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

Przytycki 1994: f rational ñ 1 ˆ ˙ 1 p f n q ˚ p z q n log p f n q ˚ p z q lim sup n log sup “ sup lim sup n Ñ8 n Ñ8 z P C z P C loooooooooooomoooooooooooon “ : χ p f , z q 1 n log p f n q ˚ p z q “ sup lim n Ñ8 z P Per p f q loooooooooomoooooooooon “ : χ p f , z q χ p f , z q “ Lyapunov exponent, Per p f q “ set of periodic points. χ p f , z q “ log | λ | f p p z q “ z , λ “ p f p q 1 p z q z P Per p f q : ñ p Eremenko, Levin 1990: f polynomial ñ D z P Per p f q : χ p f , z q ě log deg p f q Zdunik 2014: f rational ñ D z P Per p f q : χ p f , z q ą 1 2 log deg p f q 3 / 12

Przytycki 1994: f rational ñ 1 ˆ ˙ 1 p f n q ˚ p z q n log p f n q ˚ p z q lim sup n log sup “ sup lim sup n Ñ8 n Ñ8 z P C z P C loooooooooooomoooooooooooon “ : χ p f , z q 1 n log p f n q ˚ p z q “ sup lim n Ñ8 z P Per p f q loooooooooomoooooooooon “ : χ p f , z q χ p f , z q “ Lyapunov exponent, Per p f q “ set of periodic points. χ p f , z q “ log | λ | f p p z q “ z , λ “ p f p q 1 p z q z P Per p f q : ñ p Eremenko, Levin 1990: f polynomial ñ D z P Per p f q : χ p f , z q ě log deg p f q Zdunik 2014: f rational ñ D z P Per p f q : χ p f , z q ą 1 2 log deg p f q Barrett, Eremenko 2013: 3 / 12

Przytycki 1994: f rational ñ 1 ˆ ˙ 1 p f n q ˚ p z q n log p f n q ˚ p z q lim sup n log sup “ sup lim sup n Ñ8 n Ñ8 z P C z P C loooooooooooomoooooooooooon “ : χ p f , z q 1 n log p f n q ˚ p z q “ sup lim n Ñ8 z P Per p f q loooooooooomoooooooooon “ : χ p f , z q χ p f , z q “ Lyapunov exponent, Per p f q “ set of periodic points. χ p f , z q “ log | λ | f p p z q “ z , λ “ p f p q 1 p z q z P Per p f q : ñ p Eremenko, Levin 1990: f polynomial ñ D z P Per p f q : χ p f , z q ě log deg p f q Zdunik 2014: f rational ñ D z P Per p f q : χ p f , z q ą 1 2 log deg p f q Barrett, Eremenko 2013: The constant 1 2 is best possible here. 3 / 12

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Theorem: 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , z 0 P C with χ p f , z 0 q “ 8 , 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , z 0 P C with χ p f , z 0 q “ 8 , z n “ f n p z 0 q 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , z 0 P C with χ p f , z 0 q “ 8 , z n “ f n p z 0 q ñ 1 ď p f n q # p z 0 q 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , z 0 P C with χ p f , z 0 q “ 8 , z n “ f n p z 0 q ś n j “ 1 | z j | ñ 1 ď p f n q # p z 0 q “ 1 ` | z n | 2 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , z 0 P C with χ p f , z 0 q “ 8 , z n “ f n p z 0 q ś n ś n ´ 1 j “ 1 | z j | j “ 1 | z j | ñ 1 ď p f n q # p z 0 q “ 1 ` | z n | 2 ď | z n | 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , z 0 P C with χ p f , z 0 q “ 8 , z n “ f n p z 0 q ś n ś n ´ 1 j “ 1 | z j | j “ 1 | z j | ñ 1 ď p f n q # p z 0 q “ 1 ` | z n | 2 ď | z n | n ´ 1 ź ñ | z n | ď | z j | j “ 1 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , z 0 P C with χ p f , z 0 q “ 8 , z n “ f n p z 0 q ś n ś n ´ 1 j “ 1 | z j | j “ 1 | z j | ñ 1 ď p f n q # p z 0 q “ 1 ` | z n | 2 ď | z n | n ´ 1 ź ñ | z n | ď | z j | j “ 1 ñ | z n | ď exp p c ¨ 2 n q 4 / 12

Theorem: f transcendental entire ñ D z P J p f q : χ p f , z q “ 8 Idea of proof: There exists a sequence p z k q of periodic points with that χ p f , z k q Ñ 8 . Suppose for simplicity the z k are fixed points. Choose z near z 1 such that f n p z q stays near z 1 for a long time, then “jumps” close to z 2 , stays there, jumps close to z 3 , etc. How fast can p f n q # p z q tend to 8 ? Example: f p z q “ e z , z 0 P C with χ p f , z 0 q “ 8 , z n “ f n p z 0 q ś n ś n ´ 1 j “ 1 | z j | j “ 1 | z j | ñ 1 ď p f n q # p z 0 q “ 1 ` | z n | 2 ď | z n | n ´ 1 ź ñ | z n | ď | z j | j “ 1 ñ | z n | ď exp p c ¨ 2 n q 1 n log log p f n q # p z 0 q ď log 2. ñ lim sup n Ñ8 4 / 12

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B “ Eremenko-Lyubich class 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values Theorem: 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f log log M p r , f q λ p f q “ lim inf log r r Ñ8 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f log log M p r , f q λ p f q “ lim inf = lower order of f log r r Ñ8 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f log log M p r , f q λ p f q “ lim inf = lower order of f log r r Ñ8 Fact: 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f log log M p r , f q λ p f q “ lim inf = lower order of f log r r Ñ8 Fact: f P B ñ λ p f q ě 1 2. 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f log log M p r , f q λ p f q “ lim inf = lower order of f log r r Ñ8 Fact: f P B ñ λ p f q ě 1 2. Theorem: 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f log log M p r , f q λ p f q “ lim inf = lower order of f log r r Ñ8 Fact: f P B ñ λ p f q ě 1 2. Theorem: 1 n log log p f n q # p z q ě log p 1 ` λ p f qq . f P B ñ D z P J p f q : lim inf n Ñ8 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f log log M p r , f q λ p f q “ lim inf = lower order of f log r r Ñ8 Fact: f P B ñ λ p f q ě 1 2. Theorem: 1 n log log p f n q # p z q ě log p 1 ` λ p f qq . f P B ñ D z P J p f q : lim inf n Ñ8 The exponential function shows that this is sharp if λ p f q “ 1 5 / 12

B “ Eremenko-Lyubich class = transcendental entire functions with bounded set of critical and asymptotic values n log log p f n q # p z q ě log 3 1 Theorem: f P B ñ D z P J p f q : lim inf 2. n Ñ8 This result is sharp. More generally: M p r , f q “ max | z |“ r | f p z q| = maximum modulus of f log log M p r , f q λ p f q “ lim inf = lower order of f log r r Ñ8 Fact: f P B ñ λ p f q ě 1 2. Theorem: 1 n log log p f n q # p z q ě log p 1 ` λ p f qq . f P B ñ D z P J p f q : lim inf n Ñ8 The exponential function shows that this is sharp if λ p f q “ 1 – and there are examples for all other values of λ p f q . 5 / 12

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p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . Theorem: 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . Theorem: U open, U X J p f q ‰ H , R ą 0 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . Theorem: U open, U X J p f q ‰ H , R ą 0 p f n q # p z q ě log M n ´ m p R , f q ñ D m P N : sup z P U 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . Theorem: U open, U X J p f q ‰ H , R ą 0 p f n q # p z q ě log M n ´ m p R , f q ñ D m P N : sup z P U E.g. for f p z q “ e z this growth is much faster than that given by 1 n log log p f n q # p z q ě log p 1 ` λ p f qq . lim inf n Ñ8 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . Theorem: U open, U X J p f q ‰ H , R ą 0 p f n q # p z q ě log M n ´ m p R , f q ñ D m P N : sup z P U E.g. for f p z q “ e z this growth is much faster than that given by 1 n log log p f n q # p z q ě log p 1 ` λ p f qq . lim inf n Ñ8 Ideas of proof: 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . Theorem: U open, U X J p f q ‰ H , R ą 0 p f n q # p z q ě log M n ´ m p R , f q ñ D m P N : sup z P U E.g. for f p z q “ e z this growth is much faster than that given by 1 n log log p f n q # p z q ě log p 1 ` λ p f qq . lim inf n Ñ8 Ideas of proof: Sketch that there exists z satisfying last equation 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . Theorem: U open, U X J p f q ‰ H , R ą 0 p f n q # p z q ě log M n ´ m p R , f q ñ D m P N : sup z P U E.g. for f p z q “ e z this growth is much faster than that given by 1 n log log p f n q # p z q ě log p 1 ` λ p f qq . lim inf n Ñ8 Ideas of proof: Sketch that there exists z satisfying last equation, for f P B . 6 / 12

p f n q # p z q grow, for U open, U X J p f q ‰ H ? How fast does sup z P U In contrast to Przytycki’s result for rational f , for entire f this grows much faster than p f n q # p z q for individual points z . Let M n p r , f q be the iterate of M p r , f q with respect to r : M 1 p r , f q “ M p r , f q M n ` 1 p r , f q “ M p M n p r , f q , f q . and For large R we have M n p R , f q Ñ 8 . Theorem: U open, U X J p f q ‰ H , R ą 0 p f n q # p z q ě log M n ´ m p R , f q ñ D m P N : sup z P U E.g. for f p z q “ e z this growth is much faster than that given by 1 n log log p f n q # p z q ě log p 1 ` λ p f qq . lim inf n Ñ8 Ideas of proof: Sketch that there exists z satisfying last equation, for f P B . Use logarithmic change of variable. 6 / 12

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Let f P B , with critical and asymptotic values in t z : | z | ă R u . 7 / 12

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . 7 / 12

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . Let A be a component of f ´ 1 pt z : | z | ą R uq . 7 / 12

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . Let A be a component of f ´ 1 pt z : | z | ą R uq . W A f Put W “ exp ´ 1 p A q and H “ t z : Re z ą log R u . 7 / 12

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . Let A be a component of f ´ 1 pt z : | z | ą R uq . W exp exp A f Put W “ exp ´ 1 p A q and H “ t z : Re z ą log R u . 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

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . Let A be a component of f ´ 1 pt z : | z | ą R uq . H W exp exp A f Put W “ exp ´ 1 p A q and H “ t z : Re z ą log R u . 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

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . Let A be a component of f ´ 1 pt z : | z | ą R uq . H W exp exp A f Put W “ exp ´ 1 p A q and H “ t z : Re z ą log R u . 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

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . Let A be a component of f ´ 1 pt z : | z | ą R uq . H F W exp exp A f Put W “ exp ´ 1 p A q and H “ t z : Re z ą log R u . 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

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . Let A be a component of f ´ 1 pt z : | z | ą R uq . H F W exp exp A f Put W “ exp ´ 1 p A q and H “ t z : Re z ą log R u . Can lift f to map F : W Ñ H , and F maps every component of W univalently onto H . 7 / 12

Let f P B , with critical and asymptotic values in t z : | z | ă R u . Assume | f p 0 q| ă R . Let A be a component of f ´ 1 pt z : | z | ą R uq . H F W exp exp A f Put W “ exp ´ 1 p A q and H “ t z : Re z ą log R u . 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