The geometry of the Weil-Petersson metric in complex dynamics

Transcription

1 The geometry of the Weil-Petersson metric in complex dynamics Oleg Ivrii Apr. 23, 2014

2 The Main Cardioid Mandelbrot Set Conjecture: The Weil-Petersson metric is incomplete and its completion attaches the geometrically finite parameters.

3 Blaschke products Let B d = { Blaschke products of degree d with an attracting fixed point } / Aut D e.g B 2 = D: a D : z f a (z) = z z + a 1 + az. All these maps are q.s. conjugate to each other on S 1 and except for for the special map z z 2, are q.c. conjugate on the entire disk.

4 a = 0.5

5 a = 0.95

6 Mating Let f a, f b be Blaschke products. Exists a rational map f a,b and a Jordan curve γ s.t f a,b Ω = fa, f a,b Ω+ = fb. f a,b, γ change continuously with a, b. ( In degree 2, f a,b = z ) z + a 1 + bz

7 McMullen s paper on thermodynamics Let f a(t) be a curve in B d. Can form f a(0),a(t). The function t H. dim γ 0,t satisfies: H. dim γ 0,0 = 1. d dt H. dim γ 0,t = 0. t=0 Definition (McMullen). d 2 dt 2 H. dim γ 0,t =: ḟ a(t) 2 WP. t=0 f 0 f t

8 McMullen s paper on thermodynamics (ctd) Let H t denote the conformal conjugacy from D to Ω (f 0,t ). The initial map H 0 is the identity. Let v = d dt H t t=0 be the holomorphic vector field of the deformation. McMullen showed that ḟ a(t) 2 WP = 4 3 lim r 1 z =r v ρ 2 (z) 2 dθ 2π.

9 Example: Weil-Petersson metric at z 2 Lacunary series v z + z 2 + z 4 + z Can evaluate integral average explicitly due to orthogonality 1 z k z 2π l dθ = δ kl. S 1 Obtain Ruelle s formula H. dim J(z 2 + c) 1 + c 2 16 log 2 + O( c 3 ).

10 Beltrami Coefficients For an o.p. homeomorphism w : C C, we can compute its dilatation µ(w) = w w. If µ < 1, we say w is quasiconformal. Conversely, given µ with µ < 1, there exists a q.c. map w µ with dilatation µ. Dynamics: Given f Rat d and µ M(D) f, can construct new rational maps by: f tµ (z) = w tµ f (w tµ ) 1.

11 Upper bounds on quadratic differentials Suppose µ is supported on the exterior unit disk, µ 1. Then, v (z) = 6 µ(ζ) π (ζ z) 4 dζ 2. Theorem: lim sup r 1 z =r ζ >1 v ρ 2 (z) 2 dθ 2π where S R is the circle {z : z = R}. lim sup R 1 + supp µ S R

12 a = 0.5

13 a = 0.95

14 Incompleteness with a precise rate of decay Petal counting hypothesis As a e(p/q) radially, the WP metric is proportional to the petal count.

15 Incompleteness with a precise rate of decay Petal counting hypothesis As a e(p/q) radially, the WP metric is proportional to the petal count. Renewal theory: Given a point z D, let N (z, R) be the number of w satisfying f k (w) = z, for some k 0, that lie in B hyp (0, R). Then, N (z, R) 1 log 1/z e R as R 2 h(f a ) where h(f a ) = log f (z) dθ is the entropy of Lebesgue S 1 2π measure.

16 Incompleteness with a precise rate of decay (cont.) If lim v /ρ 2 2 dθ was proportional to the number of r 1 z =r da petals, then it would be asymptotically C p/q (1 a ) 3/4.

17 Incompleteness with a precise rate of decay (cont.) If lim v /ρ 2 2 dθ was proportional to the number of r 1 z =r da petals, then it would be asymptotically C p/q (1 a ) 3/4. WARNING! We might have correlations v P ρ 2 v Q ρ 2. P Q Schwarz lemma: The petals are separated in the hyperbolic metric. Indeed, d D (P, Q) d D (P 1, P 2 ) d D (0, a).

18 Decay of Correlations Fact: if d D (z, supp µ + ) > R, then v /ρ 2 e R. Triangle inequality: For any z D, C(z) v v P Q (z) ρ2 ρ 2 (z) e R1 e R 2 = e R. P Q As e d D(0,a) 1 a, correlations decay like 1 a. REMARK! This is neligible to the diagonal term 1 a.

19 a 1

20 a 1

21 a e(1/3)

22 a e(1/3)

23 a 1 horocyclically

24 a 1 horocyclically

25 Rescaling Limits Critically centered versions f a = m c,0 f a m 0,c a 1 radially: In H, this is just w w 1/w. f a z2 + 1/ /3z 2.

26 Rescaling Limits Critically centered versions f a = m c,0 f a m 0,c a 1 radially: In H, this is just w w 1/w. a 1 along a horocycle: f a z2 + 1/ /3z 2. f a w 1/w + T with T > 0 (clockwise) and T < 0 (counter-clockwise).

27 Rescaling Limits (ctd) Amazingly, if a e(p/q) along a horocycle, then f a q the same class of maps, i.e converges to f q a w 1/w + T

28 Rescaling Limits (ctd) Amazingly, if a e(p/q) along a horocycle, then f a q the same class of maps, i.e converges to f q a Lavaurs-Epstein boundary: w 1/w + T The WP metric is asymptotically periodic along horocycles Lavaurs phase We attach a punctured disk to every cusp with the same analytic and metric structure that models the limiting behaviour along horocycles.

29 a 1 horocyclically

30 a 1 horocyclically

31 A quasi-blaschke product Horizontal direction

32 A quasi-blaschke product Vertical direction

33 A quasi-blaschke product Vertical direction

34 Beyond degree 2: Spinning in B3