Notes for a graduate lecture course, University of Bristol, Spring 2007.
They are intended to be accessed only from within the University of Bristol
domain.
If you are not at the University of Bristol but would like to read the notes,
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| Lecture 1 | postscript | Background complex analysis. | |
| Lecture 2 | postscript | Normal families, definition of the Fatou and Julia sets. | |
| Lecture 3 | postscript | Complete invariance of the Fatou and Julia sets. The Fatou set contains the basins of attraction of attracting periodic cycles. Statement of Montel's theorem, introduction to the hyperbolic metric. |
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| Lecture 4 | postscript | The uniformization theorem. Explicit uniformization of the triply-punctured sphere, using the elliptic modular function. Proof of Montel's theorem. |
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| Lecture 5 | postscript | Topology of the Julia set: It is non-empty and perfect, and it is the minimal closed backward-invariant set with at least 3 points. Topological transitivity of the dynamics on the Julia set. |
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| Lecture 6 | postscript | Holomorphic normal forms near attracting and repelling fixed
points: Koenigs linearization and Böttcher's theorem. |
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| Lecture 7 | postscript | Repelling periodic points are dense in the Julia set. Julia's proof : use the rational fixed point theorem to show there is a repelling or parabolic fixed point, then topological transitivity gives a homoclinic orbit. Fatou's proof: use Montel's theorem to show the density of periodic points, then bound the number of attracting or indifferent cycles. |
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| Lecture 8 | postscript | Local behaviour at indifferent fixed points: The Leau-Fatou flower at a parabolic fixed point |
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| Lecture 9 | postscript | Linearization at non-parabolic indifferent fixed points
(Siegel and Cremer). Classification of pre-periodic Fatou components. Sullivan's no wandering domains theorem. |
© University of Bristol, 2007. Last updated: 14/03/2007.