Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Infinitely renormalizable diffeomorphisms
of the disk at the boundary of chaos

Authors: Eleonora Catsigeras, Jean-Marc Gambaudo and Fernando Moreira
Journal: Proc. Amer. Math. Soc. 126 (1998), 297-304
MSC (1991): Primary 58F13
MathSciNet review: 1403117
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that, among area contracting embeddings of the 2-disk, infinitely renormalizable maps with a bounded geometry either have positive topological entropy or correspond to a cascade of period doubling.

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  • [BF] Rufus Bowen and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337–342. MR 0431282
  • [Ca] E. Catsigeras, ``Cascades of period doubling of stable codimension one,'' IMPA's thesis, 1994.
  • [CT] P. Coullet and C. Tresser, ``Itérations d'endomorphismes et groupe de renormalisation,'' J. Phys. C5 (1978),25-28.
  • [GST] J.-M. Gambaudo, S. van Strien, and C. Tresser, Hénon-like maps with strange attractors: there exist 𝐶^{∞} Kupka-Smale diffeomorphisms on 𝑆² with neither sinks nor sources, Nonlinearity 2 (1989), no. 2, 287–304. MR 994094
  • [GT1] Jean-Marc Gambaudo and Charles Tresser, How horseshoes are created, Instabilities and nonequilibrium structures, III (Valparaíso, 1989), Math. Appl., vol. 64, Kluwer Acad. Publ., Dordrecht, 1991, pp. 13–25. MR 1177837, 10.1007/978-94-011-3442-2_2
  • [GT2] Jean-Marc Gambaudo and Charles Tresser, Self-similar constructions in smooth dynamics: rigidity, smoothness and dimension, Comm. Math. Phys. 150 (1992), no. 1, 45–58. MR 1188495
  • [Fe] Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 0501179
  • [Hu] J. Hu, ``Renormalization, rigidity and universality in bifurcation theory'', Ph. D. Dissertation, CUNY (1995).
  • [HT] J. Hu and C. Tresser, ``Period Doubling, Entropy, and Renormalization'' preprint (1995).
  • [MS] Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171
  • [Mi] Michał Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167–169 (English, with Russian summary). MR 542778
  • [Su] Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
  • [TC] Charles Tresser and Pierre Coullet, Itérations d’endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A577–A580 (French, with English summary). MR 512110

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Additional Information

Eleonora Catsigeras
Affiliation: Facultad de Ingeneria IMERL, C.C. 30, Montevideo, Uruguay

Jean-Marc Gambaudo
Affiliation: I.N.L.N., 1361, route des lucioles, Sophia-Antipolis, 06560 Valbonne, France

Fernando Moreira
Affiliation: Departamento de Matemática, Faculdade de Ciências do Porto, Praça Gomes Teixeira, 4000 Porto, Portugal

Received by editor(s): June 5, 1996
Communicated by: Linda Keen
Article copyright: © Copyright 1998 American Mathematical Society