Jordan tori and polynomial endomorphisms in $ℂ^2$

Denker, Manfred; Heinemann, Stefan

Jordan tori and polynomial endomorphisms in

ℂ^{2}

Denker, Manfred ; Heinemann, Stefan

Fundamenta Mathematicae, Tome 158 (1998), p. 139-159 / Harvested from The Polish Digital Mathematics Library

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Résumé

For a class of quadratic polynomial endomorphisms $f : ℂ^{2} \to ℂ^{2}$ close to the standard torus map $(x, y) \to (x^{2}, y^{2})$ , 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).

Publié le : 1998-01-01

Zbl 0919.58040

EUDML-ID : urn:eudml:doc:212282

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     author = {Manfred Denker and Stefan Heinemann},
     title = {Jordan tori and polynomial endomorphisms in $$\mathbb{C}$^2$
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     journal = {Fundamenta Mathematicae},
     volume = {158},
     year = {1998},
     pages = {139-159},
     zbl = {0919.58040},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv157i2p139bwm}
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Denker, Manfred; Heinemann, Stefan. Jordan tori and polynomial endomorphisms in $ℂ^2$
            . Fundamenta Mathematicae, Tome 158 (1998) pp. 139-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv157i2p139bwm/

Bibliographie

[00000] [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. | Zbl 0789.28010

[00001] [2] A. F. Beardon, Iteration of Rational Functions, Grad. Texts in Math. 132, Springer, 1991.

[00002] [3] E. Bedford and J. Smillie, Polynomial diffeomorphisms of $ℂ^{2}$ : currents, equilibrium measure and hyperbolicity, Invent. Math. 103 (1991), 69-99. | Zbl 0721.58037

[00003] [4] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. | Zbl 0127.03401

[00004] [5] J. E. Fornæss and N. Sibony, Complex dynamics in higher dimension. I, Astérisque 222 (1994), 201-231. | Zbl 0813.58030

[00005] [6] O. Forster, Riemannsche Flächen, Heidelberger Taschenbücher 184, Springer, 1977.

[00006] [7] B. A. Fuks, Special Chapters in the Theory of Analytic Functions of Several Variables, Transl. Math. Monographs 14, Amer. Math. Soc., 1991.

[00007] [8] I. R. Gelfand, D. A. Raikow und G. E. Schilow, Kommutative normierte Algebren, Deutscher Verlag der Wiss., 1964.

[00008] [9] H. Grauert und K. Fritzsche, Einführung in die Funktionentheorie mehrerer Veränderlicher, Hochschultext, Springer, 1974. | Zbl 0285.32001

[00009] [10] H. Grauert and R. Remmert, Coherent Analytic Sheaves, Springer, 1984.

[00010] [11] M. Gromov, On the entropy of holomorphic mappings, preprint, Inst. Hautes Etudes Sci.

[00011] [12] P. Hartman, Ordinary Differential Equations, Birkhäuser, 1982. | Zbl 0476.34002

[00012] [13] S. M. Heinemann, Iteration holomorpher Abbildungen in $ℂ^{n}$ , Diplomarbeit Universität Göttingen, 1993.

[00013] [14] S. M. Heinemann, Dynamische Aspekte holomorpher Abbildungen in $ℂ^{n}$ , Dissertation, Universität Göttingen, 1994.

[00014] [15] S. M. Heinemann, Julia sets for endomorphisms of $ℂ^{n}$ , Ergodic Theory Dynam. Systems 16 (1996), 1275-1296. | Zbl 0874.32008

[00015] [16] K. Knopp, Elemente der Funktionentheorie, Sammlung Göschen 2124, de Gruyter, 1978. | Zbl 0387.30001

[00016] [17] K. Krzyżewski and W. Szlenk, On invariant measures for expanding differentiable mappings, Studia Math. 33 (1969), 83-92. | Zbl 0176.00901

[00017] [18] S. Lang, Introduction to Complex Hyperbolic Spaces, Springer, 1987. | Zbl 0628.32001

[00018] [19] R. Remmert, Funktionentheorie I, Grundwissen Math. 5, Springer, 1984.

[00019] [20] G. Scheja und U. Storch, Lehrbuch der Algebra 2, Mathematische Leitfäden, Teubner, 1988.

[00020] [21] F. Schweiger, Numbertheoretical endomorphisms with σ-finite invariant measure, Israel J. Math. 21 (1975), 308-318. | Zbl 0314.10037

[00021] [22] I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1974. | Zbl 0284.14001