Parabolic implosion - from discontinuity to renormalization
Parabolic implosion - from discontinuity to renormalization Parabolic implosion - from discontinuity to renormalization
Plan Bifurcation of parabolic periodic points -- parabolic implosion (saddle-node, intermittent chaos, periodic pts that bifurcate) Local change of dynamics (“egg beater”) => Global effects Main example: f 0 (z) =z + z 2 (∼ z 2 + 1 4 ) and its perturbation Discontinuity of Julia sets Richness of bifurcated Julia sets when perturbed to “wild direction’’ “Periodicity’’ in parameter space Tools by Douady-Hubbard: Fatou coordinates, Ecalle-Voronin cylinders, horn map, phase Parabolic/Near-parabolic renormalization: study of irrationally indifferent periodic points
Basic definitions f polynomial (or just holomorphic mapping for local problems) fixed point f(z 0 )=z 0 multiplier λ = f ′ (z 0 ) attracting |λ| < 1 indifferent |λ| =1 parabolic α ∈ Q λ = e 2πiα irrationally indifferent α ∈ R Q repelling |λ| > 1 filled-in Julia set K f = {z ∈ C : {f n (z)} ∞ n=0 is bounded} Julia set (chaotic part) J f = ∂K f = closure of {repelling periodic points} We consider the bifurcation of a parabolic fixed point and its global effect.
- Page 1: Parabolic implosion from discontinu
- 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 and 14: Further results Discontinuity of st
- Page 15 and 16: Near-parabolic Renormalization (cyl
- Page 17 and 18: Applications (work in progress) The
Plan<br />
Bifurcation of parabolic periodic points -- parabolic <strong>implosion</strong><br />
(saddle-node, intermittent chaos, periodic pts that bifurcate)<br />
Local change of dynamics (“egg beater”) => Global effects<br />
Main example: f 0 (z) =z + z 2 (∼ z 2 + 1 4<br />
) and its perturbation<br />
Discontinuity of Julia sets<br />
Richness of bifurcated Julia sets<br />
when perturbed <strong>to</strong> “wild direction’’<br />
“Periodicity’’ in parameter space<br />
Tools by Douady-Hubbard:<br />
Fa<strong>to</strong>u coordinates, Ecalle-Voronin cylinders, horn map, phase<br />
<strong>Parabolic</strong>/Near-parabolic <strong>renormalization</strong>:<br />
study of irrationally indifferent periodic points
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