Mitsuhiro Shishikura Kyoto University

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1 Siegel disk boundaries and hairs for irrationally indifferent fixed points Mitsuhiro Shishikura Kyoto University Low-Dimensional Dynamics CODY Autumn in Warsaw Stefan Banach International Mathematical Center, Warsaw November 19, 2010

2 Theorem: If f(z) =e 2πiα h(z) such that α Irrat N (high type) and is a Brjuno number and h F 1, then the boundary of the Siegel disk of f is a Jordan curve. Earlier results by Herman, Petersen-Zackeri (surgery), Buff-Chéritat (function spaces) linearization Plan... Outside?? Julia set Chaotic dynamics Siegel Disk boundary In order to study tame dynamics, take advantage of expanding (chaotic) dynamics as renormalizers. How to recover the original dynamics from its renormalizations in the case of rotation-like dynamics (overlapping domains). -- solution via Dynamical Charts

3 Infinitely renormalizable map f 0 = f f 1 = Rf 0 f 2 = Rf 1 f 3 = Rf 2 g 0 g 1 g 2 Tame maps f n are maps to be renormalized, and expanding maps g n are those which renormalize f n s. Take advantage of expansion of g n (i.e. contraction by local inverse gn 1 ), to study the dynamics of f n s. How to use Exapansion: Example. J(z 2 + ε) is a Jordan curve. Construct approximating curves γ n Show that {γ n } is a Cauchy sequence and the limit is the object we want γ 1 γ 2 γ3 d(γ n, γ n+1 ) Cβ n (0 < β < 1) J f n 1 expanding γ n γ n+1 γ 2 γ 1

4 How to recover f from R n f s f 0 = f f 1 = Rf 0 f 2 = Rf 1 f 3 = Rf 2 g 0 g 1 g 2 Feigenbaum Circle map Near-parabolic proper subintervals partition of interval commuting pairs covering by sector or croissant-like domains gluing/identification needed to define the renormalization

5 Yoccoz renormalization for Siegel-Bruno Theorem f n (z) = e 2πiα n z +... f n+1 (z) = e 2πiα n+1 z +... f n f n+1 first return map glue uniformize The first return time and the first return map are discontinuous on the sector. The first return map becomes continuous after gluing. It is better to think that we are taking the quotient of an open set containing the sector. The punctured neighborhood is covered by the iterates of the open set. Due to the overlapping, there are problems when we want to recover f from the renormalization. For example, if the renormalization has a fixed point, what is the period for the corresponding periodic point for f?

6 Dynamical charts for irrational rotation-like dynamics f U i ϕ i gluing X i... dynamics F i Open sets U i (i I), which cover a punctured nbd of fixed pt Maps ϕ i : U i X, asmallnumberofmodelspaces AsmallnumberofmodeldynamicsF = ϕ r(i) f ϕ 1 i (for example, F can and id) Index set I with induced dynamics r : I I (represents the combinatorics of the dynamics) Gluing ϕ i,j = ϕ i ϕ 1 j on overlaps, compatible with model dynamics (absorbs the difference for particular maps) Special for irrational dynamics: Refinements one system of charts anewsystemofrefinedcharts Convention z 0 in U i Imw + in X i

7 Cylinder/Near-parabolic renormalization f first return map C = C {0} C/Z Rf glue & uniformize Exp (z) = exp(2πiz) Theorem (Inou & S.): For some V and N, the near-parabolic renormalization R from {e 2πiα f : α Irrat N,f F 1 } = Irrat N F 1 is well defined and expanding along α direction and uniformly contracting along F 1 direction. Moreover R(e 2πiα z + z 2 ) belong to the above set for α Irrat N. If f(z) =e 2πiα z+z 2 with α irrational of high type, then Rf,R 2 f,r 3 f,... are defined and in a nice class.

8 A Consequence from the proof of Inou-S. theorem There exist a canonical domain Ω can and a canonical dynamics F can : Ω can Ω can such that f,rf,r 2 f,... admit the following description: A punctured neighborhood is isomorphic to Ω can cut off at a certain width ( 4 ) with left and right edges glued by a homeomorphism 3 α θ f. The dynamics is induced by F can. f θ f F can f near 0 Model space Ω can and model map F can modulo gluing

9 Model space and map: Checkerboard pattern parabolic fixed point z + z 2 Φ attr F can ( =fixedpt) irrat. indiff. (near-parabolic) θ f Need to cut off at a certain width and glue left and right edges

10 0-th Successive construction (refinement of dynamical charts) F can single chart with gluing n-th to (n+1)-st U i ϕ i... U i,j ϕ i,j...

11 Construction of successive charts Ω f Ω Rf Ω (2) f,k 1,k 2 Ω (1) f,k 1 Ω (1) f,k 1 Ω (0) Ω (0) f f Ω (0) Rf f Rf Ω (0) Ω (1) k 1 Ω (n) k 1,k 2,...,k n Ω (n+1) k 1,k 2,...,k n,k n+1... each Ω (n) k 1,k 2,...,k n is isomorphic to truncated checkerboard pattern Ω Rn f they are glued via θ Rn f

12 Theorem: If f(z) =e 2πiα z + z 2 with α of sufficiently high type and Brjuno, then the boundary of Siegel disk is a Jordan curve. Idea of the proof: Construct approximating curves and show that they converge. Ω (1) f,k 1 Ω (0) f Ω (0) Rf approximation of bdry of Siegel disk Construct approximate boundary curves by joining segments in Ω (n) f,k 1,...,k n in its canonical coordinate. They converge exponentially. height of the approx. curve: h n = B(α n ) mapping Ω (1) f,k 1 Ω (0) Rf is uniformly expanding (with respect to Poincaré metrics of Ω (0) f and Ω (0) Rf ) expanding γ n γ n+1 γ 2 γ 1 One only needs to work in model space (i.e. truncated checkerboard pattern)? f n 1

13 Thank you!