Parabolic implosion - from discontinuity to renormalization

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Return map and Renormalization f g Rf Renormalization f Rf = first return map f (after rescaling) If f is infinitely renormalizable, ... f 0 = f f 1 = Rf 0 f 2 = Rf 1 h 0 g 0 g 1 h 1 h 2 ˜g 0 ˜g 1 f Rf as a “meta dynamical system” on a space of dynamical systems Often one expects a hyperbolic f ↔ (α, f dynamics 0 ) Rf(z) = e −2πi 1 α R α f 0 (z) R : (α, f 0 α α ↦→ − 1 α mod Z R 0 R R 0 contracting? R hyperbolic? (R α contrac f = P ◦ ϕ −1 ˜f 0 = ˜f ˜f1 = R ˜f 0 ˜f2 = R ˜f 1 rigidity: weak conj. upgraded to nicer one f = Q ◦ ϕ −1 0 ∞ (−∞, −1]
Near-parabolic Renormalization (cylinder renorm.) (1) w = − c z f (z) = z + O(z 2 ) χ f mod Z E f0 E f near 0 near ∞ T (w) = w + 1 Φ attr Φ rep C/Z C e 2πiβ z + O(z 2 ) β = − 1 Φα mod Z attr (f(z)) = Φ attr (z) + 1 Φ rep (f(z)) = Φ rep (z) + 1 Φ .. (z) + 1 Φ rep (f(z)) = Φ rep (z) + 1 Φ ... (f 0 (z)) = Φ ... (z) + 1 E f0 The (z) following = Φ attr ◦text Φ −1 rep is drawn χ f inχwhite. f (z) = z − 1 α 1 ep χ f χ f (z) E f = z − 1 Φ˜Rf attr χR f1 = ◦ E239 f ˜Rf Rf = χf ◦ E f ≃ αE f0 (z) = z + o(1) (Im z → +∞) (Im z → +∞) Π : C/Z ≃ F 0 (w) = w + 1 + o(1) w = − z near 0 near ∞ T ( f 0 R 0 f 0 R 0 f 0 (z) = z + O(z 2 ) f Rf Rf(z) = e 2πiβ z + O(z 2 ) β = − 1 α mod Z first return map e −2π = 0.00186 . . . h −→ 1 = C ∗ Exp ♯ ◦ E e 2πiα h ◦ ( Exp ♯) −1 Exp ♯ Exp ♯ (z) = e 2πiz Exp ♯ = R : C/Z −→ ≃ α h C ∗ Π Π(z) = e 2πiz Π : C/Z ≃ −→ C ∗ R 0 fR 0 = Π ◦ E f0 ◦ Π −1 R 0 f 0 Π ◦ E f0 ◦ −1 0 f 0 = Π ◦ E f0 ◦ Π −1 Rf = Π ◦ ˜Rf ◦ Π −1 = RΠ 0 f◦ 0 χ= f ◦ E f cylinders and Rf defined R 0 f 0 = Exp ♯ ◦E f0 ◦ (Exp ♯ ) −1 R 0 f 0 = Exp ♯ ◦E f0 ◦ (Exp ♯ ) −1 where β = − 1 when 1 h ′′ (0) ≠ 0 and (mod Z) or α = α sufficiently (m ∈small N) α m − β f 0 R −→ f1 −1 R −→ f2 Exp ♯ : C/Z ≃ −→ C ∗ ,z↦→ e 2πiz Rf is conjugate to the return map on red croissant via repelling Fatou coordinate and Exp ♯ R −→ f3 f = e 2πiα h, h = z + O(z 2 ) f ↔ (α,h) R −→ . . . ˜Rf = χ f ◦ E Rf = e −2πi 1 α h 1 , h 1 = z + O(z 2 ) Rf ↔ (− 1 α ,h 1) R 0 f 0 = Π ◦ E f0 ◦ Π −
Page 1 and 2: Parabolic implosion from discontinu
Page 3 and 4: Basic definitions f polynomial (or
Page 5 and 6: Egg beater or Douady-Hubbard’s ge
Page 7 and 8: Pick a point w in the box on the ri
Page 9 and 10: α z + . . . e 2πiα z + λ . f(z)
Page 11 and 12: morphic near 0 f 0 (0) = 0 f ′ 0
Page 13: Further results Discontinuity of st
Page 17 and 18: Applications (work in progress) The

Parabolic implosion - from discontinuity to renormalization

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