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The Real Dynamics of Bieberbach’s Example

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Abstract

Bieberbach constructed, in 1933, domains in \({\mathbb {C}}^2\) which were biholomorphic to \({\mathbb {C}}^2\) but not dense. The existence of such domains was unexpected. The special domains Bieberbach considered are basins of attraction of a cubic Hénon map. This classical method of construction is one of the first applications of dynamical systems to complex analysis. In this paper, the boundaries of the real sections of Bieberbach’s domains will be calculated explicitly as the stable manifolds of the saddle points. The real filled Julia sets and the real Julia sets of Bieberbach’s map will also be calculated explicitly and illustrated with computer generated graphics. Basic differences between real and the complex dynamics will be shown.

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References

  1. Bedford, E., Smillie, J.: Fatou-Bieberbach domains arising from polynomial automorphisms. Indiana Univ. Math. J. 40, 789–792 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bedford, E., Smillie, J.: Polynomial automorphisms of \(\mathbb{C}2\). II: stable manifolds and recurrence. J. Am. Math. Soc. 4, 657–679 (1991)

    MATH  MathSciNet  Google Scholar 

  3. Friedland, S., Milnor, J.: Dynamical properties of plane polynomial automorphisms. Ergod. Theory Dyn. Syst. 9(1), 67–99 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fornaess, J.E., Sibony, N.: Complex Hénon mappings in \(\mathbb{C}2\) and Fatou-Bieberbach domains. Duke Math. J. 65(2), 345–380 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hayes, S.: Fatou-Bieberbach Gebiete in \(\mathbb{C}2\). Deutsche Mathematiker Vereinigung Mitteilungen 1, 14–18 (1995).

  6. Kimura, T.: On Fatou-Bieberbach domains in \(\mathbb{C}2\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35(1), 103–148 (1988).

  7. Robinson, C.: Dynamical Systems (Stability, Symbolic Dynamics, and Chaos). CRC Press, Boca Raton (1999)

    MATH  Google Scholar 

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Acknowledgments

We thank Jeff Galas, who generated Figs. 1–7, and Korrigan Clark for their contributions.

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Authors and Affiliations

  1. Department of Mathematics, City College of CUNY, New York, NY, 10031, USA

    Sandra Hayes

  2. Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada

    Axel Hundemer

  3. Department of Mathematics, University of Florida, Gainesville, FL, USA

    Evan Milliken

  4. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    Tasos Moulinos

Authors
  1. Sandra Hayes
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  2. Axel Hundemer
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  3. Evan Milliken
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  4. Tasos Moulinos
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Corresponding author

Correspondence to Sandra Hayes.

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Communicated by John Fornæss.

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Hayes, S., Hundemer, A., Milliken, E. et al. The Real Dynamics of Bieberbach’s Example. J Geom Anal 25, 2386–2406 (2015). https://doi.org/10.1007/s12220-014-9518-x

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  • DOI: https://doi.org/10.1007/s12220-014-9518-x

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