Thurston's theorems in complex dynamics Mitsuhiro Shishikura (Kyoto - - PowerPoint PPT Presentation

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Feb 28, 2023 175 likes •342 views

Thurston's theorems in complex dynamics Mitsuhiro Shishikura (Kyoto University) Whats Next? The mathematical legacy of Bill Thurston Cornell University, June 26, 2014 Understanding Dynamical Systems Cantor set Douadys Rabbit Airplane

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Thurston's theorems in complex dynamics

Mitsuhiro Shishikura

(Kyoto University)

What’s Next? The mathematical legacy of Bill Thurston

Cornell University, June 26, 2014

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Understanding Dynamical Systems

Cantor set Douady’s Rabbit Airplane Dendrite Douady-like Rabbit

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Formulation

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We start from the idea of the proof, rather than the statement of the theorem.

Reformulating the question in terms of Teichmüller space

holomorphic (rational map)

So the previous question is equivalent to:

“Always look for a fixed point’’ compare: classification of surface homeos

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compare: skinning map for the hyperbolization of 3-manifolds

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Characterizing the obstruction

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Characterizing the obstruction 2

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Thurston matrix

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Applications

Construct rational maps from branched coverings. Need to check the non- existence of Thurston obstructions, this means checking for infinitely many multicurves.

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Quadratic polynomials and the Mandelbrot set

Douady-Hubbard gave a combinatorial description of the Mandelbrot set in terms of the PCF parameters.

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Mating

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Further developments

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