The geometry of the Weil-Petersson metric in complex dynamics Oleg - PowerPoint PPT Presentation

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

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

Blaschke products � � � Blaschke products of degree d Let B d = Aut D with an attracting fixed point e.g B 2 ∼ = D : z → f a ( z ) = z · z + a a ∈ D : 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.

a = 0 . 5

a = 0 . 95

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 − ∼ = f a , ◮ f a , b | Ω + ∼ = f b . ◮ f a , b , γ change continuously with a , b . � � f a , b = z · z + a In degree 2, 1 + bz

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 . f t � d � H . dim γ 0 , t = 0 . � dt � t =0 Definition (McMullen). f 0 d 2 � H . dim γ 0 , t =: � ˙ f a ( t ) � 2 � WP . � dt 2 � t =0

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 � H t � dt � t =0 be the holomorphic vector field of the deformation. McMullen showed that 2 d θ v ′′′ WP = 4 � � � � ˙ f a ( t ) � 2 � � 3 · lim ρ 2 ( z ) 2 π. � � r → 1 � � | z | = r

Example: Weil-Petersson metric at z 2 v ′ ∼ z + z 2 + z 4 + z 8 + . . . Lacunary series Can evaluate integral average explicitly due to orthogonality 1 � S 1 z k z l d θ = δ kl . 2 π Obtain Ruelle’s formula | c | 2 H . dim J ( z 2 + c ) ∼ 1 + 16 log 2 + O ( | c | 3 ) .

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 t µ ( z ) = w t µ ◦ f ◦ ( w t µ ) − 1 .

Upper bounds on quadratic differentials Suppose µ is supported on the exterior unit disk, � µ � ∞ ≤ 1. Then, v ′′′ ( z ) = − 6 µ ( ζ ) � ( ζ − z ) 4 · | d ζ | 2 . π | ζ | > 1 Theorem: 2 d θ v ′′′ � � � � � � � lim sup ρ 2 ( z ) 2 π � lim sup � supp µ ∩ S R � � � � � R → 1 + r → 1 − | z | = r where S R is the circle { z : | z | = R } .

a = 0 . 5

a = 0 . 95

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.

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, 2 · log | 1 / z | N ( z , R ) ∼ 1 · e R as R → ∞ h ( f a ) � S 1 log | f ′ ( z ) | · d θ where h ( f a ) = 2 π is the entropy of Lebesgue measure.

Incompleteness with a precise rate of decay (cont.) � | v ′′′ /ρ 2 | 2 d θ was proportional to the number of If lim r → 1 − | z | = r | da | petals, then it would be asymptotically ∼ C p / q · (1 − | a | ) 3 / 4 .

Incompleteness with a precise rate of decay (cont.) � | v ′′′ /ρ 2 | 2 d θ was proportional to the number of If lim r → 1 − | z | = r | da | petals, then it would be asymptotically ∼ C p / q · (1 − | a | ) 3 / 4 . WARNING! We might have correlations v ′′′ v ′′′ � � � Q P � � ρ 2 · � . � � 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 ).

Decay of Correlations Fact: if d D ( z , supp µ + ) > R , then | v ′′′ /ρ 2 | � e − R . Triangle inequality: For any z ∈ D , v ′′′ v ′′′ � � � � e − R 1 · e − R 2 = e − R . � Q � P � C ( z ) ≤ ρ 2 ( z ) · ρ 2 ( z ) � � � 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 | .

a → − 1

a → − 1

a → e (1 / 3)

a → e (1 / 3)

a → 1 horocyclically

a → 1 horocyclically

Rescaling Limits ˜ “Critically centered versions” f a = m c , 0 ◦ f a ◦ m 0 , c a → 1 radially: f a → z 2 + 1 / 3 ˜ 1 + 1 / 3 z 2 . In H , this is just w → w − 1 / w .

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

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

Rescaling Limits (ctd) Amazingly, if a → e ( p / q ) along a horocycle, then ˜ f ◦ q converges to a the same class of maps, i.e ˜ f ◦ q → w − 1 / w + T a Lavaurs-Epstein boundary: 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.

a → − 1 horocyclically

a → − 1 horocyclically

A quasi-Blaschke product – Horizontal direction

A quasi-Blaschke product – Vertical direction

A quasi-Blaschke product – Vertical direction

Beyond degree 2: Spinning in B 3