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1998 | 157 | 2-3 | 139-159
Tytuł artykułu

Jordan tori and polynomial endomorphisms in $ℂ^2$

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For a class of quadratic polynomial endomorphisms $f: ℂ^2 → ℂ^2$ close to the standard torus map $(x,y) → (x^2,y^2)$, we show that the Julia set J(f) is homeomorphic to the torus. We identify J(f) as the closure ℛ of the set of repelling periodic points and as the Shilov boundary of the set K(f) of points with bounded forward orbit. Moreover, it turns out that (J(f),f) is a mixing repeller and supports a measure of maximal entropy for f which is uniquely determined as the harmonic measure for K(f).
Słowa kluczowe
Kategorie tematyczne
Rocznik
Tom
157
Numer
2-3
Strony
139-159
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-10-22
poprawiono
1998-04-14
Twórcy
  • Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, Lotzestraße 13, D-37083 Göttingen, Germany, denker@math.uni-goettingen.de
  • Institut für Mathematische Stochastik, Georg-August-Universität Göttingen, Lotzestraße 13, D-37083 Göttingen, Germany, sheinema@math.uni-goettingen.de
Bibliografia
  • [1] J. Aaronson, M. Denker, and M. Urbański, Ergodic theory of Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-548.
  • [2] A. F. Beardon, Iteration of Rational Functions, Grad. Texts in Math. 132, Springer, 1991.
  • [3] E. Bedford and J. Smillie, Polynomial diffeomorphisms of $ℂ^2$: currents, equilibrium measure and hyperbolicity, Invent. Math. 103 (1991), 69-99.
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  • [6] O. Forster, Riemannsche Flächen, Heidelberger Taschenbücher 184, Springer, 1977.
  • [7] B. A. Fuks, Special Chapters in the Theory of Analytic Functions of Several Variables, Transl. Math. Monographs 14, Amer. Math. Soc., 1991.
  • [8] I. R. Gelfand, D. A. Raikow und G. E. Schilow, Kommutative normierte Algebren, Deutscher Verlag der Wiss., 1964.
  • [9] H. Grauert und K. Fritzsche, Einführung in die Funktionentheorie mehrerer Veränderlicher, Hochschultext, Springer, 1974.
  • [10] H. Grauert and R. Remmert, Coherent Analytic Sheaves, Springer, 1984.
  • [11] M. Gromov, On the entropy of holomorphic mappings, preprint, Inst. Hautes Etudes Sci.
  • [12] P. Hartman, Ordinary Differential Equations, Birkhäuser, 1982.
  • [13] S. M. Heinemann, Iteration holomorpher Abbildungen in $ℂ^n$, Diplomarbeit Universität Göttingen, 1993.
  • [14] S. M. Heinemann, Dynamische Aspekte holomorpher Abbildungen in $ℂ^n$, Dissertation, Universität Göttingen, 1994.
  • [15] S. M. Heinemann, Julia sets for endomorphisms of $ℂ^n$, Ergodic Theory Dynam. Systems 16 (1996), 1275-1296.
  • [16] K. Knopp, Elemente der Funktionentheorie, Sammlung Göschen 2124, de Gruyter, 1978.
  • [17] K. Krzyżewski and W. Szlenk, On invariant measures for expanding differentiable mappings, Studia Math. 33 (1969), 83-92.
  • [18] S. Lang, Introduction to Complex Hyperbolic Spaces, Springer, 1987.
  • [19] R. Remmert, Funktionentheorie I, Grundwissen Math. 5, Springer, 1984.
  • [20] G. Scheja und U. Storch, Lehrbuch der Algebra 2, Mathematische Leitfäden, Teubner, 1988.
  • [21] F. Schweiger, Numbertheoretical endomorphisms with σ-finite invariant measure, Israel J. Math. 21 (1975), 308-318.
  • [22] I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1974.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv157i2p139bwm
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