Parabolic implosion - from discontinuity to renormalization

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} f(z) = e z + O(z ) πi p small and q z + a 2 z 2 | arg α| α< ∈ π C {0} small and | arg α| < + Perturbation . f(z) = e 2πiα . . 4 4 z + O(z 2 α ∈f R ′ (0) Q ) = e 2πiα , α small | arg α| < π 4 f ′ (0) = e 2πiα , α small ff(z) = e 2πiα z + O(z 2 0 | R arg 0 f α| 0 < R π 0 f 0 (z)) = z + O(z 2 (wild direction) ) α ∈ R Q 4 + a 2 z 2 + f 0 ′ . . . (0) = e 2πiα a , α small i ∈α N∈ Ca i ≥{0} Nf small and | arg α| < π 4 α ∈ C {0} smallf and ′ (0) Rf | = arg e 2πiα α| Rf(z) , < α π small | arg α| < π 4 = e 2πiβ z + O(z 2 ) 4 β = − 1 ≥ N a i ∈ N a i ≥ N α ∈ C {0} small and | arg α ∈ α| R < π α λ = 1 a 2 ≠ 0 F 0 (w) = w + Q1 + o(1) w = − c near 0 near ∞ α ∈ R Q α ∈ C 4 {0} small and w| arg α| < π 4 + iα 1 + o(1) w = −F c 0 (w) near = mod w 0+ 1 Znear + Φo(1) attr E∞ f0 (f(z)) w ET (w) f = −Φ = c attr w (z) + near 1+ 10 Φ attr Φnear rep (f(z)) Φ∞ rep = T C/Z (w) Φ rep = ( z + . . . z z a i ∈ N a i ≥ N f 0 Eα f mod Z E ∈ R Q f0 E α f a i ∈ N a i ≥ ∈ R Q f N E iα z + O(z 2 f0 (z) = Φ attr ◦ Φ −1 0 R 0 f 0 R 0 f 0 (z) = z rep + O(z 2 ) ) R F 0 (w) = w + 1 + o(1) w = − c 0 f near 0 near a 0 i (z) ∈ N= z a+ i F ≥O(z f N 0 2 ) R 0 f 0 R 0 f a 0 (z) = z + O(z i ∈ N a i ≥ N 0 (w) = w + 1 + w mod o(1) Z w E = − c 2 ) near 0 near ∞ T (w) = w πiα f0 , α small | arg F 0 (w) = w + 1 + o(1) w = −F c w mod α| < Z π f Rf f = Rf(z) P ◦ ϕ −1 = e 2πiβ z + O(z 2 ) β E f0 E f 0 (w) near = w 0 1 + near o(1) ∞ w T = (w) f 0 wR 0 f 0 R 0 f 0 (z) = z + O(z 2 − = c = − 1 4 α mod Z Rf(z) = e 2πiβ z + O(z f 2 ) Rfβ = Rf(z) − 1 w + near 1 0Φ att n ) w 0} small modand Z | arg Ef f0 0 α| < R π α f mod = e 2πiβ Q Z z ◦ ϕ −1 + O(z 2 ) β = − 1 α mod Z Φ 0 4f attr (f(z)) 0 R 0 f 0 (z) mod= ZΦ z + attr O(z E(z) 2 f0 + ) E1 f Φ rep (f(z)) = Φ rep (z) + 1 = Φ attr (z) + 1 Φ rep Φ attr (f(z)) (f(z)) = Φ= rep Φ(z) attr 0 (z) + ∞ 1+ 1Φ (−∞, ... (f Φ rep 0 (z)) −1] (f(z)) = Φ = f f Rf Rf(z) = e 2πiβ ... (z) Φ rep + (z) 1 + z + O(z 2 ) β = − 1 α mo 0 R 0 f 0 R 0 f 0 (z) = z + O(z f 2 0 ) R 0 f 0 R 0 f 0 (z) = z + O(z 2 ) f Rf Rf(z) E e 2πiβ z + O(z 2 ) β = − 1 f0 (z) = Φ attr ◦ Φ −1 rep χ f χ f (z) = C R Z Q f Rf Rf(z) = e 2πiβ z Φ + attr O(z f (f(z)) 2 ) Rf = β = Φ Rf(z) attr − 1 D H α R mod z − 1 1 Φ ... (f 0 ( Q Z Rf = χ α C/Z C ∗ attr ◦ Φ −1 rep χ f χ f E(z) = z − 1 f0 (z) = Φ attr ◦Rf Φ −1 rep= χ f χ f ◦ E f χ f (z) ≃ = z − 1 ˜Rf = χ f ◦ E f i ≥ N α (z) + 1 Φ rep (f(z)) = Φ rep (z) + α mod = e 2πiβ Z α z + O(z 2 ) β = − 1 Φ attr (f(z)) = f = Φ P ◦ attr (z) ϕ −1 first return map + 1 Φ rep (f(z)) = Φ rep (z) + 1 Φ ... (f 0 (z)) α + 1 + o(1) w = − c z ↦→ e 2πiz z ↦→ exp(2πiz) + o(1) (Im z → +∞) E f0 (z) = z + o(1) (Im z → +∞) near 0E f0 near (z) = ∞Φ attr T ◦(w) Φ −1 rep = w + 1 Φ attr (f(z)) Φw attr (z) + 1 Φ rep Φ attr (f(z)) = = Φ rep Φ attr (z) (z) + 1+ 1Φ ... Φ(f rep 0 (z)) (f(z)) = Φ = ... Φ(z) rep + ( depends Econtinuously f0 (z) = Φ attr ◦on Φ −1 rep χ f χ f (z) = z − Φ attr Φ rep C/Z f = Q ◦ ϕ −1 E f0 E f f Rf R 0 f R α f Rf Rf = χ f ◦ E f e 2πiz Π : C/Z −→ ≃ Π C ∗ Π(z) = e f 2πiz Π = P ◦ ϕ −1 : C/Z −→ ≃ C ∗ αR 0 f 0 = Π ◦ E f0 ◦ Π −1 (after a suitable normalization) −1 −1 −1
morphic near 0 f 0 (0) = 0 f ′ 0 (0) = λ z + a 2 z 2 + . . . πi p q z + a 2 z 2 + . . . + a 2 z 2 + . . . λ = 1 a 2 ≠ 0 f 0 Lavaurs map Lavaurs map g β (z) =Φ −1 rep (Φ attr(z)+β) describes how the orbits entering attracting fundamental strip go out <strong>from</strong> repelling fundamental strip. πiα z + . . . 1near Φ attr ∞ T Φ(w) rep = C/Z w + 1 C Φ attr Φ rep C/Z C πiα , α small 0} small and | arg α| < π 4z ↦→ z + β or 1 α mod Z i ≥ N ∞+ 1 o(1) rep(z) + T (w) 1+ = 1 w + = Φ(f(z)) 1− c Φ attr w = ΦΦ(z) rep C/Z + 1 C E f0 πiα z + O(z 2 ) g β = lim n→∞ f k n near 0 near ∞ T (w) = w + 1 Φ attr Φ rep C/Z C n in int K(f 0) if f n (z) =e 2πiα n ( z + ..., 1 ) − k n = −β, lim n→∞ α n if f n (z) ( =z + z 2 + ε n , π ) − k n = −β. lim n→∞ √ εn
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: α z + . . . e 2πiα z + λ . f(z)
Page 13 and 14: Further results Discontinuity of st
Page 15 and 16: Near-parabolic Renormalization (cyl
Page 17 and 18: Applications (work in progress) The

Parabolic implosion - from discontinuity to renormalization

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