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

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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.

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Notes

  1. Almost the same age as the IMS at Stony Brook.

  2. Please see the electronic version of this paper to distinguish the color of the curves in Figs. 2 and 3.

  3. Connectivity of \(\mathcal {S}_d\) is implicitly assumed in Blanchard et al. (1991). Its proof can be found in Corollary 6.2 of DeMarco and Pilgrim (2011).

  4. He was nicknamed “Mr. Counterexample” with admiration. He also gave a counterexample to the 14th problem of David Hilbert. See the paper by M. Miyanishi, Masayoshi Nagata (1927–2008) and his mathematics, Kyoto J. Math. 50 (2010), no. 4, 645–659.

  5. We are also interested in the set \(J_f^{*}\) defined as the support of the unique maximal entropy measure (Bedford and Smillie 1991a; Bedford et al. 1993a) (see Sect. 3.1). We see that \(J_f^{*}\subset J_f\) holds in general, and \(J_f^{*}=J_f\) can be shown when f is hyperbolic (Bedford and Smillie 1991a).

  6. By courtesy of Mathematics Library at Kyoto University.

  7. According to J. H. Przytycki, most likely Wada published the first paper in Japan devoted to topology in 1911/1912. See his article Notes to the early history of the knot theory in Japan, arXiv:math/0108072.

  8. In this article the word planar in Definition 5.1 is a special usage; it means complex one-dimensional (real two-dimensional).

  9. This is what we called the limit-quotient model in Ishii (2014).

  10. This is a 4D visualization project launched by K. Anjyo, Z. Arai, H. Inou, Y. Ishii, S. Kaji and K. Tateiri.

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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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  1. Department of Mathematics, Kyushu University, Motooka, Fukuoka, 819-0395, Japan

    Yutaka Ishii

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Correspondence to Yutaka Ishii.

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Dedicated to Professor Shigehiro Ushiki for his retirement from Kyoto University.

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Ishii, Y. Dynamics of Polynomial Diffeomorphisms of \(\mathbb {C}^2\): Combinatorial and Topological Aspects. Arnold Math J. 3, 119–173 (2017). https://doi.org/10.1007/s40598-017-0066-x

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