Limits of sub semigroups of C and Siegel enrichments Ismael Bachy - PowerPoint PPT Presentation

Limits of closed sub semigroups of C ∗ Conformal Enrichments Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy 22 novembre 2010 Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Conformal Enrichments Table of contents Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Topological model for { Γ z | z ∈ C ∗ } Conformal Enrichments Conformal dynamic in the sens of Douady-Epstein Enrichments Siegel enrichments ∆-LLC maps The result Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } SG ( C ∗ ) = { Γ ∪{ 0 , ∞} | Γ closed sub semigroup of C ∗ } ⊂ Comp ( P 1 ) SG ( C ∗ ) has naturally the Hausdorff topology on compact subsets of P 1 . Limits of closed semigroups are closed semigroups. For z ∈ C ∗ , Γ z = { z , z 2 , ..., z k , ... } . SG 1 ( C ∗ ) = { Γ z ∪ { 0 , ∞} | z ∈ C ∗ } . Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } Topological model for { Γ z | z ∈ C ∗ } ⊂ SG ( C ∗ ) r s ∈ Q / Z with gcd( r , s ) = 1 let : s 2 and tangent to S 1 at e 2 i π r s ⊂ C \ D : the open disc of radius 1 s . D r ∂ D r s : p �→ z r s ( p ) the point intersection of ∂ D r s and the half line through e 2 i π r s making slope p ∈ [ −∞ , + ∞ ] with the line θ = r s .     � �  C \  / ∼ 1 , X 2 =  C \  / ∼ 2 X 1 = D r S · D r s s r r s s ◮ the non trivial ∼ 1 -classes consist of D and for all p ∈ [ −∞ , + ∞ ], all s ∈ N ∗ the set { z r s ( p ) | r ∈ { 1 , .., q − 1 } s.t. gcd( r , s ) = 1 } . ◮ ∼ 2 is defined by z ∼ 2 z ′ if and only if S ( z ) ∼ 1 S ( z ′ ), where S : z �→ 1 z . Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

� Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } The model for { Γ z | z ∈ C \ D } π 1 D 1 3 D 1 -1 1 D 1 D 2 D 2 C \ � s D r X 1 3 r s Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } Topological model for { Γ z | z ∈ C ∗ } ⊂ SG ( C ∗ ) Let X be the disjoint union of X 1 , X 2 and N = N ∪ {∞} endowed with the discrete topology on N making ∞ its unique accumulation point. Then Theorem The topological space X is compact and homeomorphic to SG 1 ( C ∗ ) = { Γ z | z ∈ C ∗ } ⊂ SG ( C ∗ ) . Let π 1 : C → X 1 , π 2 : C → X 2 be the canonical projections and denote S : X → X the involution induced by S . Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } Differents notions of convergence to S 1 Definition Let ( z j = ρ j e 2 i πθ j ) ⊂ C \ S 1 and e 2 i πθ ∈ S 1 , we say 1. | z j | → 1 with infinite slope w.r.t the rationals if ∀ r ∈ Q / Z � θ j − r � is unbounded. ln( ρ j ) � θ j − θ � 2. z j → e 2 i πθ tangentially if is unbounded. ln( ρ j ) � θ j − θ � 3. z j → e 2 i πθ non tangentially if is bounded. ln( ρ j ) 4. z j → e 2 i πθ with slope p ∈ R if θ j − θ ln( ρ j ) → p. Observation Given any | z j | → 1 up to a subsequence ( z j ) falls in one of the 3 above cases. Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } tangentially non tangentially e 2 i π p 1 q 1 e 2 i π p 2 q 2 ϕ e 2 i π p 3 with slope p = tan( ϕ ) q 3 D Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } Accumulation points of (Γ z j ) K > 1 = { j | ρ j > 1 } , K < 1 = { j | ρ j < 1 } , K =1 = { j | ρ j = 1 } . � � θ 1 S p : ρ = (logarithmic spiral based at 1). p Proposition (Possible accumulation points of (Γ z j )) 1. Either Acc ( z j ) ∩ S 1 = ∅ then Acc (Γ z j ) ⊂ { Γ z | z ∈ C \ S 1 } . 2. Either ∃ e 2 i πθ ∈ Acc ( z j ) ∩ S 1 and 2.1 either | K < 1 ∪ K =1 | < ∞ and then Γ z j → C \ D iff | z j | → 1 with infinite slope w.r.t the rationals or � s − 1 � � e 2 i π k 2 i π s S p Γ z j → ∩ C \ D iff Acc ( z j ) ⊂ < e > and all s k =0 the limits accumulates with slope p ∈ R , Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } Accumulation points of (Γ z j ) 2.2 either | K > 1 ∪ K =1 | < ∞ and then � q − 1 � � e 2 i π k q S p Γ z j → D or Γ z j → ∩ D (with symmetric cond.), k =0 2.3 either | K > 1 ∪ K < 1 | < ∞ and then Γ z j → S 1 iff | z j | → 1 with 2 i π q > iff infinite slope w.r.t the rationals or Γ z j → < e |{ j | z j ∈ < e 2 i π 1 q > }| = ∞ . In all other cases the sequence Γ z j does not converge ! But all the possible accumulation points are precisely those describe above. Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } Key Lemma Lemma If ( z j ) ⊂ C \ D and ∃ θ ∈ ( R \ Q ) / Z s.t e 2 i πθ ∈ lim inf Γ z j , then Γ z j → C \ D . D´ emonstration. 1. lim sup Γ z j ⊂ C \ D , 2. e 2 i πθ ∈ lim inf Γ z j , thus S 1 ⊂ lim inf Γ z j 3. Need to prove [1 , + ∞ [ ⊂ lim inf Γ z j . � � n k → e 2 i πθ and take ρ � 1. Suppose z n k ρ j k e 2 i πθ jk j k = 2 3 ln( ρ ) � � 4 5 „ ρ nk « ln ρ n k Then → ρ . jk j k Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } Definition of Φ : SG 1 ( C ∗ ) → X �� � � �� �� ≃ π 1 π 1 1. C \ s ∂ D r C \ s ∂ D r =: int ( X 1 ). r r �� s � s �� D r � ≃ ϕ 1 C \ D because D ∪ C \ s ∂ D r is comp., r s s conn, loc conn and full (choose ϕ 1 tangent to id at ∞ ). C \ D ≃ ι 1 { Γ z | z ∈ C \ D } : ι 1 : z �→ Γ z is cont. and ι − 1 1 ( z ) = z Γ where | z Γ | = inf | Γ | is also cont. Φ(Γ) := π 1 ◦ ϕ − 1 ◦ ι − 1 1 (Γ) . 1 �� � � q − 1 � e 2 i π k q S p ∩ C \ D s ( p )) for p ∈ R . 2. Φ := π 1 ( z r k =0 3. Φ( C \ D ) := p 1 the point corresponding to the ∼ 1 -class of D . 4. Φ ◦ S = S ◦ Φ, 2 i π q > and Φ( ∞ ) = S 1 . 5. Φ( q ) := < e Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Limits of closed sub semigroups of C ∗ Topology on the space of closed sub semigroups Conformal Enrichments Topological model for { Γ z | z ∈ C ∗ } Φ is a homeomorphism Theorem Φ : SG 1 ( C ∗ ) → X is a homeomorphism. D´ emonstration. 1. Φ | : { Γ z | z ∈ C \ D } → int ( X 1 ) homeo ok � � � � q − 1 q − 1 � � e 2 i π k e 2 i π k q S p q S p 2. at ∩ C \ D or ∩ D : k =0 k =0 2.1 on ∂ X : slope moves continuously = > spirals moves continuously in Hausdorff topology ok 2.2 Γ z j cv to the spiral < = > z j → e 2 i π r s with slope p ∈ R = > ok 3. Γ z j → p i ∈ X i i = 1 , 2 < = > | z j | → 1 with infinite slope w.r.t 1 ( z j ) enters all the neighbourhoods of S 1 = > rationals = > ϕ − 1 Φ(Γ z j ) → p i . Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy

Conformal dynamic in the sens of Douady-Epstein Limits of closed sub semigroups of C ∗ Enrichments Conformal Enrichments Siegel enrichments Conformal dynamic in the sens of Douady-Epstein Definition A conformal dynamic on C is a set G = { ( g , U ) | U ⊂ C open and g : U → C holomorphic } which is closed under restrictions and compostions. Let P oly d be the space of monic centered polynomials of degre d > 1. Conformal dynamic generated by a polynomial f ∈ P oly d [ f ] = { ( f n , U ) | U ⊂ C , n > 0 } . Limits of sub semigroups of C ∗ and Siegel enrichments Ismael Bachy