Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic

Smania, Daniel

doi:10.1090/S0894-0347-07-00550-4

Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic
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by Daniel Smania

J. Amer. Math. Soc. 20 (2007), 629-673

DOI: https://doi.org/10.1090/S0894-0347-07-00550-4

Published electronically: January 17, 2007

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Abstract:

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and “complex bounds”, two generalized polynomial-like maps which admit a topological conjugacy, quasiconformal outside the filled-in Julia set, are indeed quasiconformally conjugate. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen and Koskela and of Kallunki and Koskela, optimized for applications in complex dynamics. We prove, as the first application of this new method, that, for even criticalities distinct from two, the period two cycle of the Fibonacci renormalization operator is hyperbolic with $1$-dimensional unstable manifold.

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