Mating quadratic maps with the modular group Luna Lomonaco IMPA - - PowerPoint PPT Presentation

Mating quadratic maps with the modular group

Luna Lomonaco IMPA

Joint work with Shaun Bullett, QMUL

August 24, 2020

Matings

◮ A mating between 2 objects A and B is an object C behaving as A

• n an invariant subset of its domain, and as B on the complement

◮ They exist in both worlds of rational maps and of Kleinian groups. ◮ Both rat. & Klein. when iterated on C divide C in 2 invariant sets:

◮ domain of normality (Fatou set, ordinary set resp), ◮ the complement (Julia set, limit set resp).

◮ Can we mate a rational map (on an invariant component of its Fatou set) and a Kleinian group (on an invariant component of its

• rdinary) set?
• 1. Do rational maps and kleinian groups fit together? In some object C?
• 2. Is C a mating? Is a family of C a family of matings?

Bullett-Penrose: PSL(2, Z) and quadratic maps fit together

Modular group PSL(2, Z): Kleinian group with generators: τ1(z) = z + 1, and τ2(z) = z 1 + z Minkowski map h+ : [0, ∞) → [0, 1]: homeo x ∈ R, written [x0; x1, x2, . . .] = x0 + 1 x1 + 1 x2 + . . . h+(x) = 0. 1 . . . 1

x0

0 . . . 0

x1

1 . . . 1

x2

. . . It conjugates the action of τ1(z) and τ2(z) on (−∞, 0] with the doubling map, and on [0, ∞) with the halving map.

Holomorphic correspondences

A 2 : 2 holomorphic correspondence F on C: is a multi-valued map F : z → w defined by a polynomial relation P(z, w) = 0 of deg 2 in z and 2 in w. Rational map f , deg(f)=2, ← → (2:1) F : z → w f (z) = p(z)

q(z)

with P(z, w) = wq(z) − p(z) Modular group ← → (2:2) F : z → w with generators P(z, w) = (w − (z + 1))(w(z + 1) − z) τ1(z) = z + 1, τ2(z) =

z 1+z

Quadratic maps and PSL(2, Z) fit in a correspondence

Theorem(Bullett-Penrose, ’94) The matings between Qc, c ∈ M and PSL(2, Z) lie in the family Fa : z → w given by aw − 1 w − 1 2 + aw − 1 w − 1 az + 1 z + 1

• +

az + 1 z + 1 2 = 3. Moreover, for all a ∈ [4, 7] ⊂ R the correspondence Fa is conjugate to the generators of PSL(2, Z) on an invariant open set.

Figure: Limit set of Fa, a = (3 +

√ 33)/2

MΓ and some limit sets for Fa

Which correspondences in Fa are matings? Conjecture and state of the art

◮ Conjecture (B-P, ’94) The family Fa contains matings between PSL(2, Z) and every quadratic polynomial with connected Julia set, and the connectedness locus MΓ of Fa is homeomorphic to M. ◮ Theorem (B-P,’94 + B-Harvey,Electron.Res.Announc.AMS,’00 + B-Haissinsky, Conform.Geom.Dyn.,’07) There exists a mating between PSL(2, Z) and Qc for a large class of value of c

The family Fa = Ja ◦ Cov 0

Q Q(z) = z3 − 3z, Cov 0

Q is its correspondence ’deck transformation’:

Q(x) = Q(y) = Q(z) ⇒ Cov 0

Q(x) = {y, z}

Ja involution having fixed points at 1 (critical point of Q) and a. Q(−2) = Q(1), so Cov Q sends each blue line to the other two.

1 −2 ∆ CovQ fundamental domain for CovQ CovQ a ∆Ja −2 1 2 ∆ CovQ a ∆Ja F−1 a (∆Ja ) F−2 a (∆Ja ) (2 : 1) (1 : 2) (1 : 1)

Dynamics of the family Fa

∆a Fa| : F−1 a (∆a) 2:1 − → ∆a Fa| : C \ ∆a 1:2 − → Fa( C \ ∆a) Λa,− = ∞ 1 (Fa)−n(∆a) Λa,+ = ∞ 1 (Fa)n( C \ ∆a)

Facts: for every a ◮ Fa has a parabolic fixed point at z = 0. ◮ F−1

a (0) = {0, Sa}.

◮ Fa|∆a is a deg 2 holomorphic map at every z ∈ ∆a, z = Sa

The family Per1(1)

◮ Per1(1) = {PA(z) = z + 1/z + A | A ∈ C}, ◮ ∞ parabolic fixed point, with basin ΛA, KA = C \ ΛA ◮ ∀A, external class given by h2(z) = z2+1/3

1+z2/3, and

h2(z)|S1 ∼top P0(z) = z2|S1 ◮ M1: connectedness locus (M1 ≈ M by Petersen-Roesch).

Main Theorem

We say that Fa is a mating between the rational quadratic map PA : z → z + 1/z + A and the modular group Γ = PSL(2, Z) if

• 1. the 2-to-1 branch of Fa for which Λ− is invariant, is hybrid

equivalent to PA on Λ− (i.e., it is quasiconformally conjugate to PA in a nbh of Λ− by a map which is conformal on the interior of Λ−),

• 2. when restricted to a (2 : 2) correspondence from Ω(Fa) to itself, Fa

is conformally conjugate to the pair of M¨

• bius transformations

{τ1, τ2} from the complex upper half plane H to itself. Theorem (Bullett-L, Invent.Math. 220, (2020)) For every a ∈ MΓ the correspondence Fa is a mating between some rational map PA : z → z + 1/z + A and Γ.

M1 and MΓ

Proof of the Thm, part 1

A (deg 2) parabolic-like map is an object that locally behaves like a map PA, A ∈ C. Thm A deg 2 parabolic-like map is hybrid conjugated to a member of the family Per1(1), a unique such member if the filled Julia set is connected. So, if we prove that for all a ∈ MΓ, the branch of Fa which fixes Λ− restricts to a parabolic-like map, we are done!

Parabolic-like maps

Parabolic-like maps

Parabolic-like maps

Parabolic-like maps

From correspondences to parabolic-like maps

From correspondences to parabolic-like maps

From correspondences to parabolic-like maps: qc surgery to kill one image

F (1:2) U V

Surgery construction (valid also out of MΓ)

Λ− ⊂ Hl, but we need it to be contained in a top. disc making at the parabolic fixed point an angle < π, to have space to put our new Beltrami forms! ’Easy’ to do individually

φ 3πi ψ (5π/2 + θ1)i (3π/2 − θ2)i πi πi z2 F−1 (2:1) 2πi 2πi 3π/2 − θ2 π/2 + θ1 π − 2θ2 π + 2θ1 log log 2z − 4πi qc interpolation 2z − 2πi ˆ T1 ˆ T2

Part 2: B¨

• ttcher map (valid just in MΓ)

◮ ∀a ∈ MΓ ∃ a Riemann map φ : Ω → H. We prove that φ conjugates Fa|Ω(Fa) to the generators of the modular groups on H. ◮ This is: the map φ : Ω → H plays for our family Fa the role the B¨

• ttcher map plays for the quadratic family!

φ ✲

Proving MΓ ≈ M1

The straightening construction induces a map χ : MΓ → M1 The map χ is injective (punch-line: Rickmann Lemma). Theorem (Bullett-L) The map χ : MΓ \ {4, 7} → M1 \ {−3, 1} is a homeomorphism, which extends to a doubly pinched neighbourhood of MΓ.

Main tool: lunes and fundamental croissants

MΓ ⊂ lune L ⇒ for all a ∈ MΓ, Λ−,a is contained in a lune Va which moves holomorphically with the parameter. For a ∈ L, set V ′

a := F−1 a (Va). We call Aa := Va \ V ′ a the fundamental

croissant. The fundamental croissant moves holomorphically for a ∈ ˚ L: fix a0 ∈ MΓ \ {4, 7}, we have a holomorphic motion τa : Aa0 → Aa We can extend χ : ˚ L \ MΓ → C by using τ: the extension is qc.

Holomorphic motion of Beltrami forms

Do surgery at a0 ∈ MΓ, obtaining µa0 on Aa0, move it by τ : Aa0 → Aa µa = (τa)∗(µa0), and µa = (Fa)−n(µa) on F−n

a

(Aa) for all n ≥ 1 ⇒ µa is holomorphic in a wherever Λa,− moves holomorphically with a.

z2 fa0 (qc interpolation) µa0 µa = (τa)∗(µa0 ) τ

Technical: we can construct γa holomorphic with a for a in a doubly pinched neighbourhood U(MΓ) of MΓ (pinched at 4 and 7)

Last steps

◮ U(MΓ) \ ∂MΓ is the set of parameters where Λa,− moves holomorphically (by MSS decomposition). ◮ χ holomorphic on ˚ MΓ by holomorphic motion arguments ◮ + qc. out of MΓ ⇒ χ continuous on U(MΓ) \ ∂MΓ ◮ sequences of qc maps have convergent subs. + rigidity on ∂M1 ⇒ χ continuous on ∂MΓ. ◮ χ is a branched covering, and as it is injective on MΓ, it is a homeo

• n U(MΓ).

Dictionary between Qc and Fa

Quadratic polynomials Qc Quadratic correspondences Fa

ϕc : ˆ C \ K(Qc) ∼ = ˆ C \ D ϕa : ˆ C \ Λ(Fa) ∼ = H external rays external geodesics ‘periodic rays land’ ‘periodic geodesics land’ every repelling fixed point every repelling fixed point is the landing point is the landing point

• f a periodic ray
• f a periodic geodesic

Yoccoz inequality Yoccoz inequality ∼ 1/q ∼ (log q)/q2 M ⊂ D(0, 2) MΓ ⊂ Lθ

Thanks for your attention!