is a Cantor bouquet (Aarts–Oversteegen, 1993): functions Conjugacies Every connected component C of J ( f ) is an arc to infinity (a ”hair” ); 1 J ( f ) is “topologically straight” , i.e. there is a homeomorphism ϕ : C → C 2 such that the image of every hair is a straight horizontal ray.

Criniferous entire Cantor bouquets functions Lasse Rempe f ( z ) = e z − 2 Hairs Criniferous J ( f ) is a Cantor bouquet (Aarts–Oversteegen, 1993): functions Conjugacies Every connected component C of J ( f ) is an arc to infinity (a ”hair” ); 1 J ( f ) is “topologically straight” , i.e. there is a homeomorphism ϕ : C → C 2 such that the image of every hair is a straight horizontal ray.

Criniferous entire Cantor bouquets functions Lasse Rempe f ( z ) = e z − 2 Hairs Criniferous J ( f ) is a Cantor bouquet (Aarts–Oversteegen, 1993): functions Conjugacies Every connected component C of J ( f ) is an arc to infinity (a ”hair” ); 1 J ( f ) is “topologically straight” , i.e. there is a homeomorphism ϕ : C → C 2 such that the image of every hair is a straight horizontal ray.

Criniferous entire Cantor bouquets functions Lasse Rempe f ( z ) = e z − 2 Hairs Criniferous J ( f ) is a Cantor bouquet (Aarts–Oversteegen, 1993): functions Conjugacies Every connected component C of J ( f ) is an arc to infinity (a ”hair” ); 1 J ( f ) is “topologically straight” , i.e. there is a homeomorphism ϕ : C → C 2 such that the image of every hair is a straight horizontal ray.

Criniferous entire Cantor bouquets functions Lasse Rempe f ( z ) = e z − 2 Hairs Criniferous functions Conjugacies

Criniferous entire Cantor bouquets functions Lasse Rempe f ( z ) = e z − 2 Hairs Criniferous functions Conjugacies (Image courtesy of A. Dezotti)

Criniferous entire The Eremenko-Lyubich class functions Lasse Rempe Hairs S ( f ) = { singular values of f } Criniferous functions = { “singularities of f − 1 ” } Conjugacies = { critical values } ∪ { asymptotic values } = { points over which f is not a covering } . Eremenko-Lyubich class: B . . = { f : C → C transcendental entire : S ( f ) is bounded } ⊃ S . Expansion: Any f ∈ B is strongly expanding where | f ( z ) | is large; � f n ( z ) → ∞ ⇒ z ∈ J ( f ) .

Criniferous entire The Eremenko-Lyubich class functions Lasse Rempe Hairs S ( f ) = { singular values of f } Criniferous functions = { “singularities of f − 1 ” } Conjugacies = { critical values } ∪ { asymptotic values } = { points over which f is not a covering } . Eremenko-Lyubich class: B . . = { f : C → C transcendental entire : S ( f ) is bounded } ⊃ S . Expansion: Any f ∈ B is strongly expanding where | f ( z ) | is large; � f n ( z ) → ∞ ⇒ z ∈ J ( f ) .

Criniferous entire The Eremenko-Lyubich class functions Lasse Rempe Hairs S ( f ) = { singular values of f } Criniferous functions = { “singularities of f − 1 ” } Conjugacies = { critical values } ∪ { asymptotic values } = { points over which f is not a covering } . Eremenko-Lyubich class: B . . = { f : C → C transcendental entire : S ( f ) is bounded } ⊃ S . Expansion: Any f ∈ B is strongly expanding where | f ( z ) | is large; � f n ( z ) → ∞ ⇒ z ∈ J ( f ) .

Criniferous entire The Eremenko-Lyubich class functions Lasse Rempe Hairs S ( f ) = { singular values of f } Criniferous functions = { “singularities of f − 1 ” } Conjugacies = { critical values } ∪ { asymptotic values } = { points over which f is not a covering } . Eremenko-Lyubich class: B . . = { f : C → C transcendental entire : S ( f ) is bounded } ⊃ S . Expansion: Any f ∈ B is strongly expanding where | f ( z ) | is large; � f n ( z ) → ∞ ⇒ z ∈ J ( f ) .

Criniferous entire The Eremenko-Lyubich class functions Lasse Rempe Hairs S ( f ) = { singular values of f } Criniferous functions = { “singularities of f − 1 ” } Conjugacies = { critical values } ∪ { asymptotic values } = { points over which f is not a covering } . Eremenko-Lyubich class: B . . = { f : C → C transcendental entire : S ( f ) is bounded } ⊃ S . Expansion: Any f ∈ B is strongly expanding where | f ( z ) | is large; � f n ( z ) → ∞ ⇒ z ∈ J ( f ) .

Criniferous entire The Eremenko-Lyubich class functions Lasse Rempe Hairs S ( f ) = { singular values of f } Criniferous functions = { “singularities of f − 1 ” } Conjugacies = { critical values } ∪ { asymptotic values } = { points over which f is not a covering } . Eremenko-Lyubich class: B . . = { f : C → C transcendental entire : S ( f ) is bounded } ⊃ S . Expansion: Any f ∈ B is strongly expanding where | f ( z ) | is large; � f n ( z ) → ∞ ⇒ z ∈ J ( f ) .