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 on 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 ordinary) 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: z τ 1 ( z ) = z + 1 , and τ 2 ( z ) = 1 + z Minkowski map h + : [0 , ∞ ) → [0 , 1]: homeo 1 x ∈ R , written [ x 0 ; x 1 , x 2 , . . . ] = x 0 + 1 x 1 + x 2 + . . . h + ( x ) = 0 . 1 . . . 1 0 . . . 0 1 . . . 1 . . . � �� � � �� � � �� � x 0 x 1 x 2 It conjugates the action of τ 1 ( z ) and τ 2 ( 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 ) with P ( z , w ) = wq ( z ) − p ( z ) q ( z ) Modular group ← → (2:2) F : z → w with generators P ( z , w ) = ( w − ( z + 1))( w ( z + 1) − z ) z τ 1 ( z ) = z + 1 , τ 2 ( z ) = 1+ z

Quadratic maps and PSL (2 , Z ) fit in a correspondence Theorem(Bullett-Penrose, ’94) The matings between Q c , c ∈ M and PSL (2 , Z ) lie in the family F a : z → w given by � aw − 1 � 2 � aw − 1 � � az + 1 � � az + 1 � 2 + + = 3 . w − 1 w − 1 z + 1 z + 1 Moreover, for all a ∈ [4 , 7] ⊂ R the correspondence F a is conjugate to the generators of PSL (2 , Z ) on an invariant open set. √ Figure: Limit set of F a , a = (3 + 33) / 2

M Γ and some limit sets for F a

Which correspondences in F a are matings? Conjecture and state of the art ◮ Conjecture (B-P, ’94) The family F a contains matings between PSL (2 , Z ) and every quadratic polynomial with connected Julia set, and the connectedness locus M Γ of F a 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 Q c for a large class of value of c

The family F a = J a ◦ Cov 0 Q Q ( z ) = z 3 − 3 z , Cov 0 Q is its correspondence ’deck transformation’: Q ( x ) = Q ( y ) = Q ( z ) ⇒ Cov 0 Q ( x ) = { y , z } J a involution having fixed points at 1 (critical point of Q ) and a . CovQ 0 Q ( − 2) = Q (1), so Cov Q sends each blue fundamental domain for CovQ 0 ∆ CovQ 0 − 2 line to the other two. 1 0 ∆ Ja ∆ Ja F− 1 (∆ Ja ) a a a F− 2 (∆ Ja ) − 2 a 2 1 (1 : 1) (1 : 2) ∆ (2 : 1) CovQ 0

Dynamics of the family F a ∆ a 1:2 F a | : � → F a ( � C \ ∆ a − C \ ∆ a ) 2:1 F a | : F− 1 (∆ a ) − → ∆ a a Facts: for every a ◮ F a has a parabolic fixed point at z = 0. ◮ F − 1 a (0) = { 0 , S a } . ◮ F a | ∆ a is a deg 2 Λ a , − = � ∞ Λ a , + = � ∞ ( F a ) − n (∆ a ) ( F a ) n ( � holomorphic map C \ ∆ a ) 1 1 at every z ∈ ∆ a , z � = S a

The family Per 1 (1) ◮ Per 1 (1) = { P A ( z ) = z + 1 / z + A | A ∈ C } , ◮ ∞ parabolic fixed point, with basin Λ A , K A = � C \ Λ A ◮ ∀ A , external class given by h 2 ( z ) = z 2 +1 / 3 1+ z 2 / 3 , and h 2 ( z ) | S 1 ∼ top P 0 ( z ) = z 2 | S 1 ◮ M 1 : connectedness locus ( M 1 ≈ M by Petersen-Roesch).

Main Theorem We say that F a is a mating between the rational quadratic map P A : z → z + 1 / z + A and the modular group Γ = PSL (2 , Z ) if 1. the 2-to-1 branch of F a for which Λ − is invariant, is hybrid equivalent to P A on Λ − (i.e., it is quasiconformally conjugate to P A 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( F a ) to itself, F a is conformally conjugate to the pair of M¨ obius transformations { τ 1 , τ 2 } from the complex upper half plane H to itself. Theorem (Bullett-L, Invent.Math. 220, (2020) ) For every a ∈ M Γ the correspondence F a is a mating between some rational map P A : z → z + 1 / z + A and Γ.

M 1 and M Γ

Proof of the Thm, part 1 A (deg 2) parabolic-like map is an object that locally behaves like a map P A , A ∈ C . Thm A deg 2 parabolic-like map is hybrid conjugated to a member of the family Per 1 (1), a unique such member if the filled Julia set is connected. So, if we prove that for all a ∈ M Γ , the branch of F a 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) V U

Surgery construction (valid also out of M Γ ) F− 1 (2:1) ˆ T 1 ˆ T 2 φ ψ z 2 π/ 2 + θ 1 π − 2 θ 2 Λ − ⊂ H l , but we need it to π + 2 θ 1 3 π/ 2 − θ 2 be contained in a top. disc log log 2 z − 4 π i making at the parabolic fixed 3 π i 2 π i (5 π/ 2 + θ 1) i point an angle < π , to have qc interpolation space to put our new Beltrami 2 π i π i 2 z − 2 π i forms! (3 π/ 2 − θ 2) i ’Easy’ to do individually 0 π i

Part 2: B¨ ottcher map (valid just in M Γ ) ◮ ∀ a ∈ M Γ ∃ a Riemann map φ : Ω → H . We prove that φ conjugates F a | Ω( F a ) to the generators of the modular groups on H . ◮ This is: the map φ : Ω → H plays for our family F a the role the B¨ ottcher map plays for the quadratic family! φ ✲

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

Main tool: lunes and fundamental croissants M Γ ⊂ lune L ⇒ for all a ∈ M Γ , Λ − , a is contained in a lune V a which moves holomorphically with the parameter. For a ∈ L , set V ′ a := F − 1 a ( V a ). We call A a := V a \ V ′ a the fundamental croissant . The fundamental croissant moves holomorphically for a ∈ ˚ L : fix a 0 ∈ M Γ \ { 4 , 7 } , we have a holomorphic motion τ a : A a 0 → A a We can extend χ : ˚ L \ M Γ → C by using τ : the extension is qc.

Holomorphic motion of Beltrami forms Do surgery at a 0 ∈ M Γ , obtaining µ a 0 on A a 0 , move it by τ : A a 0 → A a µ a = ( τ a ) ∗ ( µ a 0 ), and µ a = ( F a ) − n ( µ a ) on F − n ( A a ) for all n ≥ 1 ⇒ µ a is a holomorphic in a wherever Λ a , − moves holomorphically with a . τ fa 0 (qc interpolation) µ a = µ a 0 ( τ a ) ∗ ( µ a 0 ) z 2 Technical: we can construct γ a holomorphic with a for a in a doubly pinched neighbourhood U ( M Γ ) of M Γ (pinched at 4 and 7)

Last steps ◮ U ( M Γ ) \ ∂ M Γ is the set of parameters where Λ a , − moves holomorphically (by MSS decomposition). ◮ χ holomorphic on ˚ M Γ by holomorphic motion arguments ◮ + qc. out of M Γ ⇒ χ continuous on U ( M Γ ) \ ∂ M Γ ◮ sequences of qc maps have convergent subs. + rigidity on ∂ M 1 ⇒ χ continuous on ∂ M Γ . ◮ χ is a branched covering, and as it is injective on M Γ , it is a homeo on U ( M Γ ).

Dictionary between Q c and F a Quadratic correspondences F a Quadratic polynomials Q c C \ K ( Q c ) ∼ C \ Λ( F a ) ∼ ϕ c : ˆ = ˆ ϕ a : ˆ C \ D = 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 of a periodic ray of a periodic geodesic Yoccoz inequality Yoccoz inequality ∼ (log q ) / q 2 ∼ 1 / q M ⊂ D (0 , 2) M Γ ⊂ L θ

Thanks for your attention!