Reversible complex Hénon maps. (English) Zbl 1117.32300

Summary: We identify and investigate a class of complex Hénon maps \(H\colon\mathbb C^ 2\rightarrow \mathbb C^ 2\) that are reversible, that is, each \(H\) can be factorized as \(RU\) where \(R^ 2=U^ 2=\text{Id}_{\mathbb C^ 2}\). Fixed points and periodic points of order two and three are classified in terms of symmetry, with respect to \(R\) or \(U\), and as either elliptic or saddle points. We report on experimental investigation, using a Java applet, of the bounded orbits of \(H\).


32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
Full Text: DOI Euclid EuDML


[1] Bedford E., J. Amer. Math. Soc. 4 pp 657– (1991)
[2] Bedford E., Invent. Math. 114 pp 277– (1993) · Zbl 0799.58039
[3] Devaney R., Trans. Amer. Math. Soc. 218 pp 89– (1976)
[4] Devaney R., J. Differential Equations 51 pp 254– (1984) · Zbl 0527.58029
[5] Devaney R., An Introduction to Chaotic Dynamical Systems,, 2. ed. (1989) · Zbl 0695.58002
[6] Fornæss, J. E. Dynamics in Several Complex Variables. CBMS Regional Conference Series in Mathematics. pp.87Providence: Amer. Math. Soc. [Fornæss 96] · Zbl 0840.58016
[7] Friedland S., Ergod. Th.& Dynam. Sys. 9 pp 67– (1989) · Zbl 0651.58027
[8] Giarrusso D., Proc. Amer. Math. Soc. 123 pp 3731– (1995)
[9] Hale J., Dynamics and Bifurcations. (1991) · Zbl 0745.58002
[10] Jordan D. A., J. Algebra 156 pp 194– (1993) · Zbl 0809.16032
[11] Hubbard J. H., Publ. Math. IHES 79 pp 5– (1994) · Zbl 0839.54029
[12] Oberste-Vorth R. W., Nonlinear analysis, Methods & Applications 30 pp 2143– (1997) · Zbl 0901.58044
[13] Smillie J., Flavors of Geometry pp 117– (1997)
[14] Zehnder E., Geometry and Topology 597 pp 855– (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.