Hyperbolic polynomial diffeomorphisms of C2. I: A non-planar map

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Abstract

In this paper we develop a general framework for verifying hyperbolicity of holomorphic dynamical systems in C2. Our framework in particular enables us to construct the first example of a hyperbolic Hénon map of C2 which is non-planar, i.e. which is not topologically conjugate on its Julia set to a small perturbation of any expanding polynomial in one variable. The key ideas in its proof are: the Poincaré box, which is a building block to apply our criterion for hyperbolicity, an operation called fusion, to merge two polynomials in one variable to obtain essentially two-dimensional dynamics, and rigorous computation by using interval arithmetic. Some conjectures and problems are also presented.

Original languageEnglish
Pages (from-to)417-464
Number of pages48
JournalAdvances in Mathematics
Volume218
Issue number2
DOIs
Publication statusPublished - Jun 1 2008

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Hyperbolic Polynomial
Hyperbolicity
Diffeomorphisms
Interval Arithmetic
Polynomial
Julia set
Small Perturbations
Building Blocks
Fusion
Dynamical system
Framework

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Hyperbolic polynomial diffeomorphisms of C2. I : A non-planar map. / Ishii, Yutaka.

In: Advances in Mathematics, Vol. 218, No. 2, 01.06.2008, p. 417-464.

Research output: Contribution to journalArticle

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