Conjugacy classes of disjoint-type functions Simon Albrecht ( joint - - PowerPoint PPT Presentation

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Conjugacy classes of disjoint-type functions Simon Albrecht ( joint - - PowerPoint PPT Presentation

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Conjugacy classes of disjoint-type functions Simon Albrecht ( joint work with Anna Benini and Lasse Rempe-Gillen ) University of Liverpool TCD 2017 Barcelona, 2 October 2017 S. Albrecht (UoL) Conjugacies in class B 2 October 2017 1 / 12

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Conjugacy classes of disjoint-type functions

Simon Albrecht ( joint work with Anna Benini and Lasse Rempe-Gillen )

University of Liverpool

TCD 2017 Barcelona, 2 October 2017

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 1 / 12

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SLIDE 2

Fatou set and Julia set

Let f : C → C be entire.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 2 / 12

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Fatou set and Julia set

Let f : C → C be entire. We define the n-th iterate of f by f n := f ◦ f ◦ . . . ◦ f

  • n-times

.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 2 / 12

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SLIDE 4

Fatou set and Julia set

Let f : C → C be entire. We define the n-th iterate of f by f n := f ◦ f ◦ . . . ◦ f

  • n-times

. F(f ) = Fatou set of f = set of stability

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 2 / 12

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SLIDE 5

Fatou set and Julia set

Let f : C → C be entire. We define the n-th iterate of f by f n := f ◦ f ◦ . . . ◦ f

  • n-times

. F(f ) = Fatou set of f = set of stability = {z ∈ C : {f n : n ∈ N} is equicontinuous in z}

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 2 / 12

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SLIDE 6

Fatou set and Julia set

Let f : C → C be entire. We define the n-th iterate of f by f n := f ◦ f ◦ . . . ◦ f

  • n-times

. F(f ) = Fatou set of f = set of stability = {z ∈ C : {f n : n ∈ N} is equicontinuous in z} J (f ) = Julia set of f = C \ F(f )

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 2 / 12

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Conjugacy

We say that two entire functions f and g are conjugate if there exists a homeomorphism T : C → C with f ◦ T = T ◦ g.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 3 / 12

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Conjugacy

We say that two entire functions f and g are conjugate if there exists a homeomorphism T : C → C with f ◦ T = T ◦ g. Then: f n ◦ T = T ◦ gn for all n ∈ N.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 3 / 12

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Conjugacy

We say that two entire functions f and g are conjugate if there exists a homeomorphism T : C → C with f ◦ T = T ◦ g. Then: f n ◦ T = T ◦ gn for all n ∈ N. F(f ) = T(F(g)) and J (f ) = T(J (g)).

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 3 / 12

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Conjugacy

We say that two entire functions f and g are conjugate if there exists a homeomorphism T : C → C with f ◦ T = T ◦ g. Then: f n ◦ T = T ◦ gn for all n ∈ N. F(f ) = T(F(g)) and J (f ) = T(J (g)). Example: f (z) = z2, g(z) = 2z2 − 2z + 1, T(z) = 2z − 1

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 3 / 12

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SLIDE 11

Examples of Julia sets

Example 1: f (z) = z2 − 1.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 4 / 12

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Examples of Julia sets

Example 1: f (z) = z2 − 1.

Source of image: Prokofiev, Wikimedia commons, http://commons.wikimedia.org/wiki/File:Julia_z2-1.png

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 4 / 12

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SLIDE 13

Examples of Julia sets

Example 2: f (z) = ez − 2.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 5 / 12

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SLIDE 14

Examples of Julia sets

Example 2: f (z) = ez − 2.

Image created by Lasse Rempe-Gillen.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 5 / 12

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SLIDE 15

Examples of Julia sets

Example 2: f (z) = ez − 2.

Image created by Lasse Rempe-Gillen.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 5 / 12

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SLIDE 16

Cantor bouquets

Cantor bouquet: (informal definition)

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 6 / 12

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SLIDE 17

Cantor bouquets

Cantor bouquet: (informal definition) Collection of injective curves to ∞.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 6 / 12

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SLIDE 18

Cantor bouquets

Cantor bouquet: (informal definition) Collection of injective curves to ∞. Each curve has a finite endpoint.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 6 / 12

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SLIDE 19

Cantor bouquets

Cantor bouquet: (informal definition) Collection of injective curves to ∞. Each curve has a finite endpoint. The set of endpoints is dense in the Cantor bouquet.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 6 / 12

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Cantor bouquets

Cantor bouquet: (informal definition) Collection of injective curves to ∞. Each curve has a finite endpoint. The set of endpoints is dense in the Cantor bouquet. Fact: All Cantor bouquets are homeomorphic to each other by ambient homeomorphisms (that is by homeomorphisms which can be extended to the whole plane).

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 6 / 12

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SLIDE 21

Cantor bouquets

Example 3: f (z) = − 3

4 cos(z) + 3 4.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 7 / 12

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SLIDE 22

Cantor bouquets

Example 3: f (z) = − 3

4 cos(z) + 3 4.

Image created by Lasse Rempe-Gillen.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 7 / 12

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SLIDE 23

Disjoint-type functions

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 8 / 12

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Disjoint-type functions

f

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 8 / 12

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Disjoint-type functions

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 8 / 12

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Disjoint-type functions

f f f f

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 8 / 12

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Disjoint-type functions

f f f f Example: ez − 2 is of disjoint type,

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 8 / 12

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Disjoint-type functions

f f f f Example: ez − 2 is of disjoint type, ez is not of disjoint type.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 8 / 12

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Order of f

We say that an entire function f has finite order, if there exist c, ρ > 0 so that |f (z)| ≤ c · exp(|z|ρ) for all z ∈ C.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 9 / 12

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Order of f

We say that an entire function f has finite order, if there exist c, ρ > 0 so that |f (z)| ≤ c · exp(|z|ρ) for all z ∈ C. The infimum of the possible ρ is called order of f , denoted ρ(f ).

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 9 / 12

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Order of f

We say that an entire function f has finite order, if there exist c, ρ > 0 so that |f (z)| ≤ c · exp(|z|ρ) for all z ∈ C. The infimum of the possible ρ is called order of f , denoted ρ(f ). In fact, ρ(f ) = lim sup

r→∞

log log max|z|=r |f (z)| log r .

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 9 / 12

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Order of f

We say that an entire function f has finite order, if there exist c, ρ > 0 so that |f (z)| ≤ c · exp(|z|ρ) for all z ∈ C. The infimum of the possible ρ is called order of f , denoted ρ(f ). In fact, ρ(f ) = lim sup

r→∞

log log max|z|=r |f (z)| log r . Examples: ρ(P) = 0 for every polynomial P.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 9 / 12

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SLIDE 33

Order of f

We say that an entire function f has finite order, if there exist c, ρ > 0 so that |f (z)| ≤ c · exp(|z|ρ) for all z ∈ C. The infimum of the possible ρ is called order of f , denoted ρ(f ). In fact, ρ(f ) = lim sup

r→∞

log log max|z|=r |f (z)| log r . Examples: ρ(P) = 0 for every polynomial P. ρ(exp(zn)) = n.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 9 / 12

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Order of f

We say that an entire function f has finite order, if there exist c, ρ > 0 so that |f (z)| ≤ c · exp(|z|ρ) for all z ∈ C. The infimum of the possible ρ is called order of f , denoted ρ(f ). In fact, ρ(f ) = lim sup

r→∞

log log max|z|=r |f (z)| log r . Examples: ρ(P) = 0 for every polynomial P. ρ(exp(zn)) = n. ρ(exp(exp(z))) = ∞.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 9 / 12

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SLIDE 35

Functions with Cantor bouquet Julia sets

Fact: Every disjoint-type transcendental entire function of finite order has a Cantor bouquet Julia set.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 10 / 12

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Functions with Cantor bouquet Julia sets

Fact: Every disjoint-type transcendental entire function of finite order has a Cantor bouquet Julia set. Hence, the Julia sets of two such functions f and g are homeomorphic (by ambient homeomorphisms).

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 10 / 12

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Functions with Cantor bouquet Julia sets

Fact: Every disjoint-type transcendental entire function of finite order has a Cantor bouquet Julia set. Hence, the Julia sets of two such functions f and g are homeomorphic (by ambient homeomorphisms). Question Are f and g conjugate on their Julia sets?

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 10 / 12

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SLIDE 38

Functions with Cantor bouquet Julia sets

Fact: Every disjoint-type transcendental entire function of finite order has a Cantor bouquet Julia set. Hence, the Julia sets of two such functions f and g are homeomorphic (by ambient homeomorphisms). Question Are f and g conjugate on their Julia sets?

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 10 / 12

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Results

Theorem Let f and g be disjoint-type transcendental entire functions of order less than 1.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 11 / 12

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Results

Theorem Let f and g be disjoint-type transcendental entire functions of order less than 1. Then f and g are conjugate on their Julia sets by an ambient homeomorphism.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 11 / 12

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Results

Theorem Let f and g be disjoint-type transcendental entire functions of order less than 1. Then f and g are conjugate on their Julia sets by an ambient homeomorphism. However, there exist uncountably many disjoint-type functions of order 1 which are pairwise not conjugate on their Julia sets.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 11 / 12

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Results

Theorem Let f and g be disjoint-type transcendental entire functions of order less than 1. Then f and g are conjugate on their Julia sets by an ambient homeomorphism. However, there exist uncountably many disjoint-type functions of order 1 which are pairwise not conjugate on their Julia sets. The conjugacy in general only holds on the Julia sets, not on the entire complex plane.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 11 / 12

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The End

Thank you very much for your attention.

  • S. Albrecht (UoL)

Conjugacies in class B 2 October 2017 12 / 12