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Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire functions

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 Conjugacies

Criniferous

• Adjective. Having hair; hairy.

(From Latin crinis (“hair”) + ferre ("to bear").)

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Crinifer piscator

Western plantain-eater

(Crinifer piscator - Lotherton Hall, West Yorkshire by snowmanradio / CC BY-SA 2.0)

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Veturius criniferous

(Fonseca and Reyes-Castillo, Zootaxa 789 (2004), 1–26)

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics of entire functions

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; Escaping set: I(f) .

.= {z ∈ C: f n(z) → ∞}.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics of entire functions

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; Escaping set: I(f) .

.= {z ∈ C: f n(z) → ∞}.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics of entire functions

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; Escaping set: I(f) .

.= {z ∈ C: f n(z) → ∞}.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics of entire functions

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; Escaping set: I(f) .

.= {z ∈ C: f n(z) → ∞}.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Hairs

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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

Hairs

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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

Hairs

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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

f(z) = ez − 2

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

f(z) = ez − 2

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f g

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

f(z) = ez − 2

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

f(z) = ez − 2

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

f(z) = ez − 2

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamic rays of polynomials

Devaney–Goldberg–Hubbard (1980s): Think of hairs as analogues (and limits)

• f dynamic rays of polynomials.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Cantor bouquets

f(z) = ez − 2

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Cantor bouquets

f(z) = ez − 2 J(f) is a Cantor bouquet (Aarts–Oversteegen, 1993):

1

Every connected component C of J(f) is an arc to infinity (a ”hair”);

2

J(f) is “topologically straight”, i.e. there is a homeomorphism ϕ: C → C such that the image of every hair is a straight horizontal ray.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Cantor bouquets

f(z) = ez − 2 J(f) is a Cantor bouquet (Aarts–Oversteegen, 1993):

1

Every connected component C of J(f) is an arc to infinity (a ”hair”);

2

J(f) is “topologically straight”, i.e. there is a homeomorphism ϕ: C → C such that the image of every hair is a straight horizontal ray.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Cantor bouquets

f(z) = ez − 2 J(f) is a Cantor bouquet (Aarts–Oversteegen, 1993):

1

Every connected component C of J(f) is an arc to infinity (a ”hair”);

2

J(f) is “topologically straight”, i.e. there is a homeomorphism ϕ: C → C such that the image of every hair is a straight horizontal ray.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Cantor bouquets

f(z) = ez − 2 J(f) is a Cantor bouquet (Aarts–Oversteegen, 1993):

1

Every connected component C of J(f) is an arc to infinity (a ”hair”);

2

J(f) is “topologically straight”, i.e. there is a homeomorphism ϕ: C → C such that the image of every hair is a straight horizontal ray.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Cantor bouquets

f(z) = ez − 2

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Cantor bouquets

f(z) = ez − 2

(Image courtesy of A. Dezotti)

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

The Eremenko-Lyubich class

S(f) = {singular values of f} = {“singularities of f −1”} = {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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

The Eremenko-Lyubich class

S(f) = {singular values of f} = {“singularities of f −1”} = {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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

The Eremenko-Lyubich class

S(f) = {singular values of f} = {“singularities of f −1”} = {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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

The Eremenko-Lyubich class

S(f) = {singular values of f} = {“singularities of f −1”} = {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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

The Eremenko-Lyubich class

S(f) = {singular values of f} = {“singularities of f −1”} = {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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

The Eremenko-Lyubich class

S(f) = {singular values of f} = {“singularities of f −1”} = {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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

The Eremenko-Lyubich class

S(f) = {singular values of f} = {“singularities of f −1”} = {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 functions Lasse Rempe Hairs Criniferous functions Conjugacies

Hairs in the Eremenko-Lyubich class

Rottenfußer–Rückert–R–Schleicher, Ann. of Math., 2011

Theorem (RRRS, 2011)

Let f ∈ B have finite order; i.e. log log|f(z)| = O(log |z|). (More generally, let f be a finite composition of such functions.) Then every point of I(f) can be connected to infinity by an arc in I(f).

Theorem (RRRS, 2011)

There is f ∈ B such that J(f) and hence I(f) contains no arc.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Hairs in the Eremenko-Lyubich class

Rottenfußer–Rückert–R–Schleicher, Ann. of Math., 2011

Theorem (RRRS, 2011)

Let f ∈ B have finite order; i.e. log log|f(z)| = O(log |z|). (More generally, let f be a finite composition of such functions.) Then every point of I(f) can be connected to infinity by an arc in I(f).

Theorem (RRRS, 2011)

There is f ∈ B such that J(f) and hence I(f) contains no arc.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Hairs in the Eremenko-Lyubich class

Rottenfußer–Rückert–R–Schleicher, Ann. of Math., 2011

Theorem (RRRS, 2011)

Let f ∈ B have finite order; i.e. log log|f(z)| = O(log |z|). (More generally, let f be a finite composition of such functions.) Then every point of I(f) can be connected to infinity by an arc in I(f).

Theorem (RRRS, 2011)

There is f ∈ B such that J(f) and hence I(f) contains no arc.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Hairs in the Eremenko-Lyubich class

Rottenfußer–Rückert–R–Schleicher, Ann. of Math., 2011

Theorem (RRRS, 2011)

Let f ∈ B have finite order; i.e. log log|f(z)| = O(log |z|). (More generally, let f be a finite composition of such functions.) Then every point of I(f) can be connected to infinity by an arc in I(f).

Theorem (RRRS, 2011)

There is f ∈ B such that J(f) and hence I(f) contains no arc.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Two responses

So, in general, there need not be any hairs (even in class B). How should we proceed?

1

Approach 1: Forget about hairs and use more general sets to partition the Julia set (“dreadlocks”, work joint with Benini).

2

Approach 2: Restrict to classes of functions where hairs exist.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Two responses

So, in general, there need not be any hairs (even in class B). How should we proceed?

1

Approach 1: Forget about hairs and use more general sets to partition the Julia set (“dreadlocks”, work joint with Benini).

2

Approach 2: Restrict to classes of functions where hairs exist.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Two responses

So, in general, there need not be any hairs (even in class B). How should we proceed?

1

Approach 1: Forget about hairs and use more general sets to partition the Julia set (“dreadlocks”, work joint with Benini).

2

Approach 2: Restrict to classes of functions where hairs exist.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire functions

Benini–R, GAFA, to appear

Definition (Benini-R, 2020)

A transcendental entire function f is criniferous if the following holds for all z ∈ I(f). For all sufficiently large n there exists an arc γn ⊂ I(f) connecting f n(z) to ∞ in such a way that

1

f(γn) = γn+1;

2

minz∈γn |z| → ∞;

3

f : γn → γn+1 is injective.

Theorem (RRRS, 2011)

Finite-order functions in B are criniferous.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire functions

Benini–R, GAFA, to appear

Definition (Benini-R, 2020)

A transcendental entire function f is criniferous if the following holds for all z ∈ I(f). For all sufficiently large n there exists an arc γn ⊂ I(f) connecting f n(z) to ∞ in such a way that

1

f(γn) = γn+1;

2

minz∈γn |z| → ∞;

3

f : γn → γn+1 is injective.

Theorem (RRRS, 2011)

Finite-order functions in B are criniferous.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire functions

Benini–R, GAFA, to appear

Definition (Benini-R, 2020)

A transcendental entire function f is criniferous if the following holds for all z ∈ I(f). For all sufficiently large n there exists an arc γn ⊂ I(f) connecting f n(z) to ∞ in such a way that

1

f(γn) = γn+1;

2

minz∈γn |z| → ∞;

3

f : γn → γn+1 is injective.

Theorem (RRRS, 2011)

Finite-order functions in B are criniferous.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire functions

Benini–R, GAFA, to appear

Definition (Benini-R, 2020)

A transcendental entire function f is criniferous if the following holds for all z ∈ I(f). For all sufficiently large n there exists an arc γn ⊂ I(f) connecting f n(z) to ∞ in such a way that

1

f(γn) = γn+1;

2

minz∈γn |z| → ∞;

3

f : γn → γn+1 is injective.

Theorem (RRRS, 2011)

Finite-order functions in B are criniferous.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire functions

Benini–R, GAFA, to appear

Definition (Benini-R, 2020)

A transcendental entire function f is criniferous if the following holds for all z ∈ I(f). For all sufficiently large n there exists an arc γn ⊂ I(f) connecting f n(z) to ∞ in such a way that

1

f(γn) = γn+1;

2

minz∈γn |z| → ∞;

3

f : γn → γn+1 is injective.

Theorem (RRRS, 2011)

Finite-order functions in B are criniferous.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous entire functions

Benini–R, GAFA, to appear

Definition (Benini-R, 2020)

A transcendental entire function f is criniferous if the following holds for all z ∈ I(f). For all sufficiently large n there exists an arc γn ⊂ I(f) connecting f n(z) to ∞ in such a way that

1

f(γn) = γn+1;

2

minz∈γn |z| → ∞;

3

f : γn → γn+1 is injective.

Theorem (RRRS, 2011)

Finite-order functions in B are criniferous.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

A landing theorem for criniferous functions

Benini–R, GAFA, to appear

Theorem (Benini-R, 2020)

Let f ∈ B be criniferous, and suppose that the post-singular set P(f) =

∞

• j=0

f j(S(f)) is bounded. Then for every repelling or parabolic periodic point z of f, there is an arc γ ⊂ I(f) connecting z to ∞, such that f n(γ) = γ for some n ≥ 1.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Disjoint-type functions

Bara´ nski-Karpi´ nska, 2007

Definition

The entire function f ∈ B is of disjoint-type if the Fatou set F(f)

1

is connected,

2

contains an attracting fixed point, and

3

contains S(f).

Observation (Bara´ nski-Karpi´ nska, 2007)

If f ∈ B, then λf is of disjoint type when |λ| is small.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Disjoint-type functions

Bara´ nski-Karpi´ nska, 2007

Definition

The entire function f ∈ B is of disjoint-type if the Fatou set F(f)

1

is connected,

2

contains an attracting fixed point, and

3

contains S(f).

Observation (Bara´ nski-Karpi´ nska, 2007)

If f ∈ B, then λf is of disjoint type when |λ| is small.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Disjoint-type functions

Bara´ nski-Karpi´ nska, 2007

Definition

The entire function f ∈ B is of disjoint-type if the Fatou set F(f)

1

is connected,

2

contains an attracting fixed point, and

3

contains S(f).

Observation (Bara´ nski-Karpi´ nska, 2007)

If f ∈ B, then λf is of disjoint type when |λ| is small.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Disjoint-type functions

Bara´ nski-Karpi´ nska, 2007

Definition

The entire function f ∈ B is of disjoint-type if the Fatou set F(f)

1

is connected,

2

contains an attracting fixed point, and

3

contains S(f).

Observation (Bara´ nski-Karpi´ nska, 2007)

If f ∈ B, then λf is of disjoint type when |λ| is small.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Disjoint-type functions

Bara´ nski-Karpi´ nska, 2007

Definition

The entire function f ∈ B is of disjoint-type if the Fatou set F(f)

1

is connected,

2

contains an attracting fixed point, and

3

contains S(f).

Observation (Bara´ nski-Karpi´ nska, 2007)

If f ∈ B, then λf is of disjoint type when |λ| is small.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous functions and Cantor bouquets

Theorem (Bara´ nski-Jarque-R, 2012)

If f is of disjoint type and of finite order, J(f) is a Cantor bouquet.

Question

If f is criniferous and of disjoint type, is J(f) a Cantor bouquet? Answer: No!

Theorem (Pardo-Simón-R)

There is f criniferous and of disjoint type such that J(f) is not a Cantor bouquet.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous functions and Cantor bouquets

Theorem (Bara´ nski-Jarque-R, 2012)

If f is of disjoint type and of finite order, J(f) is a Cantor bouquet.

Question

If f is criniferous and of disjoint type, is J(f) a Cantor bouquet? Answer: No!

Theorem (Pardo-Simón-R)

There is f criniferous and of disjoint type such that J(f) is not a Cantor bouquet.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Criniferous functions and Cantor bouquets

Theorem (Bara´ nski-Jarque-R, 2012)

If f is of disjoint type and of finite order, J(f) is a Cantor bouquet.

Question

If f is criniferous and of disjoint type, is J(f) a Cantor bouquet? Answer: No!

Theorem (Pardo-Simón-R)

There is f criniferous and of disjoint type such that J(f) is not a Cantor bouquet.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

The class CB

Definition (Pardo-Simón)

We say that f ∈ B belongs to the class CB if J(λf) is a Cantor bouquet for sufficiently small λ.

Theorem (Pardo-Simón)

All functions in CB are criniferous.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

The class CB

Definition (Pardo-Simón)

We say that f ∈ B belongs to the class CB if J(λf) is a Cantor bouquet for sufficiently small λ.

Theorem (Pardo-Simón)

All functions in CB are criniferous.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Absorbing Cantor bouquets

Pardo-Simón–R, in preparation

Definition

We say that a subset A ⊂ J(f) is absorbing if it is forward-invariant and every escaping point eventually enters A; i.e. I(f) ⊂

∞

• j=0

f −j(A).

Theorem (Pardo-Simón–R)

Let f ∈ B. Then f ∈ CB if and only if J(f) contains an absorbing Cantor bouquet.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Absorbing Cantor bouquets

Pardo-Simón–R, in preparation

Definition

We say that a subset A ⊂ J(f) is absorbing if it is forward-invariant and every escaping point eventually enters A; i.e. I(f) ⊂

∞

• j=0

f −j(A).

Theorem (Pardo-Simón–R)

Let f ∈ B. Then f ∈ CB if and only if J(f) contains an absorbing Cantor bouquet.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics on Cantor bouquets

Theorem (Aarts-Oversteegen, 1993)

Any two Cantor bouquets X, Y are ambiently homeomorphic. That is, there is a homeomorphism ϕ: C → C with ϕ(X) = ϕ(Y).

Corollary

If f, g ∈ CB are of disjoint type, then J(f) and J(g) are ambiently homeomorphic.

Question

Are f and g ambiently conjugate? I.e., is there a homeomorphism ϕ: C → C that restricts to a conjugacy between f|J(f) and g|J(g)?

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics on Cantor bouquets

Theorem (Aarts-Oversteegen, 1993)

Any two Cantor bouquets X, Y are ambiently homeomorphic. That is, there is a homeomorphism ϕ: C → C with ϕ(X) = ϕ(Y).

Corollary

If f, g ∈ CB are of disjoint type, then J(f) and J(g) are ambiently homeomorphic.

Question

Are f and g ambiently conjugate? I.e., is there a homeomorphism ϕ: C → C that restricts to a conjugacy between f|J(f) and g|J(g)?

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics on Cantor bouquets

Theorem (Aarts-Oversteegen, 1993)

Any two Cantor bouquets X, Y are ambiently homeomorphic. That is, there is a homeomorphism ϕ: C → C with ϕ(X) = ϕ(Y).

Corollary

If f, g ∈ CB are of disjoint type, then J(f) and J(g) are ambiently homeomorphic.

Question

Are f and g ambiently conjugate? I.e., is there a homeomorphism ϕ: C → C that restricts to a conjugacy between f|J(f) and g|J(g)?

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics on Cantor bouquets

Theorem (Aarts-Oversteegen, 1993)

Any two Cantor bouquets X, Y are ambiently homeomorphic. That is, there is a homeomorphism ϕ: C → C with ϕ(X) = ϕ(Y).

Corollary

If f, g ∈ CB are of disjoint type, then J(f) and J(g) are ambiently homeomorphic.

Question

Are f and g ambiently conjugate? I.e., is there a homeomorphism ϕ: C → C that restricts to a conjugacy between f|J(f) and g|J(g)?

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Dynamics on Cantor bouquets

Theorem (Aarts-Oversteegen, 1993)

Any two Cantor bouquets X, Y are ambiently homeomorphic. That is, there is a homeomorphism ϕ: C → C with ϕ(X) = ϕ(Y).

Corollary

If f, g ∈ CB are of disjoint type, then J(f) and J(g) are ambiently homeomorphic.

Question

Are f and g ambiently conjugate? I.e., is there a homeomorphism ϕ: C → C that restricts to a conjugacy between f|J(f) and g|J(g)?

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Let f ∈ B be of disjoint type. #{“tracts” of f} = #{connected components of {z : |f(z)| > R}} (R large) = #{homotopy classes of curves γ ⊂ F(f) with f(γ) bounded} = #{singularities of inner function associated to f|F(f)}.

Observation

If f and g have different numbers of tracts, f and g are not ambiently conjugate. Also, if f has finite order and g has infinite order, then f and g are not ambiently conjugate.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Let f ∈ B be of disjoint type. #{“tracts” of f} = #{connected components of {z : |f(z)| > R}} (R large) = #{homotopy classes of curves γ ⊂ F(f) with f(γ) bounded} = #{singularities of inner function associated to f|F(f)}.

Observation

If f and g have different numbers of tracts, f and g are not ambiently conjugate. Also, if f has finite order and g has infinite order, then f and g are not ambiently conjugate.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Let f ∈ B be of disjoint type. #{“tracts” of f} = #{connected components of {z : |f(z)| > R}} (R large) = #{homotopy classes of curves γ ⊂ F(f) with f(γ) bounded} = #{singularities of inner function associated to f|F(f)}.

Observation

If f and g have different numbers of tracts, f and g are not ambiently conjugate. Also, if f has finite order and g has infinite order, then f and g are not ambiently conjugate.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Let f ∈ B be of disjoint type. #{“tracts” of f} = #{connected components of {z : |f(z)| > R}} (R large) = #{homotopy classes of curves γ ⊂ F(f) with f(γ) bounded} = #{singularities of inner function associated to f|F(f)}.

Observation

If f and g have different numbers of tracts, f and g are not ambiently conjugate. Also, if f has finite order and g has infinite order, then f and g are not ambiently conjugate.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Restrictions on ambient conjugacies

Let f ∈ B be of disjoint type. #{“tracts” of f} = #{connected components of {z : |f(z)| > R}} (R large) = #{homotopy classes of curves γ ⊂ F(f) with f(γ) bounded} = #{singularities of inner function associated to f|F(f)}.

Observation

If f and g have different numbers of tracts, f and g are not ambiently conjugate. Also, if f has finite order and g has infinite order, then f and g are not ambiently conjugate.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Ambient conjugacy classes

Albrecht–Benini–R, in preparation

Question

Suppose f, g are of disjoint type and finite order, with the same number of

• tracts. Are f and g ambiently conjugate?

Recall: the order of f is ρ(f) = lim sup

z→∞

log log|f(z)| log|z| .

Theorem

There exists an uncountable collection of disjoint-type functions of order 1, no two of which are ambiently conjugate.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Ambient conjugacy classes

Albrecht–Benini–R, in preparation

Question

Suppose f, g are of disjoint type and finite order, with the same number of

• tracts. Are f and g ambiently conjugate?

Recall: the order of f is ρ(f) = lim sup

z→∞

log log|f(z)| log|z| .

Theorem

There exists an uncountable collection of disjoint-type functions of order 1, no two of which are ambiently conjugate.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Ambient conjugacy classes

Albrecht–Benini–R, in preparation

Question

Suppose f, g are of disjoint type and finite order, with the same number of

• tracts. Are f and g ambiently conjugate?

Recall: the order of f is ρ(f) = lim sup

z→∞

log log|f(z)| log|z| .

Theorem

There exists an uncountable collection of disjoint-type functions of order 1, no two of which are ambiently conjugate.

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Ambiently conjugate functions

Albrecht–Benini–R, in preparation

ρ(f) = lim sup

z→∞

log log|f(z)| log|z| .

Theorem

Suppose that f ∈ B is of disjoint type and of order ρ(f) < 1. Then f is ambiently conjugate to z → ez − 2.

Theorem

Suppose that f ∈ B is of disjoint type with n tracts, and ρ(f) < n + 1 2 . Then f is ambiently conjugate to z → λezn, for λ < 1/(e − 1).

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Ambiently conjugate functions

Albrecht–Benini–R, in preparation

ρ(f) = lim sup

z→∞

log log|f(z)| log|z| .

Theorem

Suppose that f ∈ B is of disjoint type and of order ρ(f) < 1. Then f is ambiently conjugate to z → ez − 2.

Theorem

Suppose that f ∈ B is of disjoint type with n tracts, and ρ(f) < n + 1 2 . Then f is ambiently conjugate to z → λezn, for λ < 1/(e − 1).

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Bounded gulfs

Theorem (Albrecht–Benini–R)

Suppose that f ∈ B is of disjoint type with n tracts and ρ(f) < ∞, and that f has bounded gulfs. Then f is ambiently conjugate to z → λezn, for λ < 1/(e − 1).

Criniferous entire functions Lasse Rempe Hairs Criniferous functions Conjugacies

Thank you!