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Lyapunov exponents and related concepts for entire functions Walter - - PowerPoint PPT Presentation

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Lyapunov exponents and related concepts for entire functions Walter Bergweiler Christian-Albrechts-Universitt zu Kiel 24098 Kiel, Germany (joint work with Xiao Yao and Jianhua Zheng, Tsinghua University) Parameter Problems in Analytic

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SLIDE 1

Lyapunov exponents and related concepts for entire functions

Walter Bergweiler

Christian-Albrechts-Universität zu Kiel 24098 Kiel, Germany (joint work with Xiao Yao and Jianhua Zheng, Tsinghua University)

Parameter Problems in Analytic Dynamics

London, June 27 – July 1, 2016

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SLIDE 2

Lyapunov exponents and related concepts for entire functions

Walter Bergweiler

Christian-Albrechts-Universität zu Kiel 24098 Kiel, Germany (joint work with Xiao Yao and Jianhua Zheng, Tsinghua University)

Parameter Problems in Analytic Dynamics

London, June 27 – July 1, 2016

1 / 12

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SLIDE 3

Lyapunov exponents and related concepts for entire functions

Walter Bergweiler

Christian-Albrechts-Universität zu Kiel 24098 Kiel, Germany (joint work with Xiao Yao and Jianhua Zheng, Tsinghua University)

Parameter Problems in Analytic Dynamics

London, June 27 – July 1, 2016

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SLIDE 4

Introduction

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SLIDE 5

Introduction

f entire,

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SLIDE 6

Introduction

f entire, Jpf q Julia set,

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SLIDE 7

Introduction

f entire, Jpf q Julia set, f # “ |f 1| 1 ` |f |2 spherical derivative

2 / 12

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SLIDE 8

Introduction

f entire, Jpf q Julia set, f # “ |f 1| 1 ` |f |2 spherical derivative Marty:

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SLIDE 9

Introduction

f entire, Jpf q Julia set, f # “ |f 1| 1 ` |f |2 spherical derivative Marty: U open, U X Jpf q ‰ H

2 / 12

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SLIDE 10

Introduction

f entire, Jpf q Julia set, f # “ |f 1| 1 ` |f |2 spherical derivative Marty: U open, U X Jpf q ‰ H ñ sup

zPU

pf nq#pzq unbounded

2 / 12

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SLIDE 11

Introduction

f entire, Jpf q Julia set, f # “ |f 1| 1 ` |f |2 spherical derivative Marty: U open, U X Jpf q ‰ H ñ sup

zPU

pf nq#pzq unbounded Questions: 1. How fast will sup

zPU

pf nq#pzq tend to 8?

2 / 12

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SLIDE 12

Introduction

f entire, Jpf q Julia set, f # “ |f 1| 1 ` |f |2 spherical derivative Marty: U open, U X Jpf q ‰ H ñ sup

zPU

pf nq#pzq unbounded Questions: 1. How fast will sup

zPU

pf nq#pzq tend to 8?

  • 2. How fast can pf nq#pzq tend to 8 for a point z?

2 / 12

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SLIDE 13

Introduction

f entire, Jpf q Julia set, f # “ |f 1| 1 ` |f |2 spherical derivative Marty: U open, U X Jpf q ‰ H ñ sup

zPU

pf nq#pzq unbounded Questions: 1. How fast will sup

zPU

pf nq#pzq tend to 8?

  • 2. How fast can pf nq#pzq tend to 8 for a point z?

It is more systematical to consider f ˚pzq “ |f 1pzq| 1 ` |z|2 1 ` |f pzq|2 “ f #pzqp1 ` |z|2q instead of f #

2 / 12

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SLIDE 14

Introduction

f entire, Jpf q Julia set, f # “ |f 1| 1 ` |f |2 spherical derivative Marty: U open, U X Jpf q ‰ H ñ sup

zPU

pf nq#pzq unbounded Questions: 1. How fast will sup

zPU

pf nq#pzq tend to 8?

  • 2. How fast can pf nq#pzq tend to 8 for a point z?

It is more systematical to consider f ˚pzq “ |f 1pzq| 1 ` |z|2 1 ` |f pzq|2 “ f #pzqp1 ` |z|2q instead of f #, but this does not affect the growth rates considered.

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SLIDE 15

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SLIDE 16

Przytycki 1994:

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SLIDE 17

Przytycki 1994: f rational ñ

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SLIDE 18

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq

3 / 12

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SLIDE 19

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq

3 / 12

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SLIDE 20

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq

3 / 12

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SLIDE 21

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq

3 / 12

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SLIDE 22

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq

3 / 12

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SLIDE 23

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq

3 / 12

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SLIDE 24

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq

3 / 12

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SLIDE 25

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq

3 / 12

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SLIDE 26

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent,

3 / 12

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SLIDE 27

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points.

3 / 12

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SLIDE 28

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q:

3 / 12

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SLIDE 29

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q: f ppzq “ z, λ “ pf pq1pzq

3 / 12

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SLIDE 30

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q: f ppzq “ z, λ “ pf pq1pzq ñ χpf , zq “ log |λ| p

3 / 12

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SLIDE 31

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q: f ppzq “ z, λ “ pf pq1pzq ñ χpf , zq “ log |λ| p Eremenko, Levin 1990:

3 / 12

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SLIDE 32

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q: f ppzq “ z, λ “ pf pq1pzq ñ χpf , zq “ log |λ| p Eremenko, Levin 1990: f polynomial ñ Dz P Perpf q: χpf , zq ě log degpf q

3 / 12

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SLIDE 33

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q: f ppzq “ z, λ “ pf pq1pzq ñ χpf , zq “ log |λ| p Eremenko, Levin 1990: f polynomial ñ Dz P Perpf q: χpf , zq ě log degpf q Zdunik 2014:

3 / 12

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SLIDE 34

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q: f ppzq “ z, λ “ pf pq1pzq ñ χpf , zq “ log |λ| p Eremenko, Levin 1990: f polynomial ñ Dz P Perpf q: χpf , zq ě log degpf q Zdunik 2014: f rational ñ Dz P Perpf q: χpf , zq ą 1 2 log degpf q

3 / 12

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SLIDE 35

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q: f ppzq “ z, λ “ pf pq1pzq ñ χpf , zq “ log |λ| p Eremenko, Levin 1990: f polynomial ñ Dz P Perpf q: χpf , zq ě log degpf q Zdunik 2014: f rational ñ Dz P Perpf q: χpf , zq ą 1 2 log degpf q Barrett, Eremenko 2013:

3 / 12

slide-36
SLIDE 36

Przytycki 1994: f rational ñ lim sup

nÑ8

1 nlog ˆ sup

zPC

pf nq˚pzq ˙ “ sup

zPC

lim sup

nÑ8

1 n logpf nq˚pzq loooooooooooomoooooooooooon “: χpf , zq “ sup

zPPerpf q

lim

nÑ8

1 n logpf nq˚pzq loooooooooomoooooooooon “: χpf , zq χpf , zq “ Lyapunov exponent, Perpf q “ set of periodic points. z P Perpf q: f ppzq “ z, λ “ pf pq1pzq ñ χpf , zq “ log |λ| p Eremenko, Levin 1990: f polynomial ñ Dz P Perpf q: χpf , zq ě log degpf q Zdunik 2014: f rational ñ Dz P Perpf q: χpf , zq ą 1 2 log degpf q Barrett, Eremenko 2013: The constant 1 2 is best possible here.

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SLIDE 37

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SLIDE 38

Theorem:

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SLIDE 39

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8

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SLIDE 40

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof:

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SLIDE 41

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8.

4 / 12

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SLIDE 42

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points.

4 / 12

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SLIDE 43

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time

4 / 12

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SLIDE 44

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2

4 / 12

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SLIDE 45

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there

4 / 12

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SLIDE 46

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3

4 / 12

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SLIDE 47

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc.

4 / 12

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SLIDE 48

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc.

4 / 12

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SLIDE 49

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8?

4 / 12

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SLIDE 50

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez,

4 / 12

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SLIDE 51

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez, z0 P C with χpf , z0q “ 8,

4 / 12

slide-52
SLIDE 52

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez, z0 P C with χpf , z0q “ 8, zn “ f npz0q

4 / 12

slide-53
SLIDE 53

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez, z0 P C with χpf , z0q “ 8, zn “ f npz0q ñ 1 ď pf nq#pz0q

4 / 12

slide-54
SLIDE 54

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez, z0 P C with χpf , z0q “ 8, zn “ f npz0q ñ 1 ď pf nq#pz0q “ śn

j“1 |zj|

1 ` |zn|2

4 / 12

slide-55
SLIDE 55

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez, z0 P C with χpf , z0q “ 8, zn “ f npz0q ñ 1 ď pf nq#pz0q “ śn

j“1 |zj|

1 ` |zn|2 ď śn´1

j“1 |zj|

|zn|

4 / 12

slide-56
SLIDE 56

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez, z0 P C with χpf , z0q “ 8, zn “ f npz0q ñ 1 ď pf nq#pz0q “ śn

j“1 |zj|

1 ` |zn|2 ď śn´1

j“1 |zj|

|zn| ñ |zn| ď

n´1

ź

j“1

|zj|

4 / 12

slide-57
SLIDE 57

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez, z0 P C with χpf , z0q “ 8, zn “ f npz0q ñ 1 ď pf nq#pz0q “ śn

j“1 |zj|

1 ` |zn|2 ď śn´1

j“1 |zj|

|zn| ñ |zn| ď

n´1

ź

j“1

|zj| ñ |zn| ď exppc¨2nq

4 / 12

slide-58
SLIDE 58

Theorem: f transcendental entire ñ Dz P Jpf q: χpf , zq “ 8 Idea of proof: There exists a sequence pzkq of periodic points with that χpf , zkq Ñ 8. Suppose for simplicity the zk are fixed points. Choose z near z1 such that f npzq stays near z1 for a long time, then “jumps” close to z2, stays there, jumps close to z3, etc. How fast can pf nq#pzq tend to 8? Example: f pzq “ ez, z0 P C with χpf , z0q “ 8, zn “ f npz0q ñ 1 ď pf nq#pz0q “ śn

j“1 |zj|

1 ` |zn|2 ď śn´1

j“1 |zj|

|zn| ñ |zn| ď

n´1

ź

j“1

|zj| ñ |zn| ď exppc¨2nq ñ lim sup

nÑ8

1 n log logpf nq#pz0q ď log 2.

4 / 12

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SLIDE 59

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SLIDE 60

B “ Eremenko-Lyubich class

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SLIDE 61

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values

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SLIDE 62

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem:

5 / 12

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SLIDE 63

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2.

5 / 12

slide-64
SLIDE 64

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp.

5 / 12

slide-65
SLIDE 65

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally:

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slide-66
SLIDE 66

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq|

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slide-67
SLIDE 67

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

5 / 12

slide-68
SLIDE 68

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

λpf q “ lim inf

rÑ8

log log Mpr, f q log r

5 / 12

slide-69
SLIDE 69

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

λpf q “ lim inf

rÑ8

log log Mpr, f q log r = lower order of f

5 / 12

slide-70
SLIDE 70

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

λpf q “ lim inf

rÑ8

log log Mpr, f q log r = lower order of f Fact:

5 / 12

slide-71
SLIDE 71

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

λpf q “ lim inf

rÑ8

log log Mpr, f q log r = lower order of f Fact: f P B ñ λpf q ě 1 2.

5 / 12

slide-72
SLIDE 72

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

λpf q “ lim inf

rÑ8

log log Mpr, f q log r = lower order of f Fact: f P B ñ λpf q ě 1 2. Theorem:

5 / 12

slide-73
SLIDE 73

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

λpf q “ lim inf

rÑ8

log log Mpr, f q log r = lower order of f Fact: f P B ñ λpf q ě 1 2. Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě logp1 ` λpf qq.

5 / 12

slide-74
SLIDE 74

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

λpf q “ lim inf

rÑ8

log log Mpr, f q log r = lower order of f Fact: f P B ñ λpf q ě 1 2. Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě logp1 ` λpf qq. The exponential function shows that this is sharp if λpf q “ 1

5 / 12

slide-75
SLIDE 75

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě log 3 2. This result is sharp. More generally: Mpr, f q “ max

|z|“r |f pzq| = maximum modulus of f

λpf q “ lim inf

rÑ8

log log Mpr, f q log r = lower order of f Fact: f P B ñ λpf q ě 1 2. Theorem: f P B ñ Dz P Jpf q: lim inf

nÑ8

1 n log logpf nq#pzq ě logp1 ` λpf qq. The exponential function shows that this is sharp if λpf q “ 1 – and there are examples for all other values of λpf q.

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slide-76
SLIDE 76

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SLIDE 77

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ?

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slide-78
SLIDE 78

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z.

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slide-79
SLIDE 79

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r:

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slide-80
SLIDE 80

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q.

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slide-81
SLIDE 81

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8.

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slide-82
SLIDE 82

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8. Theorem:

6 / 12

slide-83
SLIDE 83

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8. Theorem: U open, U X Jpf q ‰ H, R ą 0

6 / 12

slide-84
SLIDE 84

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8. Theorem: U open, U X Jpf q ‰ H, R ą 0 ñ Dm P N: sup

zPU

pf nq#pzq ě log Mn´mpR, f q

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slide-85
SLIDE 85

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8. Theorem: U open, U X Jpf q ‰ H, R ą 0 ñ Dm P N: sup

zPU

pf nq#pzq ě log Mn´mpR, f q E.g. for f pzq “ ez this growth is much faster than that given by lim inf

nÑ8

1 n log logpf nq#pzq ě logp1 ` λpf qq.

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slide-86
SLIDE 86

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8. Theorem: U open, U X Jpf q ‰ H, R ą 0 ñ Dm P N: sup

zPU

pf nq#pzq ě log Mn´mpR, f q E.g. for f pzq “ ez this growth is much faster than that given by lim inf

nÑ8

1 n log logpf nq#pzq ě logp1 ` λpf qq. Ideas of proof:

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slide-87
SLIDE 87

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8. Theorem: U open, U X Jpf q ‰ H, R ą 0 ñ Dm P N: sup

zPU

pf nq#pzq ě log Mn´mpR, f q E.g. for f pzq “ ez this growth is much faster than that given by lim inf

nÑ8

1 n log logpf nq#pzq ě logp1 ` λpf qq. Ideas of proof: Sketch that there exists z satisfying last equation

6 / 12

slide-88
SLIDE 88

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8. Theorem: U open, U X Jpf q ‰ H, R ą 0 ñ Dm P N: sup

zPU

pf nq#pzq ě log Mn´mpR, f q E.g. for f pzq “ ez this growth is much faster than that given by lim inf

nÑ8

1 n log logpf nq#pzq ě logp1 ` λpf qq. Ideas of proof: Sketch that there exists z satisfying last equation, for f P B.

6 / 12

slide-89
SLIDE 89

How fast does sup

zPU

pf nq#pzq grow, for U open, U X Jpf q ‰ H ? In contrast to Przytycki’s result for rational f , for entire f this grows much faster than pf nq#pzq for individual points z. Let Mnpr, f q be the iterate of Mpr, f q with respect to r: M1pr, f q “ Mpr, f q and Mn`1pr, f q “ MpMnpr, f q, f q. For large R we have MnpR, f q Ñ 8. Theorem: U open, U X Jpf q ‰ H, R ą 0 ñ Dm P N: sup

zPU

pf nq#pzq ě log Mn´mpR, f q E.g. for f pzq “ ez this growth is much faster than that given by lim inf

nÑ8

1 n log logpf nq#pzq ě logp1 ` λpf qq. Ideas of proof: Sketch that there exists z satisfying last equation, for f P B. Use logarithmic change of variable.

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slide-90
SLIDE 90

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slide-91
SLIDE 91

Let f P B, with critical and asymptotic values in tz : |z| ă Ru.

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slide-92
SLIDE 92

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R.

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slide-93
SLIDE 93

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R. Let A be a component of f ´1ptz : |z| ą Ruq.

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slide-94
SLIDE 94

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R. Let A be a component of f ´1ptz : |z| ą Ruq. f A W Put W “ exp´1pAq and H “ tz : Re z ą log Ru.

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slide-95
SLIDE 95

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R. Let A be a component of f ´1ptz : |z| ą Ruq. f exp exp A W Put W “ exp´1pAq and H “ tz : Re z ą log Ru. Can lift f to map F : W Ñ H, and F maps every component of W univalently onto H: this is the logarithmic change of variable.

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slide-96
SLIDE 96

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R. Let A be a component of f ´1ptz : |z| ą Ruq. f exp exp H A W Put W “ exp´1pAq and H “ tz : Re z ą log Ru. Can lift f to map F : W Ñ H, and F maps every component of W univalently onto H: this is the logarithmic change of variable.

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slide-97
SLIDE 97

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R. Let A be a component of f ´1ptz : |z| ą Ruq. f exp exp H A W Put W “ exp´1pAq and H “ tz : Re z ą log Ru. Can lift f to map F : W Ñ H, and F maps every component of W univalently onto H: this is the logarithmic change of variable.

7 / 12

slide-98
SLIDE 98

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R. Let A be a component of f ´1ptz : |z| ą Ruq. F f exp exp H A W Put W “ exp´1pAq and H “ tz : Re z ą log Ru. Can lift f to map F : W Ñ H, and F maps every component of W univalently onto H: this is the logarithmic change of variable.

7 / 12

slide-99
SLIDE 99

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R. Let A be a component of f ´1ptz : |z| ą Ruq. F f exp exp H A W Put W “ exp´1pAq and H “ tz : Re z ą log Ru. Can lift f to map F : W Ñ H, and F maps every component of W univalently onto H.

7 / 12

slide-100
SLIDE 100

Let f P B, with critical and asymptotic values in tz : |z| ă Ru. Assume |f p0q| ă R. Let A be a component of f ´1ptz : |z| ą Ruq. F f exp exp H A W Put W “ exp´1pAq and H “ tz : Re z ą log Ru. Can lift f to map F : W Ñ H, and F maps every component of W univalently onto H. This is the logarithmic change of variable.

7 / 12

slide-101
SLIDE 101

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slide-102
SLIDE 102

Without loss of generality R “ 1, so H “ tz : Re z ą 0u.

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slide-103
SLIDE 103

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

slide-104
SLIDE 104

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-105
SLIDE 105

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-106
SLIDE 106

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

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slide-107
SLIDE 107

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U z w “Fpzq Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-108
SLIDE 108

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-109
SLIDE 109

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-110
SLIDE 110

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-111
SLIDE 111

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-112
SLIDE 112

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re wď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-113
SLIDE 113

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUqď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-114
SLIDE 114

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-115
SLIDE 115

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-116
SLIDE 116

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-117
SLIDE 117

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-118
SLIDE 118

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-119
SLIDE 119

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z p Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-120
SLIDE 120

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z p Fppq Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

8 / 12

slide-121
SLIDE 121

Without loss of generality R “ 1, so H “ tz : Re z ą 0u. F H U w z w z p Fppq Koebe’s theorem: 1 4|pF ´1q1pwq| Re w ď distpz, BUq ď π. ñ |F 1pzq| ě 1 4π Re Fpzq. Lemma: For p in the straight line connecting z to BU we have |F 1ppq| ě 1 8π Re Fpzq. Proof: Harnack’s inequality for positive harmonic function Re Fpzq.

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slide-122
SLIDE 122

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slide-123
SLIDE 123

There are points u0 such that Re F npu0q behaves in a prescribed way

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slide-124
SLIDE 124

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq.

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slide-125
SLIDE 125

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq.

slide-126
SLIDE 126

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

slide-127
SLIDE 127

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 128

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 129

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 130

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 131

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 132

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 133

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 134

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 135

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 136

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 137

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

slide-138
SLIDE 138

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 139

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3

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SLIDE 140

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3 u0 is here

slide-141
SLIDE 141

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3 u0 is here

slide-142
SLIDE 142

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3 u0 is here

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SLIDE 143

There are points u0 such that Re F npu0q behaves in a prescribed way; that is, Re F npu0q « ξn for a given sequence pξnq. ξ0 ξ1 ξ2 ξ3 u0 is here

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SLIDE 144

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SLIDE 145

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq.

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SLIDE 146

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn.

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SLIDE 147

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q.

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SLIDE 148

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q.

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slide-149
SLIDE 149

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q. Then log Mpr, f q ě rβ

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slide-150
SLIDE 150

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q. Then log Mpr, f q ě rβ, and this translates to max

Re z“x Re Fpzq ě eβx.

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slide-151
SLIDE 151

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q. Then log Mpr, f q ě rβ, and this translates to max

Re z“x Re Fpzq ě eβx.

The lemma yields that with uk “ F kpu0q we can achieve that |F 1pukq| Ç max

Re z“Re uk

Re Fpzq

10 / 12

slide-152
SLIDE 152

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q. Then log Mpr, f q ě rβ, and this translates to max

Re z“x Re Fpzq ě eβx.

The lemma yields that with uk “ F kpu0q we can achieve that |F 1pukq| Ç max

Re z“Re uk

Re Fpzq Ç eβ Re uk

10 / 12

slide-153
SLIDE 153

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q. Then log Mpr, f q ě rβ, and this translates to max

Re z“x Re Fpzq ě eβx.

The lemma yields that with uk “ F kpu0q we can achieve that |F 1pukq| Ç max

Re z“Re uk

Re Fpzq Ç eβ Re uk « exp ´ βp1 ` αqk¯ .

10 / 12

slide-154
SLIDE 154

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q. Then log Mpr, f q ě rβ, and this translates to max

Re z“x Re Fpzq ě eβx.

The lemma yields that with uk “ F kpu0q we can achieve that |F 1pukq| Ç max

Re z“Re uk

Re Fpzq Ç eβ Re uk « exp ´ βp1 ` αqk¯ . This implies that |pF nq1pu0q| “

n´1

ź

k“0

|F 1pukq|

10 / 12

slide-155
SLIDE 155

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q. Then log Mpr, f q ě rβ, and this translates to max

Re z“x Re Fpzq ě eβx.

The lemma yields that with uk “ F kpu0q we can achieve that |F 1pukq| Ç max

Re z“Re uk

Re Fpzq Ç eβ Re uk « exp ´ βp1 ` αqk¯ . This implies that |pF nq1pu0q| “

n´1

ź

k“0

|F 1pukq| Ç exp ˜n´1 ÿ

k“0

βp1 ` αqk ¸

10 / 12

slide-156
SLIDE 156

We want to find z0 with lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` λpf qq. The example f pzq “ ez suggests to choose z0 “ eu0 with Re F npu0q « p1 ` λpf qqn. We will actually choose Re F npu0q « p1 ` αqn with α ă λpf q. Let α ă β ă λpf q. Then log Mpr, f q ě rβ, and this translates to max

Re z“x Re Fpzq ě eβx.

The lemma yields that with uk “ F kpu0q we can achieve that |F 1pukq| Ç max

Re z“Re uk

Re Fpzq Ç eβ Re uk « exp ´ βp1 ` αqk¯ . This implies that |pF nq1pu0q| “

n´1

ź

k“0

|F 1pukq| Ç exp ˜n´1 ÿ

k“0

βp1 ` αqk ¸ « exp ˆβ αp1 ` αqn ˙ .

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SLIDE 157

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SLIDE 158

So Re F npu0q « p1 ` αqn and |pF nq1pu0q| Ç exp ˆβ αp1 ` αqn ˙ .

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SLIDE 159

So Re F npu0q « p1 ` αqn and |pF nq1pu0q| Ç exp ˆβ αp1 ` αqn ˙ . Since exp Fpuq “ f peuq we have, with z0 “ eu0, |f npz0q| « exppRe F npu0qq « exp ´ p1 ` αqn¯

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SLIDE 160

So Re F npu0q « p1 ` αqn and |pF nq1pu0q| Ç exp ˆβ αp1 ` αqn ˙ . Since exp Fpuq “ f peuq we have, with z0 “ eu0, |f npz0q| « exppRe F npu0qq « exp ´ p1 ` αqn¯ and |pf nq1pz0q| “ 1 |z0||f npz0q| ¨ |pF nq1pu0q| Ç |f npz0q| exp ˆβ αp1 ` αqn ˙ .

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slide-161
SLIDE 161

So Re F npu0q « p1 ` αqn and |pF nq1pu0q| Ç exp ˆβ αp1 ` αqn ˙ . Since exp Fpuq “ f peuq we have, with z0 “ eu0, |f npz0q| « exppRe F npu0qq « exp ´ p1 ` αqn¯ and |pf nq1pz0q| “ 1 |z0||f npz0q| ¨ |pF nq1pu0q| Ç |f npz0q| exp ˆβ αp1 ` αqn ˙ . This yields pf nq#pz0q « |pf nq1pz0q| |f npz0q|2 Ç exp ˆˆβ α ´ 1 ˙ p1 ` αqn ˙ .

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SLIDE 162

So Re F npu0q « p1 ` αqn and |pF nq1pu0q| Ç exp ˆβ αp1 ` αqn ˙ . Since exp Fpuq “ f peuq we have, with z0 “ eu0, |f npz0q| « exppRe F npu0qq « exp ´ p1 ` αqn¯ and |pf nq1pz0q| “ 1 |z0||f npz0q| ¨ |pF nq1pu0q| Ç |f npz0q| exp ˆβ αp1 ` αqn ˙ . This yields pf nq#pz0q « |pf nq1pz0q| |f npz0q|2 Ç exp ˆˆβ α ´ 1 ˙ p1 ` αqn ˙ . Hence lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` αq.

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SLIDE 163

So Re F npu0q « p1 ` αqn and |pF nq1pu0q| Ç exp ˆβ αp1 ` αqn ˙ . Since exp Fpuq “ f peuq we have, with z0 “ eu0, |f npz0q| « exppRe F npu0qq « exp ´ p1 ` αqn¯ and |pf nq1pz0q| “ 1 |z0||f npz0q| ¨ |pF nq1pu0q| Ç |f npz0q| exp ˆβ αp1 ` αqn ˙ . This yields pf nq#pz0q « |pf nq1pz0q| |f npz0q|2 Ç exp ˆˆβ α ´ 1 ˙ p1 ` αqn ˙ . Hence lim inf

nÑ8

1 n log logpf nq#pz0q ě logp1 ` αq. Actually will choose α “ αn Ñ λpf q and β “ βn Ñ λpf q.

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SLIDE 164

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SLIDE 165

Thank you very much for your attention

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SLIDE 166

Thank you very much for your attention Gelukkige verjaardag, Sebastian!

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