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Arnold Mathematical Journal

, Volume 3, Issue 1, pp 119–173 | Cite as

Dynamics of Polynomial Diffeomorphisms of \(\mathbb {C}^2\): Combinatorial and Topological Aspects

Research Contribution
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Abstract

The Fig. 1 was drawn by Shigehiro Ushiki using his software called HenonExplorer. This complicated object is the Julia set of a complex Hénon map \(f_{c, b}(x, y)=(x^2+c-by, x)\) defined on \(\mathbb {C}^2\) together with its stable and unstable manifolds, hence it is a fractal set in the real 4-dimensional space! The purpose of this paper is to survey some results, questions and problems on the dynamics of polynomial diffeomorphisms of \(\mathbb {C}^2\) including complex Hénon maps with an emphasis on the combinatorial and topological aspects of their Julia sets.

Keywords

Polynomial diffeomorphism of \(\mathbb {C}^2\) Hénon map Julia set hyperbolicity Hubbard tree automaton 

Notes

Acknowledgements

The author thanks Laurent Bartholdi, Eric Bedford, Romain Dujardin, John Hubbard, Hiroyuki Inou, Sarah Koch, Mitsuhiro Shishikura, John Smillie and Shigehiro Ushiki for their comments on the subjects of the manuscript. He is also grateful to the anonymous referee for his/her careful reading of the manuscript and detailed comments. This research is partially supported by JSPS KAKENHI Grant Numbers 25287020 and 25610020.

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© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2017

Authors and Affiliations

  1. 1.Department of MathematicsKyushu UniversityFukuokaJapan

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