Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T22:21:25.342Z Has data issue: false hasContentIssue false
  • English
  • Français

Julia sets for holomorphic endomorphisms of ℂn

Published online by Cambridge University Press:  14 October 2010

Stefan-M. Heinemann
Stefan-M. Heinemann
Affiliation:
Institut für Mathematische Stochastik, Lotzestraβe 13, D-37083 Göttingen, Germany
Get accessRights & Permissions [Opens in a new window]

Abstract

We give a definition for a Julia set J(f) for generic classes of polynomial endomorphisms f: ℂn→ ℂn. For n = 1, our definition is equivalent to the usual one, which gives the points where the iterates of f do not form a normal family. Moreover, the Julia set J(f1 × … × fn) ⊂ ℂn for a product of one-dimensional polynomials fi: ℂ → ℂ turns out to be the product J(f1) × … × J(fn) of the associated Julia sets J(fi) ⊂ ℂ. For a special class of mappings f: ℂ2 → ℂ2 which is not of this simple type, the so-called Cantor skews, we investigate topological structure as well as measure theoretic aspects of the Julia sets obtained using our definition.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996, pp. 1275 - 1296
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1] Beardon, A. F.. Iteration of Rational Functions (Graduate Texts in Mathematics 132). Springer, 1991.CrossRefGoogle Scholar
[2] Bedford, E. and Taylor, B. A.. Fine topology, Šilov boundary and dd c. J. Functional Analysis 72 (1987), 225–251.CrossRefGoogle Scholar
[3] Brolin, H.. Invariant sets under iteration of rational functions. Arkiv för Matematik 6 (1965), 103–144.CrossRefGoogle Scholar
[4] Dieudonné, J.. Grundzüge der modernen Analysis l (Logik und Grundlagen der Mathematik 8). Vieweg, 1985.Google Scholar
[5] Gaier, D.. Vorlesungen über Approximation im Komplexen. Birkhäuser, 1980.CrossRefGoogle Scholar
[6] Gelfand, I. R., Raikow, D. A. and Schilow, G. E.. Kommutative normierte Algebren. VEB Verlag der Wissenschaften, Berlin, 1964.Google Scholar
[7] Grauert, H. and Remmert, R.. Coherent Analytic Sheaves. Springer, 1984.CrossRefGoogle Scholar
[8] Gromov, M.. On the entropy of holomorphic mappings. Preprint d'Institut des Hautes Etudes scientifiques.Google Scholar
[9] Heinemann, S. M.. Iteration holomorpher Abbildungen in Cn. Diplomarbeit Universität Göttingen, 1993.Google Scholar
[10] Heinemann, S. M.. Dynamische Aspekte holomorpher Abbildungen in Cn. Dissertation Universität Göttingen, 1994.Google Scholar
[11] Jank, G. and Volkmann, L.. Meromorphe Funktionen und Differentialgleichungen. UTB: Große Reihe. Birkhäuser, 1985.Google Scholar
[12] Klimek, M.. Pluripotential Theory (London Mathematical Society Monographs 6). Oxford University Press, 1991.CrossRefGoogle Scholar
[13] Lyubich, M. Yu.. Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. & Dynam. Sys. 3 (1983), 351–385.CrossRefGoogle Scholar
[14] Rudin, W.. Function Theory in the Unit Ball of Cn (Grundlehren der mathematischen Wissenschaften 241). Springer, 1980.Google Scholar
[15] Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–108.CrossRefGoogle Scholar
[16] Shafarevich, I. R.. Basic Algebraic Geometry. (Springer study edition.) Springer, 1974.CrossRefGoogle Scholar