Limits of quadratic rational maps with degenerate parabolic fixed - PowerPoint PPT Presentation

Limits of quadratic rational maps with degenerate parabolic fixed points of multiplier e 2 π i / q → 1 Xavier Buff joint work with Jean Écalle and Adam Epstein 26 novembre 2010 X. Buff Limits of degenerate parabolics

Degenerate parabolic fixed points Let f : P 1 → P 1 be a rational map. A fixed point of f is parabolic if the multiplier is a root of unity. If the multiplier is e 2 π i p / q and ζ is a coordinate vanishing at the fixed point, then ζ ◦ f ◦ q = e 2 π i p / q ζ · ( 1 + ζ ν q ) + O ( ζ ν q + 2 ) for some integer ν ≥ 1. The fixed point is a degenerate parabolic fixed point if ν ≥ 2. X. Buff Limits of degenerate parabolics

Families of quadratic rational maps Consider the quadratic rational map z f a , p / q : z �→ e 2 π i p / q 1 + az + z 2 which fixes 0 with multiplier e 2 π i p / q . Question What can we say regarding the set A p / q of points a ∈ C for which f a , p / q has a degenerate parabolic fixed point at 0 ? X. Buff Limits of degenerate parabolics

Pictures The bifurcation locus B p / q is the closure of the set of parameters a ∈ C for which f a , p / q has a parabolic cycle of period > 1. A p / q ⊂ B p / q . X. Buff Limits of degenerate parabolics

Pictures B 0 / 1 X. Buff Limits of degenerate parabolics

Pictures B 1 / 5 X. Buff Limits of degenerate parabolics

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Pictures B 0 / 1 X. Buff Limits of degenerate parabolics

The cardinality of A p / q as z → ∞ , we have � 1 + C p / q ( a ) z q � + O ( z q + 1 ) . f ◦ q a , p / q ( z ) = z · a ∈ A p / q if and only if C p / q ( a ) = 0. Proposition C p / q is a polynomial of degree q − 2 having only simple roots. The degree q − 2 is obtained by studying the behaviour as a → ∞ . The simplicity of roots is a transversality statement which we shall not study today. X. Buff Limits of degenerate parabolics

Limits as 1 / q → 0 It is tempting to conjecture that the sets B 1 / q have a Hausdorff limit in C ∪ {∞} . This is still unknown. It is tempting to conjecture that the sets A 1 / q have a Hausdorff limit in C ∪ {∞} . This is almost known. Proposition There exists an entire function C with the following properties. C has order of growth 1 . More precisely, as b → ∞ log | C ( b ) | ∈ O ( | b | log | b | ) \ O ( | b | ) . In particular C has infinitely many zeroes. the set A of points a ∈ C such that C ( 1 / a 2 ) = 0 satisfies A ∪ { 0 } ⊆ lim inf lim sup A 1 / q ⊆ A ∪ { 0 , ∞} . q →∞ A 1 / q and q →∞ X. Buff Limits of degenerate parabolics

Changes of coordinates It is convenient to introduce the rational map G b : w �→ w + 1 + b w . If b = 1 / a 2 , then F a , 0 is conjugate to G b via w = a / z . X. Buff Limits of degenerate parabolics

Ecalle-Voronin invariants Attracting Fatou coordinates : n � 1 n → + ∞ G ◦ n lim Φ b , att ( w ) = b ( w ) − n − b · k . k = 1 Repelling Fatou parameterization : � � n � 1 n → + ∞ G ◦ n lim Ψ b , rep ( w ) = w − n + b · . b k k = 1 Voronin invariants : E ± � b ( w ) = Φ b , att ◦ Ψ b , rep ( w ) . X. Buff Limits of degenerate parabolics

The function C � c k ( b ) e 2 π i kw � E + b = Id + k ≥ 0 and � c k ( b ) e 2 π i kw � E − b = Id + k ≤ 0 with c k entire functions of b . The entire function C is the Fourier coefficient : C = c 1 . X. Buff Limits of degenerate parabolics

Hypertangents and multizetas Hypertangents : � 1 Pe 1 = π cot ( π w ) = k + w k ∈ Z and � 1 Pe n = ( k + w ) n . k ∈ Z Multizetas : � 1 · · · 1 · 1 ζ ( s 1 , . . . , s r ) = . n s r n s 2 n s 1 r 2 1 0 < n r <...< n 2 < n 1 < ∞ X. Buff Limits of degenerate parabolics

Expansion with respect to b b = id + b e 1 + b 2 e 2 + b 3 e 3 + . . . � E ± with e 1 = Pe 1 e 2 = 0 e 3 = 3 ζ ( 3 ) Pe 2 e 4 = − ζ ( 4 ) Pe 3 + 10 ζ ( 5 ) Pe 2 X. Buff Limits of degenerate parabolics

Order of growth � b in the upper half-plane ℑ ( w ) > h + b with h + b comparable E + to ℑ ( b ) log | b | . E − � b in the lower half-plane ℑ ( w ) < h − b with h − b comparable to ℑ ( b ) log | b | . This is obtained by comparing the dynamics of G b to the real flow of the vector field � � 1 + b d dw . w The Koebe 1 / 4-Theorem implies that log | C ( b ) | ≤ 1 4 · h + 2 π = O ( | b | log | b | ) . b X. Buff Limits of degenerate parabolics

Order of growth Assume ℜ ( b ) = 1 / 2. G b has a indifferent fixed point at − b and so, the basin of ∞ only contains 1 critical point. There is a univalent map χ : {ℑ ( w ) > 0 } → {ℑ ( w ) > h − b } satisfying χ ( w + 1 ) = χ ( w ) + 1 and a translation T such that E 1 / 4 = T ◦ � � E + b ◦ χ. According to the Fatou-Shishikura Inequality for Finite Type Maps, c 1 ( 1 / 4 ) � = 0. � � log | C ( b ) | ≥ 2 π h − b + log � c 1 ( 1 / 4 ) � . X. Buff Limits of degenerate parabolics

Pictures again � 1 / 4 sends each red tile univalently to a upper half-plane and E ± each yellow tile univalently to a lower half-plane. X. Buff Limits of degenerate parabolics

Pictures again E ± � 1 / 2 + 10 i sends each red tile univalently to a upper half-plane and each yellow tile univalently to a lower half-plane. X. Buff Limits of degenerate parabolics