Near-parabolic Renormalization and Rigidity

Transcription

1 Near-parabolic enormalization and igidity Mitsuhiro Shishikura Kyoto University Complex Dynamics and elated Topics esearch Institute for Mathematical Sciences, Kyoto University September 3, 2007

2 Irrationally indifferent fixed points We consider holomorphic functions of one variable with fixed point z=0. f(z) = λz + a 2 z If If λ = 1, z=0 is called indifferent fixed point. λ is a root of unity, parabolic; otherwise irrationally indifferent. λ = e 2πiα Consider α Q If conjugate to a rotation (linearizable), then it has a Siegel disk. Otherwise, very complicated invariant sets (hedgehogs). Earlier works: Siegel, Bruno, Herman, Yoccoz, Perez Marco, Petersen, McMullen, Buff, Chéritat,... f 0 (z) = z + a 2 z 2 + O(z 3 ) a 2 0 and its perturbation f(z) = e 2πiα z + a 2 z

3 Linearizability of irrationally indifferent fixed points Siegel (under Diophantine cond.), Bruno (under Bruno condition), Yoccoz (a new proof using renormalization and converse); Cremer (nonlinearizable ex.) Boundary of Siegel disks (Jordan curve in known cases) Herman (quadratic polynomial, bouded type rotation number => J. curve) Petersen (quad. poly., bouded type => locally connected J, measure 0) Herman-Yoccoz, Petersen-Zackeri (weaker cond. for J. curve w. crit. pt.) Herman (quadratic polynomial, no critical point on bdry) Buff-Cheritat (various smoothness) Universality/igidity at the boundary Manton-Nauenberg (experiments, heuristic argument) McMullen (quadratic-like map => rigidity and differentiability) This talk (a new class, high type rotation number => rigidity and differentiability)

4 Physicists motivation KAM torus Chaos last KAM torus destroyed? Physicists expect a universal phenomenon at critical parameter Simpler model (no parameter, only in the phase space) Irrationally indifferent fixed point f(z) = e 2πiα z + a 2 z linearization Siegel Disk holomorphic near 0... Outside?? Julia set Chaotic dynamics boundary Boundary of Siegel Disk is the closure of critical orbit (for polynomials) Physicists expect a universal phenomenon at the boundary of SD Manton-Nauenberg (physicists), McMullen (for bounded type)

5 Theorem (McMullen). Let f and ˆf be quadratic-like maps with Siegel disks of period one with the same rotation number α of bounded type. Then f and ˆf are conjugate by a quasiconformal mapping ϕ which is C 1+γ -conformal on the boundary of the Siegel disk, i.e. where z 0 constant. ϕ(z) = ϕ(z 0 ) + A(z z 0 ) + O( z z 0 1+γ ) as z z 0 is on the boundary of the Siegel disk and A is a non-zero Theorem. Let f = e 2πiα h and ˆf = e 2πiα ĥ where h and ĥ are in the class F 1 which will be defined later, and the rotation number α is of high type (N) with sufficiently large N (also defined later). Then f and ˆf are asymptotically conformally conjugate on the closure of critical orbit. Moreover the conjugacy is C 1+γ -conformal on the critical orbit. Furthermore there exists 0 < λ < 1 such that if the contunued fraction coefficients of α satisfies a n Cλ n with some C > 0 then the conjugacy is C 1+γ -conformal on the closure of the critical orbit. emark. The closure of critical orbit contains boundary of Siegel disk. The above theorem follows from igidity result (Theorem 5) via a differentiability result on quasiconformal mappings.

6 Differentiable functions graph looks like a line In small scale... homeomorphism: can do anything quasi-symmetric, quasi-conformal: bounded ratio asymptotically conformal: ratio -> 1 C 1+α : ratio -> 1 fast For conjugacies between dynamical systems... compare orbits to see details, need to iterate many times

7 eturn map f g f high iterates of f f = (first return map of f) after rescaling = g f k g 1 (if return time k) enormalization fine orbit structure for f fewer iterates of f large scale orbit structure for Successive construction of f, 2 f,..., helps to understand the dynamics of f (orbits, invariant sets, rigidity, bifurcation,... ) If f = f (fixed point of renormalization), then f = g f k g 1 (fixed point equation) f

8 enormalization and igidity (an oversimplified view) Suppose f and f have the same combinatorial type and admit successive construction of renormalizations. f 0 = f f 1 = f 0 f 2 = f 1 f 3 = f 2 g 0 g 1 g 2 h 0 h 1 h 2 h 3 g 0 g 1 g 2 f 0 = f f1 = f 0 f2 = f 1 f3 = f 2 {h n } bounded f and f quasi-conformally conjugate d(f n, f n ) 0 h n linear conjugacy is asymptotically conformal or smooth, etc.

9 Yoccoz renormalization for Siegel-Bruno Theorem log q n+1 q n f(z) = e 2πiα z +..., n = f is conjugate to z e 2πiα z < where p n q n α (convergents) Yoccoz s proof: construct the sequence of renormalizations f n f 0 = f, α 0 = α, α n+1 = dist( 1 α n, Z) 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

10 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 Cylinder/Near-parabolic renormalization f first return map C = C {0} C/Z f glue & uniformize Exp (z) = exp(2πiz) f can be defined when f(z) = e 2πiα z +... is a small perturbation of z + a 2 z (a 2 0) and arg α < π/4.

11 h enormalization: The Picture Write f(z) = e 2πiα z + O(z 2 ) = e 2πiα h(z) where h(z) = z + O(z 2 ). f (α, h) Then f(z) = e 2πi 1 α α h(z) where α h = Exp E (e 2πiα h) (Exp ) 1. Hence : (α, h) ( 1 α, αh) (skew product) hyperbolic? ( α contracting?) 0 α h 0 h (α 0) α 1 α mod Z α 0 contracting? YES for α small

12 Horn map and Parabolic enormalization f 0 (z) = z + a 2 z a 2 0 f 0 Horn map E f0 = Φ attr Φ 1 rep Parabolic enormalization 0 f 0 = Exp E f0 (Exp ) 1 Exp (z) = e 2πiz : C/Z C Φ attr E f0 Φ rep 0 f 0 (z) = z +... by normalization E f0 (z) = z + o(1) (Im z + ) Exp Exp 0 f 0

13 Perturbation (Douady-Hubbard-Lavaurs) f (0) = e 2πiα, α small arg α < π 4 f 0 f E f0 E f χ f f = χ f E f first return map E f depends continuously on f χ f (z) = z 1 α (after a suitable normalization)

14 Main Theorems 1-4 (with H. Inou) We define a class of functions F 1, (and F 1 F 1 ) such that if f F 1, then f is holomorphic, f(0) = 0, f (0) = 1, f has a unique critical point c f in its domain of definition and the critical value f(c f ) = 4 27 (fixed). Moreover f (0) 0. Theorem 1. F 0 1 F 1 F 1. Moreover 0 is holomorphic and 0 (z + z 2 ) F 1. Theorem 2. For small α ( ), F α 1 F 1 F 1. Hence there exists a large N such that if f = e 2πiα h with α of high type (N) and h F 1, then the sequence of renormalizations f = f 0 f 1 f 2 f 3... is defined so that f n = e 2πiα n h n (z), h n F 1. (Here α n+1 = 1 α n and h n+1 = αn h n, possibly after complex conjugation.)

15 Irrational numbers of high type (N) α = a 1 ± a 2 ± a 3 ± 1... where a i N

16 Let P (z) = z(1 + z) 2. We take specific simply connected open sets V and V with 0 V V V C. { } F 1 = f = P ϕ 1 : ϕ(v ) C ϕ : V C is univalent ϕ(0) = 0, ϕ (0) = 1 Define F 1 with V replaced by V. P (0) = 0, P (0) = 1 Definition of F 1 and F 1 critical points: 1 3 and 1 critical values: P ( 1 3 ) = 4 27 and P ( 1) = 0 P V η = e 2πη V slightly smaller domain than V 27 e2πη

17 Theorem 3. After modifying the definition slightly, F 1 is in one to one correspondence with the Teichmüller space of a punctured disk. With respect to the Teichmüller distance (which is complete), 0 is a uniform contraction. Theorem 4. The same statement for small α ( ). Hence when restricted to the subset where α is small, the renormalization is hyperbolic. Teichmüller space is like the unit disk with Poincaré metric. holomorphic self map does not expand the distance. (oyden-gardiner Theorem: Teichmüller distance = Kobayashi distance) F 0 1 F 1 F 1 Estimate of contraction of F 1 F 1 via cotangent space which is the space of integrable holomorphic quadratic differentials. + modulus-area inequality

18 Applications Theorem. Under the assumption of Theorem 2, the critical orbit stays in the domain of f and can be iterated infinitely many times. Moreover if f is (a part of) a rational map, then the critical orbit is not dense. Theorem (Buff-Chéritat). There exists an irrational number α such that the Julia set of the quadratic polynomial P α (z) = e 2πiα z + z 2 has positive Lebesgue measure. Theorem. Suppose f and f satisfy the assumption of Theorem 2, with the same rotation number α. Then they have small periodic cycles ζ n and ζ n around 0 with period q n. Let λ(ζ n ), λ(ζ n) be their multipliers. The differences λ(ζ n ) λ(ζ n) and 1 λ(ζ n) 1 1 λ(ζ n ) 1 tends to 0 exponentially fast as n with a uniform rate.

19 Application 2: igidity Theorem 5 (igidity). If h, h F 1 and α satisfies the hypothesis of Theorem 2, then there exists a quasiconformal homeomorphism ϕ which conjugates f = e 2πiα h and f = e 2πiα h along their critical orbits, and asymptotically conformal on the closure of critical orbits. f Within this class of maps, the same rotation number implies a better conjugacy. f 0 = f f 1 = f 0 f 2 = f 1 f 3 = f 2 g 0 g 1 g 2 h 0 h 1 h 2 h 3 g 0 g 1 g 2 E f f = χ f E f f 0 = f f1 = f 0 f2 = f 1 f3 = f 2 χ f Exp f g n s, g n s are exponential-like (very expanding).

20 Various enormalizations Feigenbaum Circle map Near-parabolic proper subintervals -> Cantor set partition of interval covering by sector or croissant-like domains gluing/identification needed to define the renormalization

21 eturn to Theorem 5 Theorem 5 (igidity). If h, h F 1 and α satisfies the hypothesis of Theorem 2, then there exists a quasiconformal homeomorphism ϕ which conjugates f = e 2πiα h and f = e 2πiα h along their critical orbits, and asymptotically conformal on the closure of critical orbits. f f 0 = f f 1 = f 0 f 2 = f 1 f 3 = f 2 g 0 g 1 g 2 h 0 h 1 h 2 h 3 g 0 g 1 g 2 E f f = χ f E f χ f Exp f 0 = f f1 = f 0 f2 = f 1 f3 = f 2 f g n s, g n s are exponential-like (very expanding). Need to reconstruct the dynamics of f in subdomains (with control on geometry) from f n = n f. Because the relation between f and f n = n f is less obvious.

22 Difficulty in proving rigidity for irrationally indiff. fixed pts. Knowing f, what can be said about f? How to transfer information (e.g. geometry) on n f to previous generations of renormalizations n 1 f, n 2 f,..., f? Fundamental domains (and their boundary curves) are not unique. Need to cover previous fund. regions with next generation fund. regions WITH OVELAP. (not partition) Need to reconstruct the dynamics of so that one can understand f better. f from that of f this is like...

23 Thank you!