Parabolic implosion - from discontinuity to renormalization

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Limit dynamics f 0 ◦ g β = g β ◦ f 0 = g β+1 in K f0 〈f 0 ,g β 〉 = {f0 n ◦ gm β : (m = 0 and n ≥ 0) or (m >0 and n ∈ Z)} (when m>0, each element is defined in an open subset of C) K(〈f 0 ,g β 〉)=C {z : ∃h = f n 0 ◦ gm β ∈〈f 0,g β 〉,h(z) ∈ C K f0 } J(〈f 0 ,g β 〉) = closure of {repelling fixed points of h = f n 0 ◦ gm β ∈〈f 0,g β 〉} J(f 0 ) J(〈f 0 ,g β 〉) ⊂ lim inf n→∞ J(f n) ⊂ lim sup n→∞ K(〈f 0 ,g β 〉)=J(〈f 0 ,g β 〉) =⇒ K(f n ) ⊂ K(〈f 0 ,g β 〉) K(f 0 ) lim K(f n) = lim J(f n)=J(〈f 0 ,g β 〉) n→∞ n→∞ ∃h = f n 0 ◦ g m β (∈ 〈f 0 ,g β 〉) has an attracting fixed point =⇒ lim n→∞ K(f n)=K(〈f 0 ,g β 〉) and lim J(f n)=J(〈f 0 ,g β 〉) n→∞
Further results Discontinuity of straightening of polynomial-like mappings (Douady-Hubbard, first published account on parabolic implosion) Limit parameter space for z 2 1 + c around c 0 =1/4 (wrt √ ) c−1/4 versus parameter space of 〈f 0 ,g β 〉 (Lavaurs) Non-local connectivity of connectedness locus for real/complex cubic polynomials (Lavaurs) . . . . . Parabolic/near-parabolic renormalization Renormalization for germs holomorphic functions with parabolic or near-parabolic fixed points (in wild direction) S. with H. Inou: an invariant class of maps for the renormalization. => control on irrationally indifferent fixed points when the continued fraction of the rotation number has large coefficients. => Buff-Cheritat’s result on quadratic Julia sets with positive area.
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: morphic near 0 f 0 (0) = 0 f ′ 0
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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