Stability of the DenjoyWolff theorem Argyris Christodoulou The Open - - PowerPoint PPT Presentation

Stability of the Denjoy–Wolff theorem

Argyris Christodoulou The Open University Topics in Complex Dynamics October 2017

Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 0 / 10

Preliminaries

❉ = ❢z ✷ ❈: ❥z❥ ❁ 1❣ ❍(❉) = ❢g : ❉ ✦ ❉ holomorphic❣ Endow ❍(❉) with the topology of locally uniform convergence.

Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 1 / 10

Iteration

For a function f ✷ ❍(❉), the nth iterate of f is the function f n = f ✍ f ✍ ✁ ✁ ✁ ✍ f

⑤ ④③ ⑥

n times

✿

Theorem (Denjoy–Wolff)

Let f ✷ ❍(❉). Assume that f is not an elliptic M¨

• bius map, then there

exists p ✷ ❉ such that f n ✦ p locally uniformly on ❉. The point p is called the Denjoy–Wolff point of f .

Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 2 / 10

Elliptic M¨

• bius maps

Let f (z) = ei✙✒z, where ✒ ✷ ❘. If ✒ ✷ ◗, then the set ❢f n(z0): n ✷ ◆❣ is finite. If ✒ ✷ ❘ ♥ ◗, then the set ❢f n(z0): n ✷ ◆❣ is dense in the circle of radius ❥z0❥ centred at 0.

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Composition sequences

Let ❢fn❣ be a sequence in ❍(❉). The composition sequence generated by ❢fn❣ is the sequence Fn = f1 ✍ f2 ✍ ✁ ✁ ✁ ✍ fn✿ Examples, for n = 2❀ 3❀ ✁ ✁ ✁ fn(z) =

✒

1 1 n2

✓

z❀ Fn(z) = z

n

❨

k=2

✒

1 1 k2

✓

• ✦ 1

2z

gn(z) =

✒

1 1 n

✓

z❀ Gn(z) = z

n

❨

k=2

✒

1 1 k

✓

• ✦ 0

Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 4 / 10

Composition sequences

Theorem

Let ❢fn❣ be a sequence in ❍(❉). Assume that there exists a compact set K ✚ ❉, such that fn(❉) ✚ K. Then Fn = f1 ✍ ✁ ✁ ✁ ✍ fn converges locally uniformly to a constant in ❉.

Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 5 / 10

Problem

Let ❢fn❣ be a sequence in ❍(❉), such that fn ✦ f ✷ ❍(❉) locally uniformly. f n = f ✍ ✁ ✁ ✁ ✍ f ✥Denjoy–Wolff theorem/ Fn = f1 ✍ ✁ ✁ ✁ ✍ fn ✥ ? Cases: f is an elliptic M¨

• bius map

f has its Denjoy–Wolff point inside ❉ f has its Denjoy–Wolff point on ❅❉

Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 6 / 10

Elliptic case

Examples, for n = 2❀ 3❀ ✁ ✁ ✁ fn(z) = ei✙✒

✒

1 1 n2

✓

z❀ Fn(z) = ei(n1)✙✒z

n

❨

k=2

✒

1 1 k2

✓

gn(z) = ei✙✒

✒

1 1 n

✓

z❀ Gn(z) = ei(n1)✙✒z

n

❨

k=2

✒

1 1 k

✓

Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 7 / 10

Elliptic case

Theorem

Let ❢fn❣ be a sequence in ❍(❉). Assume that there exist z1❀ z2 ✷ ❉ distinct such that

✶

❳

n=1

✚(fn(zi)❀ ei✙✒zi) ❁ ✶❀ for i = 1❀ 2 ❀ where ✒ ✷ ◗. Then Fn = f1 ✍ ✁ ✁ ✁ ✍ fn has finitely many limit functions.

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Non-elliptic case

Let ❢fn❣ be a sequence in ❍(❉), and let f be a function in ❍(❉), which is not an elliptic M¨

• bius map.

Denote by p the Denjoy–Wolff point of f .

Theorem

If f has p ✷ ❉ as its Denjoy–Wolff point, then there exists a neighbourhood ◆ of f in ❍(❉) such that if fn ✷ ◆, then Fn = f1 ✍ ✁ ✁ ✁ ✍ fn converges locally uniformly to a constant in ❉.

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Non-elliptic case

Theorem

Assume that f has p ✷ ❅❉ as its Denjoy–Wolff point. Then for every sequence of neighbourhoods ❢◆m❣ of f , there exists a sequence of functions ❢fm❣, such that fm ✷ ◆m and Fm = f1 ✍ ✁ ✁ ✁ ✍ fm diverges.

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