Double Parabolic Renormalization in the Quadratic Family Xavier - PowerPoint PPT Presentation

Double Parabolic Renormalization in the Quadratic Family Xavier Buff joint work with A. Epstein and C. Petersen X. Buff Double parabolic renormalization

Quadratic rational maps rat 2 is the space of conjugacy classes of quadratic rational maps. the elementary symmetric functions of the multipliers of the fixed points are σ 1 := µ 1 + µ 2 + µ 3 , σ 2 := µ 1 µ 2 + µ 2 µ 3 + µ 3 µ 1 and σ 3 := µ 1 µ 2 µ 3 . Proposition (Milnor) rat 2 is isomorphic to C 2 . σ 1 and σ 2 provide global coordinates. σ 3 = σ 1 − 2 . X. Buff Double parabolic renormalization

Quadratic rational maps rat 2 is the space of conjugacy classes of quadratic rational maps. the elementary symmetric functions of the multipliers of the fixed points are σ 1 := µ 1 + µ 2 + µ 3 , σ 2 := µ 1 µ 2 + µ 2 µ 3 + µ 3 µ 1 and σ 3 := µ 1 µ 2 µ 3 . Proposition (Milnor) rat 2 is isomorphic to C 2 . σ 1 and σ 2 provide global coordinates. σ 3 = σ 1 − 2 . � � Per 1 ( µ ) ⊂ rat 2 = [ f ] having a fixed point with multiplier µ . X. Buff Double parabolic renormalization

Quadratic rational maps rat 2 is the space of conjugacy classes of quadratic rational maps. the elementary symmetric functions of the multipliers of the fixed points are σ 1 := µ 1 + µ 2 + µ 3 , σ 2 := µ 1 µ 2 + µ 2 µ 3 + µ 3 µ 1 and σ 3 := µ 1 µ 2 µ 3 . Proposition (Milnor) rat 2 is isomorphic to C 2 . σ 1 and σ 2 provide global coordinates. σ 3 = σ 1 − 2 . � � Per 1 ( µ ) ⊂ rat 2 = [ f ] having a fixed point with multiplier µ . Per 1 ( µ ) is the line µ 3 − σ 1 µ 2 + σ 2 µ − σ 3 = 0. X. Buff Double parabolic renormalization

Bifurcation loci � � Bif ( µ ) := [ f ] having a cycle with multiplier 1 . Bif ( 0 ) X. Buff Double parabolic renormalization

Bifurcation loci � � Bif ( µ ) := [ f ] having a cycle with multiplier 1 . Bif ( 1 ) X. Buff Double parabolic renormalization

Germs with a parabolic fixed point Assume g ( z ) = e 2 π i p q z + O ( z 2 ) . Then g ◦ q ( z ) = z + Cz ν q + 1 + O ( z ν q + 2 ) with C � = 0 . g is formally conjugate to � 1 + z ν q + α z 2 ν q � e 2 π i p q z · with α ∈ C . Definition The résidu itératif of g is résit ( g ) = ν q + 1 − α. 2 X. Buff Double parabolic renormalization

Quadratic rational maps with parabolic fixed points z g p / q , a ( z ) = e 2 π i p q · 1 + az + z 2 . g p / q , a and g p / q , − a are conjugate via z �→ − z . X. Buff Double parabolic renormalization

Quadratic rational maps with parabolic fixed points p / q , a ( z ) = z + C p / q ( a ) z q + 1 + O ( z q + 2 ) . g ◦ q Set R p / q ( a ) := résit ( g p / q , a ) when the parabolic point is not degenerate (i.e., C p / q ( a ) � = 0). Theorem (B., Ecalle, Epstein) For q ≥ 2 , C p / q is a polynomial of degree q − 2 having simple roots. R p / q is a rational map of degree 2 q − 2 which only has double poles : infinity and the zeroes of C p / q . X. Buff Double parabolic renormalization

Quadratic rational maps with parabolic fixed points p / q , a ( z ) = z + C p / q ( a ) z q + 1 + O ( z q + 2 ) . g ◦ q Set R p / q ( a ) := résit ( g p / q , a ) when the parabolic point is not degenerate (i.e., C p / q ( a ) � = 0). Theorem (B., Ecalle, Epstein) For q ≥ 2 , C p / q is a polynomial of degree q − 2 having simple roots. R p / q is a rational map of degree 2 q − 2 which only has double poles : infinity and the zeroes of C p / q . Question How does R p / q depend on p / q ? X. Buff Double parabolic renormalization

Parabolic degeneracy p / q = 1 / 1 p / q = 1 / 4 p / q = 1 / 10 X. Buff Double parabolic renormalization

Bifurcation locus p / q = 1 / 10 X. Buff Double parabolic renormalization

Limit of bifurcation loci 1 / 1 1 / 4 1 / 50 1 / 100 X. Buff Double parabolic renormalization

Limit of résidus itératifs Consider R p / q as a function R p / q : Per 1 ( e 2 π i p / q ) → � C . Given r / s ∈ Q and k ∈ N , set p k / q k := 1 / ( k + r / s ) . Theorem (B., Ecalle, Epstein) As k → + ∞ , the sequence of meromorphic functions � p k � 2 R p k / q k : Per 1 ( e 2 π i p / q ) → � C q k converges to a meromorphic transcendental function R r / s : Per 1 ( 1 ) � { [ g 0 ] } → � C . The function R r / s has a double pole at infinity, an essential singularity at [ g 0 ] and infinitely many poles (which are double poles) accumulating this singularity. X. Buff Double parabolic renormalization

Fatou coordinates Set b = 1 / a 2 . The map z g a ( z ) = 1 + az + z 2 is conjugate via Z = 1 / ( az ) to F b ( Z ) = Z + 1 + b Z . � � F ◦ n The sequence b ( Z ) − n − b log n converges to an attracting Fatou coordinate Φ b . � � F ◦ n The sequence b ( Z − n + b log n ) converges to a repelling Fatou parametrization Ψ b . X. Buff Double parabolic renormalization

Horn maps The lifted horn map H b = Φ b ◦ Ψ b commutes with the translation by 1. It projects via Z �→ z = e 2 π i Z to a germ h b fixing 0 with multiplier e 2 π 2 b . The holomorphic map s · e − 2 π 2 b · h b f r / s , b = e 2 π i r fixes 0 with multiplier e 2 π i r s . R r / s ( b ) is the résidu itératif of f r / s , b when the parabolic point is not degenerate. X. Buff Double parabolic renormalization

Where are the poles? Proposition � � 0 < Re ( b ) < 1 / 2 The poles of R r / s belong to the strip . X. Buff Double parabolic renormalization

Where are the poles? Theorem (B., Epstein, Petersen) For each r / s, the poles of R r / s form s sequences ( b j , n ) , j ∈ [ [ 1 , s ] ] , satisfying 4 − σ j b j , n = n 2 π i + 1 2 π i + o ( 1 ) , as n → + ∞ , where the numbers µ j := e 2 π i σ j are distinct, � µ j = ( − 1 ) s and � � the set µ j , j ∈ [ [ 1 , s ] ] is invariant by the map µ �→ e 2 π i r / s /µ . X. Buff Double parabolic renormalization

Where are the poles? Corollary The poles of R 0 / 1 form a sequence ( b n ) with the asymptotic behavior b n = n 2 π i + 1 4 − 1 4 π i + o ( 1 ) . Corollary The poles of R 1 / 2 form a sequence ( b n ) with the asymptotic behavior b n = n 4 π i + 1 4 − 1 8 π i + o ( 1 ) . X. Buff Double parabolic renormalization

Where are the poles? Corollary For r / s with s odd, among the poles of R r / s , there is a sequence ( b n ) which has the asymptotic behavior b n = n 2 π i + 1 4 − 1 − r + r / s + o ( 1 ) . 4 π i Corollary For r / s with s even, among the poles of R r / s , there is a sequence ( b n ) which has the asymptotic behavior b n = n 4 π i + 1 4 − r / s 4 π i + o ( 1 ) . X. Buff Double parabolic renormalization

Elements of the proof The proof consist in controlling the asymptotic behavior of the horn maps H b as Im ( b ) → ±∞ . For a = 0, the map g 0 ( z ) = z / ( 1 + z 2 ) is semi-conjugate to g 2 via z �→ z 2 . Note that a = 2 corresponds to b = 1 / 4. X. Buff Double parabolic renormalization

Elements of the proof The proof consist in controlling the asymptotic behavior of the horn maps H b as Im ( b ) → ±∞ . For a = 0, the map g 0 ( z ) = z / ( 1 + z 2 ) is semi-conjugate to g 2 via z �→ z 2 . Note that a = 2 corresponds to b = 1 / 4. Set F b := H b + π i b . Let τ : C� ( −∞ , 0 ] → C be the holomorphic map defined by √ τ ( b ) := Φ b ( b ) − Φ 1 / 4 ( 1 / 2 ) − i π ( b − 1 / 4 ) . Set Λ( b ) := 2 π i ( b − 1 / 4 ) . X. Buff Double parabolic renormalization

Elements of the proofLimit of horn maps Proposition As Im ( b ) → + ∞ , T − τ ( b ) ◦ F b ◦ T τ ( b ) → F 1 / 4 locally uniformly in the upper half-plane. Proposition As Im ( b ) → −∞ , T − τ ( b ) ◦ F b ◦ T τ ( b ) − T Λ( b ) ◦ F 1 / 4 ◦ T − Λ( b ) ◦ F 1 / 4 → 0 locally uniformly in the upper half-plane. X. Buff Double parabolic renormalization

The dynamics when Im ( b ) → −∞ X. Buff Double parabolic renormalization

Perturbed Fatou Coordinates X. Buff Double parabolic renormalization

Elements of the proof Corollary As Im ( b ) → + ∞ , the following convergence holds locally uniformly in the unit disk: � � e − 2 π i τ ( b ) g b e 2 π i τ ( b ) z → g 1 / 4 . Corollary As Im ( b ) → + ∞ , R r / s ( b ) → R r / s ( 1 / 4 ) . Corollary The entire map R r / s has no poles with large positive imaginary part. X. Buff Double parabolic renormalization

For µ ∈ C� { 0 } , let F µ : D → � C be the finite type analytic map on � C defined by F µ ( z ) := e 2 π r / s � � · f 1 / 4 µ · f 1 / 4 ( z ) . µ Let R : C� { 0 } → C be the meromorphic transcendental function defined by R ( µ ) := résit ( F µ ) when the parabolic point is not degenerate. X. Buff Double parabolic renormalization

Elements of the proof Let λ : C → C� { 0 } be defined by λ ( b ) := e 2 π i Λ( b ) = e 4 π 2 ( b − 1 / 4 ) . Corollary As Im ( b ) → −∞ , the following convergence holds locally uniformly in the unit disk: � � e − 2 π i τ ( b ) g b e 2 π i τ ( b ) z − F λ ( b ) ( z ) → 0 . Corollary As Im ( b ) → −∞ , R r / s − R ◦ λ → 0 uniformly. X. Buff Double parabolic renormalization