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 n th 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¨ obius 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¨ obius maps Let f ( z ) = e i ✙✒ z , where ✒ ✷ ❘ . If ✒ ✷ ◗ , then the set ❢ f n ( z 0 ): n ✷ ◆ ❣ is finite. If ✒ ✷ ❘ ♥ ◗ , then the set ❢ f n ( z 0 ): n ✷ ◆ ❣ is dense in the circle of radius ❥ z 0 ❥ centred at 0. Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 3 / 10

Composition sequences Let ❢ f n ❣ be a sequence in ❍ ( ❉ ). The composition sequence generated by ❢ f n ❣ is the sequence F n = f 1 ✍ f 2 ✍ ✁ ✁ ✁ ✍ f n ✿ Examples, for n = 2 ❀ 3 ❀ ✁ ✁ ✁ n ✒ ✓ ✒ ✓ 1 � 1 1 � 1 ❨ ✦ 1 f n ( z ) = F n ( z ) = z � z ❀ 2 z n 2 k 2 k =2 n ✒ ✓ ✒ ✓ 1 � 1 1 � 1 ❨ � ✦ 0 g n ( z ) = z ❀ G n ( z ) = z n k k =2 Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 4 / 10

Composition sequences Theorem Let ❢ f n ❣ be a sequence in ❍ ( ❉ ) . Assume that there exists a compact set K ✚ ❉ , such that f n ( ❉ ) ✚ K. Then F n = f 1 ✍ ✁ ✁ ✁ ✍ f n converges locally uniformly to a constant in ❉ . Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 5 / 10

Problem Let ❢ f n ❣ be a sequence in ❍ ( ❉ ), such that f n ✦ f ✷ ❍ ( ❉ ) locally uniformly. f n = f ✍ ✁ ✁ ✁ ✍ f ✥ Denjoy–Wolff theorem/ F n = f 1 ✍ ✁ ✁ ✁ ✍ f n ✥ ? Cases: f is an elliptic M¨ obius 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 ❀ ✁ ✁ ✁ n ✒ ✓ ✒ ✓ 1 � 1 1 � 1 ❨ f n ( z ) = e i ✙✒ F n ( z ) = e i ( n � 1) ✙✒ z z ❀ n 2 k 2 k =2 n ✒ ✓ ✒ ✓ 1 � 1 1 � 1 ❨ g n ( z ) = e i ✙✒ G n ( z ) = e i ( n � 1) ✙✒ z z ❀ n k k =2 Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 7 / 10

Elliptic case Theorem Let ❢ f n ❣ be a sequence in ❍ ( ❉ ) . Assume that there exist z 1 ❀ z 2 ✷ ❉ distinct such that ✶ ❳ ✚ ( f n ( z i ) ❀ e i ✙✒ z i ) ❁ ✶ ❀ for i = 1 ❀ 2 ❀ n =1 where ✒ ✷ ◗ . Then F n = f 1 ✍ ✁ ✁ ✁ ✍ f n has finitely many limit functions. Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 8 / 10

Non-elliptic case Let ❢ f n ❣ be a sequence in ❍ ( ❉ ), and let f be a function in ❍ ( ❉ ), which is not an elliptic M¨ obius 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 f n ✷ ◆ , then F n = f 1 ✍ ✁ ✁ ✁ ✍ f n converges locally uniformly to a constant in ❉ . Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 9 / 10

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 ❢ f m ❣ , such that f m ✷ ◆ m and F m = f 1 ✍ ✁ ✁ ✁ ✍ f m diverges. Argyris Christodoulou (OU) Stability of the Denjoy–Wolff theorem TCD 2017 10 / 10