Dynamics of Newton Map and Complexity. Yuefei Wang Institute of Mathematics, CAS

Transcription

1 Dynamics o Newton Ma an Comlexity Yueei Wang Institute o Mathematics CAS

2 Newtons Ma or a holo.ma let N - ' C C. Newton' s iterates N n

3 A ero o is either suerattracting or attracting ixe oint o N ; N has only one reelling ixe oint.

4 The basin o a ixe oint o N U { C N n }. The immeiate basin U o the basin containing. the comonent U is simly connectean unboune Mayer - Schleicher 2006 or entire mas. ;

5 Shishikura 90 The Julia set J N JN is is connectei connecte. N is a rational ma. I a rational ma has only one reelling ixe oint then J is connecte. Ruekert N is 2006 rational olynomials such that has the orm quasiconormal surgeries. e an q. is a reelling or arabolic ixe oint. q

6 Ruekert 06 basin but N Let N C Cˆ Entire Case Newton ma o an entire ma C C or each ixe m 1 oint N C m N such that N'. m Two entire mas g have the same Newton mas c g. An invariant Fatou comonent o N be a meromorhic ma. It is the must be an immeiate may contain invariant comonents which o not have any interior ixe oint such as ixe Baker omains. Newton mas o entire mas have no ixe Herman rings.

7 Entire Case Baranski - Fagella- Jarque - Karinsha let be entire then J N is connecte Equivalently all comonents o the Fatou set are simly connecte. I is a transcenental meromorhic unction with a isconnecte Julia set then has at least one weakly reelling ixe oint. Methos Rational - like mas; renormaliation; counting wining numbers;...

8 Diiculties Alreay or cubic olynomials there may be oen sets o initial oints which o not lea to any root; Instea to an attracting cycle o erio greater than one; The bounaries o the basins will usually be comlicate ractals.

9 Douay - Hubbar 85 For every quaratic olynomial q there isa cubic olynomial so that in the Newton ynamics N there is a coy o the ille-in Julia set o q within the set o ba starting oints. Polynomial- like mas Renormaliation.

10 Smale Inequality 81 a i 1/ i1 K. a a 2 2 r min ' 0 1.

11 Smale Bull AMS 81 Let S = { < r 2K + }. 1 For S N n a ero o.

12 Hubbar - Schleicher - Sutherlan 's Result. Invent Math 01 P 0 with all rootsin D. 1 a set S # S 111. log 2. For each P 0 an each root o s S in the basin o uner N.

13 2 For olynomials with all rootsreal S # S 1. 3 S aroximately log circles each containing log oints at equal istances.

14 Key The global geometry o the basins. Estimate o the sies o accesses to ininity. Each immeiate basin U critical oints o N has exactly channels accessing to ininity. containing m m istinct With o Channels Every basin has a channel with moulus at least /log.

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16 Purely Iterative Algorithms Q - ' Bull.AMS k k k k k u u u u u u P u u u F J C C P Smale. Ma Newton's. ma A rational 1 0 u Q u P J F T T = = = C C iterative algorithm A urely

17 T Generally Convergent Problem generallyconvergent i oen set U o ull measure in C P s.t.or each U T n a ero o as n. ProblemSmale. I > k+1 any generally convergent urely iterative algorithm? Known = 2.

18 Classiicationo GenerallyConvergent Algorithms McMullen Ann Math 87 ; Invent. Math a There is no generally convergent urely iterative algorithm or > b For = 3 every generally convergent algorithm is obtaine by a rational ma T convergent or The algorithm T = M TM M Mobius carrying unity roots to roots o. -1 x 3-1

19 ~ 83 Mane - Sa - Sullivan Density o structurally stable rational mas. Holomorhic Motion. Thurston 82 Toology o rational mas an rigiity theorem or critical inite rational mas. A nonainelattes critically inite rational ma is uniquely etermine by its combinatorial tye. Rigiity Theorem. mas is either trivial A stable algebraic amily o or a amily o Lattes mas. rational

20 Critical oints Gauss - Lucas theorem The convex hull o the roots o a olynomial contains all its critical oints. It contains the critical oints o N.

21 Smale s Mean Value Conjecture Smale I"The Funamenta l Theorem o Algebra an Comlexity Thoery" Bull AMS 81 ; II "Mathmatical Problems or the Next Century" "Mathematic s Frontiers an Persectives" Arnol Atiyah etc AMS 2000.

22 Smale s Mean Value Conjecture For each an ' 0 ' - min = 0 - K ' hols or K = 1 or -1/. Smale K 4 Koebe 1/4 Theorem.

23 . ; Let 0. ' v V v monic = = = P. '0 min -1 / 1or For 0 K v K = = P Normalie Conjecture.

24 Bearon Mina & Ng 02 K 4-2/-1 < 4. Hyerbolic metric. Conte Fujikawa & Lakic Bieberbach's Theorem a K Fujikawa & Sugawa K + 1 Hyerbolic Metric + Bieberbach's Theorem. -2 / -1. Crane 07 K 41-2/ Extremal olynomials. 1/2.

25 Tischler 89 P 1 are on the unit circle. Then V 1/ -1 with all other eros -1 ' 0. Problem. For P with all other eros are in the close unit isk. Does one have V 1/ '0?

26 Thm W. For P eros are in the close unit isk. Then V 1/ Thm W. There is P with all other '0-1 '0. 1/ -1 eros are in the close unit isk such that V 1/ with all other

27 Newton Metho or Riemann Zeta unction Schleicher so that or every ero o its immeiate basin contains at least one o the oints 08 c n Uner the assumtion o the Riemann hyothesis c 2 icn logn. Here ξ ζ Γ / 2-1 / 2 an ξ ξ1-. 0 Conjecture. c t > 0. I there isa root α o ξ whose immeiate basin oes not contain one o the oints c hyothesis is alse an the immeiate basin o an there isa root α α 0 n = ± 2+cn/ 0 contains a oint log n then the Riemann o the critical line with α c n. 0 -α < t

28 Alying Newtons metho to the Riemann unction ierent colorsshow the basins o attraction or ierent eroes o. By Sebastian Mayer

29 Dynamical Conjectures Theorem a c The b Each non - trivial ero o is an inierent ixe oint o Riemann hyothesis has Kawahira 2015.The -. ' no attracting ixe oint. ollowings are equivalent is airmative. There is no toological isk D with D D.

30 The En