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


  1. 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

  2. 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

  3. 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

  4. 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

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

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

  7. Pictures B 1 / 10 X. Buff Limits of degenerate parabolics

  8. Pictures B 1 / 20 X. Buff Limits of degenerate parabolics

  9. Pictures B 1 / 50 X. Buff Limits of degenerate parabolics

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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

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