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Mating the Basilica with a Siegel Disc
Jonguk Yang University of Toronto Topics in Complex Dynamics, 2016 Universitat de Barcelona
Consider .
Mating the Basilica with a Siegel Disc Jonguk Yang University of - - PowerPoint PPT Presentation
Mating the Basilica with a Siegel Disc Jonguk Yang University of - - PowerPoint PPT Presentation
Mating the Basilica with a Siegel Disc Jonguk Yang University of Toronto Topics in Complex Dynamics, 2016 Universitat de Barcelona Consider . Consider . Suppose
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Consider .
Suppose has a connected and locally connected filled Julia set.
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Consider .
Suppose has a connected and locally connected filled Julia set.
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Consider .
Suppose has a connected and locally connected filled Julia set.
Carathéodory Loop:
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Consider .
Suppose has a connected and locally connected filled Julia set.
Carathéodory Loop:
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Mating Construction [Douady, Hubbard]
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Mating Construction [Douady, Hubbard]
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Mating Construction [Douady, Hubbard]
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Mating Construction [Douady, Hubbard]
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If can be realized by a rational map, we say that and are mateable.
Mating Construction [Douady, Hubbard]
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[Rees, Tan, Shishikura] Suppose and are post-critically
- finite. Then and are mateable if and only if and do
not belong in conjugate limbs.
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The Basilica Family
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Consider . is a superattracting 2-periodic orbit.
The Basilica Family
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Consider . is a superattracting 2-periodic orbit. is a free critical point, and is a free critical value.
The Basilica Family
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The Basilica Polynomial
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The Basilica Polynomial
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The Basilica Polynomial
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a-plane c-plane
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Can the basilica family be understood as the set of matings
- f the quadratic family with the basilica polynomial?
a-plane c-plane
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Known Results
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Known Results
Suppose is not trivially non-mateable with .
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Known Results
Suppose is not trivially non-mateable with .
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If is hyperbolic, then it is mateable with .
Known Results
Suppose is not trivially non-mateable with .
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If is hyperbolic, then it is mateable with .
Known Results
Suppose is not trivially non-mateable with . [Aspenberg, Yampolsky] If is finitely renormalizable, and has no non-repelling periodic orbits, then it is mateable with .
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If is hyperbolic, then it is mateable with .
Known Results
Suppose is not trivially non-mateable with . [D. Dudko] If is at least 4 times renormalizable, then it is mateable with . [Aspenberg, Yampolsky] If is finitely renormalizable, and has no non-repelling periodic orbits, then it is mateable with .
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Boundary of Hyperbolic Components
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If lives in the boundary of a hyperbolic component, then it is either: parabolic, Cremer, or Siegel.
Boundary of Hyperbolic Components
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If lives in the boundary of a hyperbolic component, then it is either: parabolic, Cremer, or Siegel.
Boundary of Hyperbolic Components
If is parabolic, then it is mateable with . (An application of transquasiconformal surgery due to Haïssinsky.)
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If lives in the boundary of a hyperbolic component, then it is either: parabolic, Cremer, or Siegel.
Boundary of Hyperbolic Components
If is parabolic, then it is mateable with . (An application of transquasiconformal surgery due to Haïssinsky.) If is Cremer, then its Julia set is non-locally connected. Hence it is non-mateable with .
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Siegel Parameters
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[Petersen, Zakeri] Suppose has an indifferent periodic
- rbit with rotation number . Then for or a.e. ,
Siegel Parameters
is Siegel, and has a locally connected Julia set.
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[Petersen, Zakeri] Suppose has an indifferent periodic
- rbit with rotation number . Then for or a.e. ,
Siegel Parameters
is Siegel, and has a locally connected Julia set. [Y.] Let be a quadratic polynomial with a fixed Siegel disk with a rotation number of bounded type. Then it is mateable with .
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[Petersen, Zakeri] Suppose has an indifferent periodic
- rbit with rotation number . Then for or a.e. ,
Siegel Parameters
is Siegel, and has a locally connected Julia set. [Y.] Let be a quadratic polynomial with a fixed Siegel disk with a rotation number of bounded type. Then it is mateable with .
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[Petersen, Zakeri] Suppose has an indifferent periodic
- rbit with rotation number . Then for or a.e. ,
Siegel Parameters
is Siegel, and has a locally connected Julia set. [Y.] Let be a quadratic polynomial with a fixed Siegel disk with a rotation number of bounded type. Then it is mateable with .
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Puzzle Partition
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Puzzle Partition
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Puzzle Partition
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Puzzle Partition
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Puzzle Partition
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Main challenge: Prove that puzzle pieces shrink to points.
Puzzle Partition
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Complex A Priori Bounds [Yampolsky]
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Complex A Priori Bounds [Yampolsky]
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Complex A Priori Bounds [Yampolsky]
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Complex A Priori Bounds [Yampolsky]
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Blaschke Product Model
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Blaschke Product Model
An adaptation of construction found in [Yampolsky, Zakeri].
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Critical Puzzle Pieces
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Critical Puzzle Pieces
Using complex a priori bounds, can show that all puzzles
- shrink. Therefore, and are mateable.
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