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t f The high type quadratic Siegel disks are Jordan domains a YANG Fei r Nanjing University joint with Mitsuhiro Shishikura D T OPICS IN C OMPLEX D YNAMICS 2019 F ROM COMBINATORICS TO TRANSCENDENTAL DYNAMICS Barcelona University, Barcelona

  1. t f The high type quadratic Siegel disks are Jordan domains a YANG Fei r Nanjing University joint with Mitsuhiro Shishikura D T OPICS IN C OMPLEX D YNAMICS 2019 F ROM COMBINATORICS TO TRANSCENDENTAL DYNAMICS Barcelona University, Barcelona March 25, 2019 YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 1 / 15

  2. t f Siegel disk and continued fractions a Let 0 < α < 1 be irrational, f non-linear holo., f ( 0 ) = 0 and f ′ ( 0 ) = e 2 π i α . The maximal region in which f is conjugate to R α ( z ) = e 2 π i α z is a simply connected domain ∆ f called the Siegel disk of f centered at 0. r D YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 2 / 15

  3. t f Siegel disk and continued fractions a Let 0 < α < 1 be irrational, f non-linear holo., f ( 0 ) = 0 and f ′ ( 0 ) = e 2 π i α . The maximal region in which f is conjugate to R α ( z ) = e 2 π i α z is a simply connected domain ∆ f called the Siegel disk of f centered at 0. Let r 1 α = [ 0; a 1 , a 2 , ··· , a n , ··· ] = 1 a 1 + a 2 + 1 D ... be the continued fraction expansion of α . Then p n 1 = [ 0; a 1 , a 2 , ··· , a n ] = 1 q n a 1 + ... + 1 a n converges to α exponentially fast. YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 2 / 15

  4. t f Siegel-Brjuno-Yoccoz a Diophantine condition of order ≤ κ : � � � � � � � > ε � α − p q κ for every rational p � � D ( κ ) : = α ∈ ( 0 , 1 ) : ∃ ε > 0 s.t. . q q r Theorem (Siegel, 1942) The holomorphic germ f has a Siegel disk at 0 if α ∈ D ( κ ) for some κ ≥ 2 . D ∩ κ > 2 D ( κ ) has full measure. D ( 2 ) has measure 0 and α ∈ D ( 2 ) is of bounded type , i.e. sup n { a n } < ∞ . α ∈ D = ∪ κ ≥ 2 D ( κ ) ⇔ sup n { log q n + 1 log q n } < ∞ . YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 3 / 15

  5. t f Siegel-Brjuno-Yoccoz a Diophantine condition of order ≤ κ : � � � � � � � > ε � α − p q κ for every rational p � � D ( κ ) : = α ∈ ( 0 , 1 ) : ∃ ε > 0 s.t. . q q r Theorem (Siegel, 1942) Theorem (Brjuno, 1965) The holomorphic germ f has a Siegel The holomorphic germ f has a Siegel disk at 0 if α ∈ D ( κ ) for some κ ≥ 2 . D disk at 0 if α belongs to � � log q n + 1 α ∈ ( 0 , 1 ) \ Q : ∑ ∩ κ > 2 D ( κ ) has full measure. B = < ∞ . q n n D ( 2 ) has measure 0 and α ∈ D ( 2 ) is of bounded type , i.e. sup n { a n } < ∞ . Remark: D � B . α ∈ D = ∪ κ ≥ 2 D ( κ ) ⇔ sup n { log q n + 1 log q n } < ∞ . YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 3 / 15

  6. t f Siegel-Brjuno-Yoccoz a Conjecture (Douady, 1986) If a non-linear holomorphic function (entire or rational) has a Siegel disk, then the rotation number is necessarily in B . r Theorem (Brjuno, 1965) The holomorphic germ f has a Siegel D disk at 0 if α belongs to � � log q n + 1 α ∈ ( 0 , 1 ) \ Q : ∑ B = < ∞ . q n n Some progresses have been made by Remark: D � B . P´ erez-Marco, Geyer, Okuyama, Manlove, Cheraghi ... YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 3 / 15

  7. t f Siegel-Brjuno-Yoccoz a Conjecture (Douady, 1986) If a non-linear holomorphic function (entire or rational) has a Siegel disk, then the rotation number is necessarily in B . r Theorem (Yoccoz, 1988) Theorem (Brjuno, 1965) If α �∈ B , then P α ( z ) = e 2 π i α z + z 2 has The holomorphic germ f has a Siegel D disk at 0 if α belongs to no Siegel disk at the origin. � � log q n + 1 Remark: Douady’s conjecture is still α ∈ ( 0 , 1 ) \ Q : ∑ B = < ∞ . open even for cubic polynomials. q n n Some progresses have been made by Remark: D � B . P´ erez-Marco, Geyer, Okuyama, Manlove, Cheraghi ... YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 3 / 15

  8. t f Siegel disks a r D The Siegel disk of f ( z ) = e 2 π i α z + z 2 , where √ √ The Siegel disk of f ( z ) = e π i ( 5 − 1 ) z 5 − 1 α = = [ 0;1 , 1 , 1 , ··· ] ( 1 − z ) 2 2 YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 4 / 15

  9. t f Siegel disks a r D √ √ 5 − 1 ) / 2 sin ( z ) The Siegel disk of f ( z ) = e π i ( 5 − 1 ) ze z The Siegel disk of f ( z ) = e π i ( YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 4 / 15

  10. t f Douady-Sullivan’s conjecture Conjecture (Douady-Sullivan, 1986) a The Siegel disk of a rational map (deg ≥ 2) is always a Jordan domain. r D YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 5 / 15

  11. t f Douady-Sullivan’s conjecture Conjecture (Douady-Sullivan, 1986) a The Siegel disk of a rational map (deg ≥ 2) is always a Jordan domain. When α ∈ D ( 2 ) is of bounded type : r Theorem (Zhang, 2011) The bounded type Siegel disk of a rational map ( deg ≥ 2 ) is a quasi-disk. D (Douady-Ghys-Herman-´ Swia ¸tek, 1987) quadratic poly (Zakeri, 1999) cubic poly (Shishikura, 2001) all poly (Yampolsky-Zakeri, 2001) some quadratic rational map YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 5 / 15

  12. t f Douady-Sullivan’s conjecture Conjecture (Douady-Sullivan, 1986) a The Siegel disk of a rational map (deg ≥ 2) is always a Jordan domain. When α ∈ D ( 2 ) is of bounded type : r Theorem (Zakeri, 2010) Theorem (Zhang, 2011) The bounded type Siegel disk of a The bounded type Siegel disk of a non-linear f ( z ) = P ( z ) e Q ( z ) is a rational map ( deg ≥ 2 ) is a quasi-disk. D quasi-disk, where P, Q are polys., f ( 0 ) = 0 , f ′ ( 0 ) = λ = e 2 π i α . (Douady-Ghys-Herman-´ Swia ¸tek, 1987) quadratic poly (Geyer, 2001) f ( z ) = λ ze z (Zakeri, 1999) cubic poly (Keen-Zhang, 2009) (Shishikura, 2001) all poly f ( z ) = ( λ z + az 2 ) e z (Yampolsky-Zakeri, 2001) some quadratic rational map YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 5 / 15

  13. t f Douady-Sullivan’s conjecture Conjecture (Douady-Sullivan, 1986) a The Siegel disk of a rational map (deg ≥ 2) is always a Jordan domain. When α ∈ D ( 2 ) is of bounded type : r Theorem (Zhang, 2011) The bounded type Siegel disk of a rational map ( deg ≥ 2 ) is a quasi-disk. (Zhang, 2005) f ( z ) = λ sin ( z ) D (Y., 2013) f ( z ) = λ sin ( z )+ a sin 3 ( z ) (Douady-Ghys-Herman-´ Swia ¸tek, 1987) (Ch´ eritat, 2006) quadratic poly some “simple” entire functions (Zakeri, 1999) cubic poly (Ch´ eritat-Epstein, 2018) (Shishikura, 2001) all poly some holo. maps with at most 3 (Yampolsky-Zakeri, 2001) some singular values. quadratic rational map YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 5 / 15

  14. t f Douady-Sullivan’s conjecture Conjecture (Douady-Sullivan, 1986) a The Siegel disk of a rational map (deg ≥ 2) is always a Jordan domain. When α ∈ PZ is of Petersen-Zakeri type : log a n = O ( √ n ) as n → ∞ , r where D ( 2 ) � PZ ⊂ ∩ κ > 2 D ( κ ) , and PZ has full measure in ( 0 , 1 ) : D Theorem (Petersen-Zakeri, 2004) For all α ∈ PZ , the Siegel disk of P α ( z ) = e 2 π i α z + z 2 is a Jordan domain. (Zhang, 2014) all polynomials (Zhang, 2016) f ( z ) = e 2 π i α sin ( z ) Some related work YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 5 / 15

  15. t f Douady-Sullivan’s conjecture Conjecture (Douady-Sullivan, 1986) a The Siegel disk of a rational map (deg ≥ 2) is always a Jordan domain. When α ∈ PZ is of Petersen-Zakeri Theorem (Avila-Buff-Ch´ eritat, type : log a n = O ( √ n ) as n → ∞ , r 2004) ∃ α s.t. the boundary of the Siegel disk where D ( 2 ) � PZ ⊂ ∩ κ > 2 D ( κ ) , and of P α is smooth. PZ has full measure in ( 0 , 1 ) : D Theorem (Buff-Ch´ eritat, 2007) Theorem (Petersen-Zakeri, 2004) ∃ α s.t. the boundary of the Siegel disk For all α ∈ PZ , the Siegel disk of of P α is C r but not C r + 1 . P α ( z ) = e 2 π i α z + z 2 is a Jordan domain. Some related work has been done by (Zhang, 2014) all polynomials P´ erez-Marco, Rogers, Shen, ... (Zhang, 2016) f ( z ) = e 2 π i α sin ( z ) YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 5 / 15

  16. t f Counter-examples √ 5 − 1 ) : Siegel disk of f ( z ) = λ e z − λ , where λ = e π i ( a r D Courtesy of A. Ch´ eritat Theorem (Ch´ eritat, 2011) There is a holomorphic germ f such that the corresponding Siegel disk ∆ f is compactly contained in Dom ( f ) but ∂ ∆ f is a pseudo-circle , which is not locally connected. YANG F. (Nanjing Univ.) The high type quadratic Siegel disks Barcelona, March 25, 2019 6 / 15

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