Admissible Covers and Rescaling Limits Xavier Buff Universit de - PowerPoint PPT Presentation

Rescaling limits Moduli spaces of rational maps Compactifications Admissible Covers and Rescaling Limits Xavier Buff Université de Toulouse after Matthieu Arfeux X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Origin of the problem Rescaling limits appear in the work of Epstein and have been studied by Kiwi, DeMarco and others. The approach we will use is inspired by the work of Epstein on the deformation space Def B A ( f ) and by work of Selinger and Koch on Thurston’s theorem. Our contribution is to use admissible covers between marked stable curves. X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications The moduli space rat D S := P 1 ( C ) is the Riemann sphere. Rat D is the space of rational maps f : S → S of degree D . The group of Möbius transformations Aut ( S ) = Rat 1 acts on Rat D by conjugation. The moduli space rat D is the quotient orbifold Rat D / Aut ( S ) . The moduli space rat D is not compact. X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Rescaling limits Let ( τ k ∈ rat D ) be a divergent sequence. A rescaling for ( τ k ) of period q ≥ 1 is a sequence of representatives ( f k ∈ Rat d ) such that f ◦ q → g k for some rational map g with deg g ≥ 2 (the convergence is locally uniform outside a finite subset of S ). X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Example τ ε = [ f ε ] ∈ rat 3 with f ε ( z ) = z 2 + ε/ z and ε → 0. ( f ε ) is a rescaling of period 1. √ ε and Set δ = 3 g ε ( w ) = 1 � w 2 + 1 � δ f ε ( δ · w ) = δ · . w Then, � 2 � w 2 + 1 w g ◦ 2 ε ( w ) = ε · + 1 + w 3 . w ( g ε ) is a rescaling of period 2. 0 is a multiple fixed point of the limit w �→ w / ( 1 + w 3 ) . X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Example √ ε = . 5 3 X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Example √ ε = . 2 3 X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Example √ ε = . 1 3 X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Example Julia sets for z �→ z 2 and w �→ w / ( 1 + w 3 ) . X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Questions How many essentially distinct rescaling limits can there be? How can one explain the presence of a multiple fixed point? X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Moduli space of marked Riemann spheres I is a finite set containing at least three points. Two injective maps x 1 : I → S and x 2 : I → S are equivalent when there is a Möbius transformation M : S → S such that x 2 = M ◦ x 1 . Mod ( I ) is the set of equivalence classes of injective maps x : I → S . Mod ( I ) may be endowed with the structure of a quasiprojective variety of dimension | B | − 3. X. Buff Admissible Covers and Rescaling Limits

� � � Rescaling limits Moduli spaces of rational maps Compactifications Sarah Koch’s space f ∈ Rat D , V ( f ) is the set of critical values of f . B , C ⊂ S are finite sets with V ( f ) ⊆ B , | B | ≥ 3 and C = f − 1 ( B ) . K B ( f ) is the set of pairs ( y , z ) ∈ Mod ( B ) × Mod ( C ) for which there are triples ( F , y , z ) ∈ Rat D × S B × S C with [ y ] = y , [ z ] = z and: z C S with deg z ( c ) F = deg c f . f F � S B y X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Sarah Koch’s space Theorem The set K B ( f ) is a smooth submanifold of Mod ( B ) × Mod ( C ) . The dimension of K B ( f ) is | B | − 3 . The projection K B ( f ) → Mod ( B ) is a finite cover. The projection K B ( f ) → Mod ( C ) is a proper embedding. X. Buff Admissible Covers and Rescaling Limits

� � � Rescaling limits Moduli spaces of rational maps Compactifications Adam Epstein’s space A , B , C ⊂ S are finite sets with | A | ≥ 3, A ⊆ B ∩ C , B ⊇ V ( f ) and C = f − 1 ( B ) . E B A ( f ) is the set of pairs ( y , z ) ∈ K B ( f ) which admits representatives y ∈ S B and z ∈ S C such that y | A = z | A : z C S with deg z ( c ) F = deg c f and y | A = z | A . f F � S B y X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Adam Epstein’s space The conjugacy class of F is determined by the point of E B A ( f ) . The space E B A ( f ) parameterizes conjugacy classes of rational maps marked by the dynamics of f on A . Theorem Assume f is not a flexible Lattès example. The set E B A ( f ) is a smooth submanifold of K B ( f ) . The dimension of E B A ( f ) is | B | − | A | . X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Compactification of Mod ( I ) Mod ( I ) is the Deligne-Mumford compactification of Mod ( I ) . An I -tree of spheres is a pair ( T , Z ) where T is a tree whose leaves are the points of I and whose nodes have valence ≥ 3 and Z is a collection of maps z v : I → S indexed by the nodes of T with z v ( i 1 ) = z v ( i 2 ) if and only if i 1 and i 2 are in the same component of T − v . Mod ( I ) parameterizes equivalence classes of I -trees spheres. X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Compactification of K B ( f ) K B ( f ) is the closure of K B ( f ) in Mod ( B ) × Mod ( C ) . An admissible cover F : ( T C , Z ) → ( T B , Y ) is a pair � ˆ � f , { f v } where ˆ f : T C → T B is a weighted tree map whose restriction to C coincides with f , f v : S − z v ( C ) → S − y ˆ f ( v ) ( B ) is a cover and � � � � f v z v ( e ) = f w z w ( e ) and deg z v ( e ) f v = deg z w ( e ) f w if v and w are linked by an edge e . K B ( f ) parameterizes equivalence classes of admissible covers. X. Buff Admissible Covers and Rescaling Limits

� � � Rescaling limits Moduli spaces of rational maps Compactifications Compactification of E B A ( f ) E B A ( f ) is the closure of E B A ( f ) in K B ( f ) . To each point of E B A ( f ) , one may associate a dynamical cover is a triple ( F , ι B , ι C ) where F : ( T C , Z ) → ( T B , Y ) is an admissible cover, ι B : ( T B , Y ) → ( T A , X ) is a contraction, ι C : ( T C , Z ) → ( T A , X ) is a contraction. ( T C , Z ) ι C F ( T A , X ) ( T B , Y ) . ι B X. Buff Admissible Covers and Rescaling Limits

Rescaling limits Moduli spaces of rational maps Compactifications Back to rescaling limits Let ( F , ι B , ι C ) represent a point in E B A ( f ) . Rescaling limits correspond to cycles of spheres for which the first return map has degree ≥ 2. Proposition (Kiwi) There are at most 2 D − 2 rescaling limits which are not postcritically finite. Proposition (Arfeux) If e ∈ T A is a periodic edge linking vertices v and w, the product of multipliers at x v ( e ) and x w ( e ) is 1 . X. Buff Admissible Covers and Rescaling Limits